Finite Group Databasebeta

← #33104350

Subgroups — #33104350

Order 18688.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988685C211yesmaximal, characteristic
2C2×#17988685C211yesmaximal, characteristic
3C2×C4×#11924015C211yesmaximal, characteristic
4#22612842C211yesmaximal, characteristic
5#22610666C211yesmaximal, characteristic, complemented
6#22610661C211yesmaximal, characteristic, complemented
7#22610199C211yesmaximal, characteristic, complemented
8#22610191C211yesmaximal, characteristic
9#22610185C211yesmaximal, characteristic, complemented
10#22610177C211yesmaximal, characteristic
11#22610097C211yesmaximal, characteristic, complemented
12#22609946C211yesmaximal, characteristic, complemented
13#22609928C211yesmaximal, characteristic
14#22609806C211yesmaximal, characteristic, complemented
15Q8×C4⋊C4×C73C211yesmaximal, characteristic, complemented
16C22×#11924015C2211yescharacteristic
17C2×C4×D292C2211yescharacteristic
18C2×C4×D292C2211yescharacteristic
19C2×C4×D292C2211yescharacteristic
20#17988685—24no—
21#17988685C418yescomplemented
22C4×#11924015—24no—
23C2×#14556524C2211yescharacteristic
24C2×#14556524C2211yescharacteristic
25C2×#14556524C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×#14556522C2211yescharacteristic
28C2×C4⋊#10596831C2211yescharacteristic
29C2×#14556515C2211yescharacteristic
30C2×#14556515C2211yescharacteristic
31C2×#14556515C2211yescharacteristic
32C2×#14556511C2211yescharacteristic
33C2×#14556511C2211yescharacteristic
34C2×#14556511C2211yescharacteristic
35C2×#14556502C2211yescharacteristic
36C2×#14556502C2211yescharacteristic
37C2×C4×#10596831C2211yescharacteristic
38#17988485C2211yescharacteristic
39#17988485C2211yescharacteristic
40#17988458—21no—
41#17988458—21no—
42#17988453C2211yescharacteristic
43#17988446—21no—
44#95×D73—21no—
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988342C2211yescharacteristic
50#17988342C2211yescharacteristic
51C2×C4×Q8×C73C2211yescharacteristic
52C2×C4×Q8×C73C2211yescharacteristic
53C2×C4×Q8×C73C2211yescharacteristic
54#389×C73—21no—
55#389×C73—21no—
56#389×C73C2211yescharacteristic
57#389×C73C2211yescharacteristic
58#384×C73—21no—
59#384×C73C2211yescharacteristic
60C4×C4⋊C4×C73—21no—
61C4×C4⋊C4×C73C2211yescharacteristic
62C22×D292C2311yescharacteristic
63C22×C4×D73—21no—
64C22×C4×D73—21no—
65C22×C4×D73—21no—
66C2×#11924015C2×C414yes—
67C2×#11924015D414yes—
68C2×#11924015Q814yes—
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71C22×#10596831C2311yescharacteristic
72C22×#10596831C2311yescharacteristic
73C4×D292—22no—
74C4×D292—22no—
75C4×D292—22no—
76C4×D292—41no—
77C4×D292—41no—
78C4×D292C2×C414yes—
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×C4⋊Dic73C2311yescharacteristic
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924005—21no—
85C2×#11924005—21no—
86C2×#11924005—21no—
87#14556524—22no—
88#14556524—41no—
89#14556524C2×C414yescomplemented
90#14556522—22no—
91#14556522—41no—
92C4⋊#10596831—22no—
93#14556515—22no—
94#14556515—22no—
95#14556515C2×C414yescomplemented
96#14556511—22no—
97#14556511—22no—
98#14556511C2×C414yes—
99#14556502—22no—
100#14556502C2×C414yescomplemented
101C4×#10596831—22no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106C22×Q8×C73C2311yescharacteristic
107C4×Q8×C73—22no—
108C4×Q8×C73—22no—
109C4×Q8×C73—22no—
110C4×Q8×C73—41no—
111C4×Q8×C73—41no—
112C4×Q8×C73C2×C414yescomplemented
113C2×C4⋊C4×C73—21no—
114C2×C4⋊C4×C73—21no—
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73—21no—
117C2×C4⋊C4×C73—21no—
118C2×C4⋊C4×C73—21no—
119C2×C4⋊C4×C73—21no—
120C2×C4⋊C4×C73D411yescharacteristic
121C2×C4⋊C4×C73D411yescharacteristic
122C2×C4⋊C4×C73C2311yescharacteristic
123C2×C4⋊C4×C73C2311yescharacteristic
124C2×C4⋊C4×C73C2311yescharacteristic
125C2×C42×C73—21no—
126C2×C42×C73—21no—
127C2×C42×C73—21no—
128C2×C42×C73C2311yescharacteristic
129C2×C42×C73C2311yescharacteristic
130C2×C42×C73C2311yescharacteristic
131#95×C73—21no—
132#95×C73—21no—
133#95×C73D411yescharacteristic
134#95×C73D411yescharacteristic
135C23×D73—21no—
136C2×D292—21no—
137C2×D292—21no—
138C2×D292—21no—
139C2×D292C22×C412yes—
140C2×D292C2×D412yes—
141C2×D292C2×Q812yes—
142C22×Dic73—21no—
143C22×Dic73—21no—
144C22×Dic73—21no—
145C2×C4×D73—24no—
146C2×C4×D73—42no—
147C2×C4×D73—42no—
148C2×C4×D73—42no—
149C2×C4×D73—42no—
150C2×C4×D73—81no—
151C2×C4×D73—81no—
152C2×C4×D73—81no—
153C2×C4×D73—81no—
154#11924015C4⋊C4116yescomplemented
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C2×#10596831—21no—
163C2×#10596831C22×C412yes—
164C2×#10596831C22×C412yes—
165C2×#10596831C2×D412yes—
166C2×#10596831C2×Q812yes—
167C4⋊Dic73—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73—21no—
170C4⋊Dic73—21no—
171C4⋊Dic73—21no—
172C4⋊Dic73C22×C412yes—
173#11924007—81no—
174#11924007—81no—
175#11924005—24no—
176#11924005—42no—
177#11924005—42no—
178C2×Q8×C73—21no—
179C2×Q8×C73—21no—
180C2×Q8×C73—21no—
181C2×Q8×C73C22×C412yes—
182C2×Q8×C73C2×D412yes—
183C2×Q8×C73C2×Q812yes—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2411yescharacteristic
195C4⋊C4×C73—24no—
196C4⋊C4×C73—22no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—41no—
203C4⋊C4×C73—42no—
204C4⋊C4×C73—42no—
205C4⋊C4×C73—42no—
206C4⋊C4×C73—81no—
207C4⋊C4×C73—81no—
208C4⋊C4×C73—81no—
209C4⋊C4×C73C22×C412yes—
210C42×C73—24no—
211C42×C73—21no—
212C42×C73—21no—
213C42×C73—21no—
214C42×C73—21no—
215C42×C73—42no—
216C42×C73—42no—
217C42×C73C22×C412yes—
218C42×C73#4112yes—
219C23×C73C22×D411yescharacteristic
220C22×D73—22no—
221C22×D73—22no—
222C22×D73—22no—
223C22×D73—41no—
224C22×D73—41no—
225C22×D73—41no—
226C22×D73—41no—
227Q8×C73—22no—
228Q8×C73—22no—
229Q8×C73—22no—
230Q8×C73C2×C4⋊C414yes—
231C2×C4×C73—22no—
232C2×C4×C73—21no—
233C2×C4×C73—22no—
234C2×C4×C73—22no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—22no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73C4×D412yes—
259C2×C4×C73C4×D412yes—
260C2×C4×C73#13712yes—
261C2×C4×C73C23×C411yescharacteristic
262C2×C4×C73C22×D411yescharacteristic
263C2×C4×C73C22×Q811yescharacteristic
264C2×C4×C73C2×#4111yescharacteristic
265C2×C4×C73H(5,𝔽2)11yescharacteristic
266C2×C4×C732−1+411yescharacteristic
267C2×Dic73—22no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276D292—22no—
277D292—22no—
278D292—22no—
279D292C2×C4⋊C414yes—
280C4×D73—84no—
281C4×D73—84no—
282C4×D73—84no—
283#10596831—22no—
284#10596831—41no—
285#10596831—41no—
286#10596831C2×C4⋊C414yes—
287C22×C73C2×C4×D411yescharacteristic
288C22×C73D4211yescharacteristic
289C22×C73D4×Q811yescharacteristic
290C22×C73C2×#13711yescharacteristic
291C22×C73#57511yescharacteristic
292C22×C73#57611yescharacteristic
293C22×C73#57711yescharacteristic
294C2×D73—24no—
295C2×D73—42no—
296C2×D73—42no—
297C2×D73—42no—
298C2×D73—42no—
299C2×D73—42no—
300C2×D73—42no—
301C4×C73—24no—
302C4×C73—22no—
303C4×C73—22no—
304C4×C73—22no—
305C4×C73—42no—
306C4×C73—41no—
307C4×C73—41no—
308C4×C73—41no—
309C4×C73—81no—
310C4×C73—81no—
311C4×C73C22×C4⋊C411yescharacteristic
312C4×C73#52011yescharacteristic
313C4×C73#54711yescharacteristic
314C4×C73#55411yescharacteristic
315Dic73—42no—
316Dic73—81no—
317Dic73—81no—
318#22678—731nomaximal, Sylow
319C2×C73D4×C4⋊C411yescharacteristic
320C2×C73C4×#13711yescharacteristic
321C2×C73#294611yescharacteristic
322C2×C73#326611yescharacteristic
323C2×C73#326811yescharacteristic
324C2×C73#338511yescharacteristic
325C2×C73#338811yescharacteristic
326D73—44no—
327C2×#440—731no—
328C2×#440—731no—
329C2×C4×SD16—731no—
330Q8×C4⋊C4D7311yescharacteristic, complemented
331#2349—731no—
332#1943—731no—
333#1942—731no—
334#1865—731no—
335#1864—731no—
336#1863—731no—
337#1862—731no—
338#1847—731no—
339#1805—731no—
340#1802—731no—
341#1771—731no—
342C73#2267811yescharacteristic, complemented, Sylow
343C22×SD16—731no—
344C2×C4×Q8C2×D7311yescharacteristic
345C2×C4×Q8C2×D7311yescharacteristic
346C2×C4×Q8C2×D7311yescharacteristic
347C2×C4×D4—731no—
348C2×C4×D4—731no—
349C2×C4×D4—731no—
350#440—738no—
351#440—1464no—
352C4×SD16—1464no—
353C2×#107—731no—
354C2×#107—731no—
355C2×#107—731no—
356C2×#106—731no—
357C2×#106—731no—
358C2×C4⋊C8—731no—
359C2×#103—731no—
360C2×#103—731no—
361C2×#103—731no—
362C2×#102—731no—
363C2×#102—731no—
364C2×#102—731no—
365C2×#97—731no—
366C2×#97—731no—
367C2×C4×C8—731no—
368#389—21no—
369#389—21no—
370#389—731no—
371#389—731no—
372#389C2×D7311yescharacteristic
373#389C2×D7311yescharacteristic
374#385—1461no—
375#385—1461no—
376#384—21no—
377#384C2×D7311yescharacteristic
378#382—731no—
379#380—1461no—
380C4×C4⋊C4—21no—
381C4×C4⋊C4C2×D7311yescharacteristic
382C2×#95—1461no—
383#340—731no—
384#340—731no—
385#340—731no—
386#340—731no—
387#336—731no—
388#336—731no—
389C22×Q8C22×D7311yescharacteristic
390C22×D4—731no—
391C23×C4—1461no—
392C23×C4—1461no—
393C23×C4—1461no—
394C2×SD16—734no—
395C2×SD16—734no—
396C2×SD16—734no—
397C22×C8—731no—
398C22×C8—731no—
399C22×C8—731no—
400C22×C8—731no—
401C4×Q8—22no—
402C4×Q8—22no—
403C4×Q8—22no—
404C4×Q8—41no—
405C4×Q8—41no—
406C4×Q8C4×D7314yescomplemented
407C4×D4—2921no—
408C4×D4—2921no—
409C4×D4—734no—
410C4×D4—1462no—
411C4×D4—1462no—
412C4×D4—1462no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—731no—
421C2×C4⋊C4—731no—
422C2×C4⋊C4—731no—
423C2×C4⋊C4—1461no—
424C2×C4⋊C4—1461no—
425C2×C4⋊C4C22⋊D7311yescharacteristic
426C2×C4⋊C4C22⋊D7311yescharacteristic
427C2×C4⋊C4C22×D7311yescharacteristic
428C2×C4⋊C4C22×D7311yescharacteristic
429C2×C4⋊C4C22×D7311yescharacteristic
430C2×C22⋊C4—1461no—
431C2×C22⋊C4—1461no—
432C2×C22⋊C4—1461no—
433C2×C42—21no—
434C2×C42—21no—
435C2×C42—21no—
436C2×C42C22×D7311yescharacteristic
437C2×C42C22×D7311yescharacteristic
438C2×C42C22×D7311yescharacteristic
439#107—2921no—
440#107—734no—
441#107—1462no—
442#106—2921no—
443#106—1462no—
444C4⋊C8—1462no—
445#103—734no—
446#103—1462no—
447#103—1462no—
448#102—734no—
449#102—1462no—
450#102—1462no—
451#97—734no—
452#97—1462no—
453C4×C8—1462no—
454#95—21no—
455#95—21no—
456#95—1461no—
457#95—1461no—
458#95—1461no—
459#95—1461no—
460#95C22⋊D7311yescharacteristic
461#95C22⋊D7311yescharacteristic
462C24—1461no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8C2×C4×D7312yes—
467C2×Q8D4×D7312yes—
468C2×Q8Q8×D7312yes—
469C2×D4—732no—
470C2×D4—732no—
471C2×D4—732no—
472C2×D4—1461no—
473C2×D4—1461no—
474C2×D4—1461no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—2922no—
480C22×C4—2922no—
481C22×C4—2922no—
482C22×C4—2922no—
483C22×C4—5841no—
484C22×C4—5841no—
485C22×C4—5841no—
486C22×C4—5841no—
487C22×C4—1461no—
488C22×C4—1461no—
489C22×C4—1461no—
490C22×C4—1464no—
491C22×C4C2×C22⋊D7311yescharacteristic
492C22×C4C2×C22⋊D7311yescharacteristic
493C22×C4C2×C22⋊D7311yescharacteristic
494C22×C4C2×C22⋊D7311yescharacteristic
495C22×C4C2×C22⋊D7311yescharacteristic
496C22×C4C2×C22⋊D7311yescharacteristic
497C22×C4C23×D7311yescharacteristic
498SD16—7316no—
499C2×C8—732no—
500C2×C8—732no—
501C2×C8—732no—
502C2×C8—732no—
503C2×C8—1461no—
504C2×C8—1461no—
505C2×C8—1461no—
506C2×C8—1461no—
507C2×C8—1461no—
508C2×C8—1461no—
509C2×C8—1461no—
510C2×C8—1461no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—41no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—81no—
523C4⋊C4—81no—
524C4⋊C4—81no—
525C4⋊C4—5841no—
526C4⋊C4—5841no—
527C4⋊C4—732no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4C2×C4×D7312yes—
534C22⋊C4—2922no—
535C22⋊C4—2922no—
536C22⋊C4—1464no—
537C42—24no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—42no—
543C42—42no—
544C42C2×C4×D7312yes—
545C42#1192402812yes—
546C23—2921no—
547C23—2921no—
548C23—2921no—
549C23—2921no—
550C23—1462no—
551C23—1462no—
552C23—1462no—
553C23C22×C22⋊D7311yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8C4⋊C4×D7314yes—
558D4—734no—
559D4—1462no—
560D4—1462no—
561D4—1462no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—5844no—
598C2×C4—5844no—
599C2×C4—5844no—
600C2×C4—1462no—
601C2×C4C4×C22⋊D7312yes—
602C2×C4C4×C22⋊D7312yes—
603C2×C4#1455662212yes—
604C2×C4C22×C4×D7311yescharacteristic
605C2×C4C2×D4×D7311yescharacteristic
606C2×C4C2×Q8×D7311yescharacteristic
607C2×C4C2×#1192402811yescharacteristic
608C2×C4#1455663611yescharacteristic
609C2×C4#1455663911yescharacteristic
610C8—2921no—
611C8—2921no—
612C8—734no—
613C8—1462no—
614C22—2922no—
615C22—2922no—
616C22—2922no—
617C22—2922no—
618C22—2922no—
619C22—2922no—
620C22—1464no—
621C22C2×C4×C22⋊D7311yescharacteristic
622C22D4×C22⋊D7311yescharacteristic
623C22Q8×C22⋊D7311yescharacteristic
624C22C2×#1455662211yescharacteristic
625C22#1798938511yescharacteristic
626C22#1798939011yescharacteristic
627C22#1798939511yescharacteristic
628C4—22no—
629C4—24no—
630C4—22no—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—42no—
636C4—81no—
637C4—81no—
638C4—2922no—
639C4—5841no—
640C4—5841no—
641C4C2×C4⋊C4×D7311yescharacteristic
642C4#1798906811yescharacteristic
643C4#1798923811yescharacteristic
644C4#1798929211yescharacteristic
645C2—2924no—
646C2C4⋊C4×C22⋊D7311yescharacteristic
647C2C4×#1455662211yescharacteristic
648C2#2261875911yescharacteristic
649C2#2262144211yescharacteristic
650C2#2262147211yescharacteristic
651C2#2262283711yescharacteristic
652C2#2262287011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingQ8×C4⋊C4×C73C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenterQ8×C4⋊C4D73
hypocenterC73#22678
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #22610666C2semidirect
6. #22610661C2semidirect
7. #22610199C2semidirect
9. #22610185C2semidirect
11. #22610097C2semidirect
12. #22609946C2semidirect
14. #22609806C2semidirect
15. Q8×C4⋊C4×C73C2semidirect
21. #17988685C4semidirect
89. #14556524C2×C4semidirect
95. #14556515C2×C4semidirect
100. #14556502C2×C4semidirect
112. C4×Q8×C73C2×C4semidirect
154. #11924015C4⋊C4semidirect
330. Q8×C4⋊C4D73semidirect
342. C73#22678semidirect
406. C4×Q8C4×D73semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179886852
C2×C4×#119240152
#226128422
#226106662
#226106612
#226101992
#226101912
#226101852
#226101772
#226100972
#226099462
#226099282
#226098062
Q8×C4⋊C4×C732
#2267873