Finite Group Databasebeta

← #33104401

Subgroups — #33104401

Order 18688.

986 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11924015C211yesmaximal, characteristic
2C2×#17988672C211yesmaximal, characteristic
3C2×#17988672C211yesmaximal, characteristic
4#22612849C211yesmaximal, characteristic, complemented
5#22612840C211yesmaximal, characteristic
6#22610658C211yesmaximal, characteristic, complemented
7#22610656C211yesmaximal, characteristic, complemented
8#22610187C211yesmaximal, characteristic, complemented
9#22610181C211yesmaximal, characteristic
10#22610175C211yesmaximal, characteristic
11#22610094C211yesmaximal, characteristic, complemented
12#22609944C211yesmaximal, characteristic, complemented
13#22609940C211yesmaximal, characteristic
14#22609792C211yesmaximal, characteristic, complemented
15C73×#1863C211yesmaximal, characteristic, complemented
16C22×#11924015C2211yescharacteristic
17C2×Q8×Dic73C2211yescharacteristic
18C2×Q8×Dic73C2211yescharacteristic
19C2×C4×D292C2211yescharacteristic
20C2×#14556553C2211yescharacteristic
21C2×#14556553C2211yescharacteristic
22C4×#11924015—24no—
23#17988672—24no—
24#17988672C418yescomplemented
25C2×#14556525C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×#14556522C2211yescharacteristic
28C2×#14556522C2211yescharacteristic
29C2×#14556520C2211yescharacteristic
30C2×#14556515C2211yescharacteristic
31C2×#14556513C2211yescharacteristic
32C2×#14556513C2211yescharacteristic
33C2×#14556511C2211yescharacteristic
34C2×C4×#10596831C2211yescharacteristic
35C2×C8×Dic73C2211yescharacteristic
36C2×C8×Dic73C2211yescharacteristic
37#17988486C2211yescharacteristic
38#17988485—21no—
39#17988485C2211yescharacteristic
40#17988483—21no—
41#17988483C2211yescharacteristic
42#17988461—21no—
43#17988458—21no—
44#17988452—21no—
45#17988452C2211yescharacteristic
46#17988452C2211yescharacteristic
47#17988444—21no—
48C4×#11924007—21no—
49C4×#11924007C2211yescharacteristic
50#95×D73—21no—
51#17988350C2211yescharacteristic
52#17988349C2211yescharacteristic
53#17988349C2211yescharacteristic
54#17988341C2211yescharacteristic
55C2×C4×Q8×C73C2211yescharacteristic
56C2×#107×C73C2211yescharacteristic
57C2×#103×C73C2211yescharacteristic
58C2×#103×C73C2211yescharacteristic
59#389×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#336×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
71Q8×Dic73—22no—
72Q8×Dic73—22no—
73Q8×Dic73—41no—
74Q8×Dic73C2×C414yescomplemented
75C4×D292—22no—
76C4×D292—41no—
77#14556553—22no—
78#14556553—22no—
79#14556553—41no—
80#14556553C2×C414yes—
81C2×C4⋊Dic73—21no—
82C2×C4⋊Dic73—21no—
83C2×C4⋊Dic73—21no—
84C2×C4⋊Dic73C2311yescharacteristic
85C2×C4⋊Dic73C2311yescharacteristic
86C2×C4⋊Dic73C2311yescharacteristic
87C2×#11924007—21no—
88C2×#11924007—21no—
89C2×#11924007D411yescharacteristic
90C2×#11924007D411yescharacteristic
91C2×#11924005—21no—
92C2×#11924005—21no—
93C2×#11924005—21no—
94C2×C4×Dic73—21no—
95C2×C4×Dic73—21no—
96C2×C4×Dic73C2311yescharacteristic
97C2×C4×Dic73C2311yescharacteristic
98#14556525—41no—
99#14556522—22no—
100#14556522—22no—
101#14556522C2×C414yescomplemented
102#14556520—22no—
103#14556515—22no—
104#14556513—22no—
105#14556513C2×C414yes—
106#14556511—22no—
107C4×#10596831—22no—
108C8×Dic73—22no—
109C8×Dic73C2×C414yescomplemented
110#14556498—21no—
111#14556498—21no—
112#14556498—21no—
113#14556498—21no—
114#14556498—21no—
115#14556498—21no—
116#14556498—21no—
117#14556498D411yescharacteristic
118#14556498D411yescharacteristic
119C22×Q8×C73C2311yescharacteristic
120C22×C8×C73C2311yescharacteristic
121C22×C8×C73C2311yescharacteristic
122C4×Q8×C73—22no—
123C4×Q8×C73—41no—
124C2×C4⋊C4×C73—21no—
125C2×C4⋊C4×C73—21no—
126C2×C4⋊C4×C73C2311yescharacteristic
127C2×C4⋊C4×C73C2311yescharacteristic
128C2×C4⋊C4×C73C2311yescharacteristic
129C2×C42×C73—21no—
130C2×C42×C73C2311yescharacteristic
131#107×C73—41no—
132#103×C73—22no—
133#103×C73C2×C414yescomplemented
134#95×C73—21no—
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—
145C22×Dic73C2×D411yescharacteristic
146C22×Dic73C2×D411yescharacteristic
147C22×Dic73C2×D411yescharacteristic
148C22×Dic73C2×D411yescharacteristic
149C2×C4×D73—24no—
150C2×C4×D73—42no—
151C2×C4×D73—42no—
152C2×C4×D73—42no—
153C2×C4×D73—42no—
154C2×C4×D73—81no—
155C2×C4×D73—81no—
156C2×C4×D73—81no—
157C2×C4×D73—81no—
158#11924015C4⋊C4116yescomplemented
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C2×#10596831C22×C412yes—
163C2×#10596831C2×D412yes—
164C2×#10596831C2×Q812yes—
165C4⋊Dic73—21no—
166C4⋊Dic73—24no—
167C4⋊Dic73—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73—21no—
170C4⋊Dic73—21no—
171C4⋊Dic73—42no—
172C4⋊Dic73—81no—
173C4⋊Dic73C22×C412yes—
174#11924007—22no—
175#11924007—22no—
176#11924007—42no—
177#11924007—42no—
178#11924005—24no—
179#11924005—42no—
180#11924005—42no—
181C4×Dic73—21no—
182C4×Dic73—21no—
183C4×Dic73—24no—
184C4×Dic73—42no—
185C4×Dic73C22×C412yes—
186C4×Dic73#4112yes—
187C2×Q8×C73—21no—
188C2×Q8×C73—21no—
189C2×Q8×C73—21no—
190C2×Q8×C73C22×C412yes—
191C2×Q8×C73C2×D412yes—
192C2×Q8×C73C2×Q812yes—
193C22×C4×C73—21no—
194C22×C4×C73—21no—
195C22×C4×C73—21no—
196C22×C4×C73—21no—
197C22×C4×C73C2×D411yescharacteristic
198C22×C4×C73C2×D411yescharacteristic
199C22×C4×C73C2411yescharacteristic
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C2×C8×C73C22×C412yes—
206C4⋊C4×C73—21no—
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73—21no—
210C4⋊C4×C73—21no—
211C4⋊C4×C73—42no—
212C4⋊C4×C73—81no—
213C4⋊C4×C73C22×C412yes—
214C42×C73—21no—
215C42×C73—21no—
216C42×C73—42no—
217C23×C73C22×D411yescharacteristic
218C22×D73—22no—
219C22×D73—22no—
220C22×D73—22no—
221C22×D73—41no—
222C22×D73—41no—
223C22×D73—41no—
224C22×D73—41no—
225Q8×C73—22no—
226Q8×C73—22no—
227Q8×C73—22no—
228Q8×C73C2×C4⋊C414yes—
229C2×C4×C73—22no—
230C2×C4×C73—21no—
231C2×C4×C73—22no—
232C2×C4×C73—22no—
233C2×C4×C73—21no—
234C2×C4×C73—22no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73#13512yes—
247C2×C4×C73C23×C411yescharacteristic
248C2×C4×C73C22×D411yescharacteristic
249C2×C4×C73C22×Q811yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C2×C4×C73H(5,𝔽2)11yescharacteristic
252C2×C4×C732−1+411yescharacteristic
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—22no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73C4×D412yes—
272C2×Dic73C4×D412yes—
273C2×Dic73#13512yes—
274D292—22no—
275D292—22no—
276D292—22no—
277D292C2×C4⋊C414yes—
278C4×D73—84no—
279C4×D73—84no—
280C4×D73—84no—
281C8×C73—22no—
282C8×C73—41no—
283#10596831—41no—
284#10596831C2×C4⋊C414yes—
285C22×C73C2×C4×D411yescharacteristic
286C22×C73D4211yescharacteristic
287C22×C73D4×Q811yescharacteristic
288C22×C73C2×#13511yescharacteristic
289C22×C73C2×#13511yescharacteristic
290C22×C73#57711yescharacteristic
291C22×C73#57711yescharacteristic
292C2×D73—24no—
293C2×D73—42no—
294C2×D73—42no—
295C2×D73—42no—
296C2×D73—42no—
297C2×D73—42no—
298C2×D73—42no—
299C4×C73—24no—
300C4×C73—22no—
301C4×C73—42no—
302C4×C73—41no—
303C4×C73—81no—
304C4×C73—81no—
305C4×C73C22×C4⋊C411yescharacteristic
306C4×C73#52011yescharacteristic
307C4×C73#54711yescharacteristic
308C4×C73#55411yescharacteristic
309Dic73—22no—
310Dic73—22no—
311Dic73—22no—
312Dic73—42no—
313Dic73—41no—
314Dic73—81no—
315Dic73—81no—
316#22683—731nomaximal, Sylow
317C2×C73D4×C4⋊C411yescharacteristic
318C2×C73C4×#13511yescharacteristic
319C2×C73#294911yescharacteristic
320C2×C73#328211yescharacteristic
321C2×C73#328211yescharacteristic
322C2×C73#338111yescharacteristic
323C2×C73#338111yescharacteristic
324D73—44no—
325C2×C4×SD16—731no—
326C2×C4×SD16—731no—
327C2×C4×SD16—731no—
328#2351—731no—
329#2349—731no—
330#1941—731no—
331#1941—731no—
332#1863—731no—
333#1863D7311yescharacteristic, complemented
334#1862—731no—
335#1862—731no—
336#1846—731no—
337#1805—731no—
338#1804—731no—
339C8×C4⋊C4—731no—
340C73#2268311yescharacteristic, complemented, Sylow
341C22×SD16—731no—
342C2×C4×Q8—731no—
343C2×C4×Q8—731no—
344C2×C4×Q8C2×D7311yescharacteristic
345C2×C4×D4—731no—
346C2×C4×D4—731no—
347C2×C4×D4—731no—
348C4×SD16—738no—
349C4×SD16—1464no—
350C4×SD16—1464no—
351C2×#107—731no—
352C2×#107C2×D7311yescharacteristic
353C2×#106—731no—
354C2×#106—731no—
355C2×#106—731no—
356C2×C4⋊C8—731no—
357C2×#103—731no—
358C2×#103C2×D7311yescharacteristic
359C2×#103C2×D7311yescharacteristic
360C2×#102—731no—
361C2×#102—731no—
362C2×#102—731no—
363C2×C4×C8—731no—
364C2×C4×C8—731no—
365C2×C4×C8—731no—
366#389—731no—
367#389—731no—
368#389—731no—
369#389—1461no—
370#389—1461no—
371#389C2×D7311yescharacteristic
372#385—1461no—
373#385—1461no—
374#382—731no—
375#382—731no—
376#382—1461no—
377#380—1461no—
378C4×C4⋊C4—731no—
379C4×C4⋊C4—1461no—
380C2×#95—1461no—
381#340—731no—
382#340—731no—
383#340—731no—
384#340C2×D7311yescharacteristic
385#336—731no—
386#336C2×D7311yescharacteristic
387C22×Q8C22×D7311yescharacteristic
388C22×D4—731no—
389C23×C4—1461no—
390C23×C4—1461no—
391C23×C4—1461no—
392C2×SD16—734no—
393C2×SD16—734no—
394C2×SD16—734no—
395C22×C8—731no—
396C22×C8—731no—
397C22×C8C22×D7311yescharacteristic
398C22×C8C22×D7311yescharacteristic
399C4×Q8—22no—
400C4×Q8—41no—
401C4×Q8—2921no—
402C4×Q8—734no—
403C4×Q8—1462no—
404C4×Q8—1462no—
405C4×D4—2921no—
406C4×D4—2921no—
407C4×D4—734no—
408C4×D4—1462no—
409C4×D4—1462no—
410C4×D4—1462no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—731no—
414C2×C4⋊C4—731no—
415C2×C4⋊C4—731no—
416C2×C4⋊C4—731no—
417C2×C4⋊C4—731no—
418C2×C4⋊C4—1461no—
419C2×C4⋊C4—1461no—
420C2×C4⋊C4—1461no—
421C2×C4⋊C4—1461no—
422C2×C4⋊C4—1461no—
423C2×C4⋊C4C22×D7311yescharacteristic
424C2×C4⋊C4C22×D7311yescharacteristic
425C2×C4⋊C4C22×D7311yescharacteristic
426C2×C22⋊C4—1461no—
427C2×C22⋊C4—1461no—
428C2×C22⋊C4—1461no—
429C2×C42—21no—
430C2×C42—731no—
431C2×C42—731no—
432C2×C42—1461no—
433C2×C42—1461no—
434C2×C42C22×D7311yescharacteristic
435#107—41no—
436#107—2921no—
437#106—734no—
438#106—1462no—
439#106—1462no—
440C4⋊C8—1462no—
441#103—22no—
442#103—1462no—
443#103C4×D7314yescomplemented
444#102—734no—
445#102—1462no—
446#102—1462no—
447C4×C8—734no—
448C4×C8—1462no—
449C4×C8—1462no—
450#95—21no—
451#95—731no—
452#95—731no—
453#95—1461no—
454#95—1461no—
455#95—1461no—
456#95—1461no—
457#95—1461no—
458#95—1461no—
459#95—1461no—
460C24—1461no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8C2×C4×D7312yes—
465C2×Q8D4×D7312yes—
466C2×Q8Q8×D7312yes—
467C2×D4—732no—
468C2×D4—732no—
469C2×D4—732no—
470C2×D4—1461no—
471C2×D4—1461no—
472C2×D4—1461no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—2922no—
478C22×C4—2922no—
479C22×C4—2922no—
480C22×C4—2922no—
481C22×C4—5841no—
482C22×C4—5841no—
483C22×C4—5841no—
484C22×C4—5841no—
485C22×C4—731no—
486C22×C4—731no—
487C22×C4—731no—
488C22×C4—731no—
489C22×C4—1464no—
490C22×C4—1461no—
491C22×C4—1461no—
492C22×C4—1461no—
493C22×C4D4×D7311yescharacteristic
494C22×C4D4×D7311yescharacteristic
495C22×C4C23×D7311yescharacteristic
496SD16—7316no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—732no—
503C2×C8—732no—
504C2×C8—732no—
505C2×C8—1461no—
506C2×C8—1461no—
507C2×C8—1461no—
508C2×C8C2×C4×D7312yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—81no—
516C4⋊C4—2922no—
517C4⋊C4—2922no—
518C4⋊C4—2922no—
519C4⋊C4—5841no—
520C4⋊C4—732no—
521C4⋊C4—1462no—
522C4⋊C4—1464no—
523C4⋊C4—1461no—
524C4⋊C4—1461no—
525C4⋊C4—1461no—
526C4⋊C4—1461no—
527C4⋊C4—1462no—
528C4⋊C4—1461no—
529C4⋊C4C2×C4×D7312yes—
530C22⋊C4—2922no—
531C22⋊C4—2922no—
532C22⋊C4—1464no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42—2922no—
537C42—732no—
538C42—732no—
539C42—1464no—
540C42—1461no—
541C42—1461no—
542C23—2921no—
543C23—2921no—
544C23—2921no—
545C23—2921no—
546C23—1462no—
547C23—1462no—
548C23—1462no—
549C23C2×D4×D7311yescharacteristic
550Q8—22no—
551Q8—22no—
552Q8—22no—
553Q8C4⋊C4×D7314yes—
554D4—734no—
555D4—1462no—
556D4—1462no—
557D4—1462no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—5844no—
584C2×C4—5844no—
585C2×C4—5844no—
586C2×C4—732no—
587C2×C4—732no—
588C2×C4—732no—
589C2×C4—1461no—
590C2×C4—1461no—
591C2×C4—1462no—
592C2×C4—1461no—
593C2×C4—1461no—
594C2×C4—1461no—
595C2×C4—1461no—
596C2×C4—1461no—
597C2×C4—1461no—
598C2×C4—1461no—
599C2×C4#1455661412yes—
600C2×C4C22×C4×D7311yescharacteristic
601C2×C4C2×D4×D7311yescharacteristic
602C2×C4C2×Q8×D7311yescharacteristic
603C2×C4#41×D7311yescharacteristic
604C2×C4#1455663611yescharacteristic
605C2×C4#1455663811yescharacteristic
606C8—22no—
607C8—41no—
608C8—2921no—
609C8—734no—
610C22—2922no—
611C22—2922no—
612C22—2922no—
613C22—2922no—
614C22—2922no—
615C22—2922no—
616C22—1464no—
617C22C4×D4×D7311yescharacteristic
618C22#1798921911yescharacteristic
619C22#1798925011yescharacteristic
620C22#135×D7311yescharacteristic
621C22C2×#1455661411yescharacteristic
622C22#1798939611yescharacteristic
623C22#1798939711yescharacteristic
624C4—24no—
625C4—22no—
626C4—42no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—2922no—
631C4—2921no—
632C4—5841no—
633C4—5841no—
634C4—1462no—
635C4—1462no—
636C4—1462no—
637C4C2×C4⋊C4×D7311yescharacteristic
638C4#1798906611yescharacteristic
639C4#1798923511yescharacteristic
640C4#1798929111yescharacteristic
641C2—2924no—
642C2#2261282911yescharacteristic
643C2C4×#1455661411yescharacteristic
644C2#2261878711yescharacteristic
645C2#2262163111yescharacteristic
646C2#2262164111yescharacteristic
647C2#2262280511yescharacteristic
648C2#2262280911yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1863C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenter#1863D73
hypocenterC73#22683
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
4. #22612849C2semidirect
6. #22610658C2semidirect
7. #22610656C2semidirect
8. #22610187C2semidirect
11. #22610094C2semidirect
12. #22609944C2semidirect
14. #22609792C2semidirect
15. C73×#1863C2semidirect
24. #17988672C4semidirect
74. Q8×Dic73C2×C4semidirect
101. #14556522C2×C4semidirect
109. C8×Dic73C2×C4semidirect
133. #103×C73C2×C4semidirect
158. #11924015C4⋊C4semidirect
333. #1863D73semidirect
340. C73#22683semidirect
443. #103C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#119240152
C2×#179886722
#226128492
#226128402
#226106582
#226106562
#226101872
#226101812
#226101752
#226100942
#226099442
#226099402
#226097922
C73×#18632
#2268373