Finite Group Databasebeta

← #33104332

Subgroups — #33104332

Order 18688.

1010 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988687C211yesmaximal, characteristic
2C2×#17988684C211yesmaximal, characteristic
3C2×#17988678C211yesmaximal, characteristic
4#22612850C211yesmaximal, characteristic, complemented
5D4×#11924007C211yesmaximal, characteristic
6#22610666C211yesmaximal, characteristic, complemented
7#22610663C211yesmaximal, characteristic, complemented
8#22610196C211yesmaximal, characteristic, complemented
9#22610188C211yesmaximal, characteristic, complemented
10#22610180C211yesmaximal, characteristic
11#22610096C211yesmaximal, characteristic, complemented
12#22609939C211yesmaximal, characteristic
13#22609935C211yesmaximal, characteristic, complemented
14#22609804C211yesmaximal, characteristic, complemented
15C73×#1864C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18C2×#14556556C2211yescharacteristic
19C2×#14556556C2211yescharacteristic
20C2×D4×Dic73C2211yescharacteristic
21C2×D4×Dic73C2211yescharacteristic
22C22×#11924007—21no—
23#17988687—24no—
24#17988684C418yescomplemented
25#17988678—24no—
26C2×#14556525C2211yescharacteristic
27C2×#14556524C2211yescharacteristic
28C2×#14556524C2211yescharacteristic
29C2×#14556522C2211yescharacteristic
30C2×#14556520C2211yescharacteristic
31C2×#14556516C2211yescharacteristic
32C2×#14556516C2211yescharacteristic
33C2×#14556514C2211yescharacteristic
34C2×#14556512C2211yescharacteristic
35C2×#14556502C2211yescharacteristic
36C2×#14556501C2211yescharacteristic
37C2×C8×Dic73C2211yescharacteristic
38#17988486C2211yescharacteristic
39#17988485C2211yescharacteristic
40#17988484—21no—
41#17988483C2211yescharacteristic
42#17988482—21no—
43#17988460—21no—
44#17988459—21no—
45#17988456C2211yescharacteristic
46#17988452C2211yescharacteristic
47#17988451—21no—
48#17988445—21no—
49C4⋊C4×Dic73—21no—
50C4×#11924007C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988349C2211yescharacteristic
53#17988349C2211yescharacteristic
54#17988341C2211yescharacteristic
55C2×C4×D4×C73C2211yescharacteristic
56C2×#106×C73C2211yescharacteristic
57C2×#102×C73C2211yescharacteristic
58C2×#102×C73C2211yescharacteristic
59#389×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#336×C73C2211yescharacteristic
62C22×Dic146C2311yescharacteristic
63C23×Dic73—21no—
64C23×Dic73—21no—
65C2×#11924016C2×C414yes—
66C2×#11924016D414yes—
67C2×#11924016Q814yes—
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C4×Dic146—22no—
71C4×Dic146—41no—
72#14556556—22no—
73#14556556—22no—
74#14556556—41no—
75#14556556C2×C414yescomplemented
76D4×Dic73—22no—
77D4×Dic73—22no—
78D4×Dic73—41no—
79D4×Dic73C2×C414yes—
80C2×C4⋊Dic73—21no—
81C2×C4⋊Dic73C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×C4⋊Dic73C2311yescharacteristic
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×#11924007—21no—
87C2×#11924007—21no—
88C2×#11924007—21no—
89C2×#11924007—21no—
90C2×#11924007—21no—
91C2×#11924007—44no—
92C2×#11924007D411yescharacteristic
93C2×#11924007D411yescharacteristic
94C2×C22⋊Dic73—21no—
95C2×C22⋊Dic73—21no—
96C2×C4×Dic73—21no—
97C2×C4×Dic73—21no—
98C2×C4×Dic73—21no—
99C2×C4×Dic73C2311yescharacteristic
100C2×C4×Dic73C2311yescharacteristic
101#14556525—41no—
102#14556524—22no—
103#14556524C2×C414yescomplemented
104#14556522—22no—
105#14556520—22no—
106#14556516—22no—
107#14556516C2×C414yescomplemented
108#14556514—22no—
109#14556512—22no—
110#14556502—22no—
111#14556501C2×C414yescomplemented
112C8×Dic73—22no—
113#14556498—21no—
114#14556498—21no—
115#14556498—21no—
116#14556498—21no—
117#14556498—21no—
118#14556498D411yescharacteristic
119#14556498D411yescharacteristic
120C22×D4×C73C2311yescharacteristic
121C23×C4×C73—21no—
122C22×C8×C73C2311yescharacteristic
123C22×C8×C73C2311yescharacteristic
124C4×D4×C73—22no—
125C4×D4×C73—41no—
126C2×C4⋊C4×C73—21no—
127C2×C4⋊C4×C73C2311yescharacteristic
128C2×C4⋊C4×C73C2311yescharacteristic
129C2×C4⋊C4×C73C2311yescharacteristic
130C2×C22⋊C4×C73—21no—
131C2×C42×C73C2311yescharacteristic
132#106×C73—41no—
133#102×C73—22no—
134#102×C73C2×C414yes—
135#95×C73—21no—
136C2×Dic146—21no—
137C2×Dic146—21no—
138C2×Dic146—21no—
139C2×Dic146C22×C412yes—
140C2×Dic146C2×D412yes—
141C2×Dic146C2×Q812yes—
142C22×Dic73—21no—
143C22×Dic73—21no—
144C22×Dic73—24no—
145C22×Dic73—21no—
146C22×Dic73—21no—
147C22×Dic73—42no—
148C22×Dic73—42no—
149C22×Dic73—42no—
150C22×Dic73—42no—
151C22×Dic73—81no—
152C22×Dic73C2×D411yescharacteristic
153C22×Dic73C2×D411yescharacteristic
154C22×Dic73C2×D411yescharacteristic
155C22×Dic73C2×D411yescharacteristic
156#11924016C4⋊C4116yescomplemented
157C2×#10596831—21no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831C22×C412yes—
161C2×#10596831C2×D412yes—
162C2×#10596831C2×Q812yes—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—21no—
166C4⋊Dic73—21no—
167C4⋊Dic73—21no—
168C4⋊Dic73—42no—
169C4⋊Dic73C22×C412yes—
170#11924007—24no—
171#11924007—22no—
172#11924007—41no—
173#11924007—42no—
174#11924007—42no—
175#11924007—42no—
176#11924007—42no—
177#11924007—42no—
178#11924007—81no—
179C22⋊Dic73—24no—
180C22⋊Dic73—42no—
181C4×Dic73—21no—
182C4×Dic73—21no—
183C4×Dic73—24no—
184C4×Dic73—42no—
185C4×Dic73—42no—
186C4×Dic73C22×C412yes—
187C4×Dic73#4112yes—
188C24×C73—21no—
189C2×D4×C73—21no—
190C2×D4×C73—21no—
191C2×D4×C73—21no—
192C2×D4×C73C22×C412yes—
193C2×D4×C73C2×D412yes—
194C2×D4×C73C2×Q812yes—
195C22×C4×C73—21no—
196C22×C4×C73—21no—
197C22×C4×C73—21no—
198C22×C4×C73—42no—
199C22×C4×C73—42no—
200C22×C4×C73—81no—
201C22×C4×C73C2×D411yescharacteristic
202C22×C4×C73C2×D411yescharacteristic
203C22×C4×C73C2411yescharacteristic
204C2×C8×C73—21no—
205C2×C8×C73—21no—
206C2×C8×C73—21no—
207C2×C8×C73—21no—
208C2×C8×C73—21no—
209C2×C8×C73C22×C412yes—
210C4⋊C4×C73—21no—
211C4⋊C4×C73—21no—
212C4⋊C4×C73—21no—
213C4⋊C4×C73—21no—
214C4⋊C4×C73—21no—
215C4⋊C4×C73—42no—
216C4⋊C4×C73C22×C412yes—
217C22⋊C4×C73—42no—
218C42×C73—21no—
219C42×C73—21no—
220C23×C73—22no—
221C23×C73—22no—
222C23×C73—22no—
223C23×C73—41no—
224C23×C73—41no—
225C23×C73—41no—
226C23×C73—41no—
227C23×C73C22×D411yescharacteristic
228D4×C73—22no—
229D4×C73—22no—
230D4×C73—22no—
231D4×C73C2×C4⋊C414yes—
232C2×C4×C73—22no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
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—41no—
247C2×C4×C73—84no—
248C2×C4×C73C23×C411yescharacteristic
249C2×C4×C73C22×D411yescharacteristic
250C2×C4×C73C22×Q811yescharacteristic
251C2×C4×C73C2×#4111yescharacteristic
252C2×C4×C73H(5,𝔽2)11yescharacteristic
253C2×C4×C732−1+411yescharacteristic
254C2×Dic73—22no—
255C2×Dic73—21no—
256C2×Dic73—22no—
257C2×Dic73—22no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—22no—
264C2×Dic73—21no—
265C2×Dic73—21no—
266C2×Dic73—21no—
267C2×Dic73—48no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—84no—
277C2×Dic73C4×D412yes—
278C2×Dic73C4×D412yes—
279C2×Dic73AGL(1,ℤ/8)12yes—
280Dic146—22no—
281Dic146—22no—
282Dic146—22no—
283Dic146C2×C4⋊C414yescomplemented
284C8×C73—22no—
285C8×C73—41no—
286#10596831—41no—
287#10596831C2×C4⋊C414yescomplemented
288C22×C73—24no—
289C22×C73—42no—
290C22×C73—42no—
291C22×C73—42no—
292C22×C73—42no—
293C22×C73—42no—
294C22×C73—42no—
295C22×C73C2×C4×D411yescharacteristic
296C22×C73D4211yescharacteristic
297C22×C73D4×Q811yescharacteristic
298C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C73#57511yescharacteristic
300C22×C73#57711yescharacteristic
301C22×C73#57811yescharacteristic
302C4×C73—42no—
303C4×C73—41no—
304C4×C73—41no—
305C4×C73—81no—
306C4×C73—81no—
307C4×C73C22×C4⋊C411yescharacteristic
308C4×C73#52011yescharacteristic
309C4×C73#54711yescharacteristic
310C4×C73#55411yescharacteristic
311Dic73—24no—
312Dic73—22no—
313Dic73—22no—
314Dic73—22no—
315Dic73—42no—
316Dic73—41no—
317Dic73—81no—
318Dic73—81no—
319#22677—731nomaximal, Sylow
320C2×C73—44no—
321C2×C73D4×C4⋊C411yescharacteristic
322C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
323C2×C73#294711yescharacteristic
324C2×C73#326511yescharacteristic
325C2×C73#326911yescharacteristic
326C2×C73#338611yescharacteristic
327C2×C73#338811yescharacteristic
328C2×#440—731no—
329C2×#440—731no—
330C2×C4×SD16—731no—
331#2351—731no—
332D4×C4⋊C4—731no—
333#1943—731no—
334#1942—731no—
335#1865—731no—
336#1864D7311yescharacteristic
337#1863—731no—
338#1862—731no—
339#1847—731no—
340#1804—731no—
341#1803—731no—
342#1771—731no—
343C73#2267711yescharacteristic, complemented, Sylow
344C22×SD16—731no—
345C2×C4×Q8—731no—
346C2×C4×Q8—731no—
347C2×C4×Q8—731no—
348C2×C4×D4—731no—
349C2×C4×D4—731no—
350C2×C4×D4C2×D7311yescharacteristic
351C22×C4⋊C4—1461no—
352#440—738no—
353#440—1464no—
354C4×SD16—1464no—
355C2×#107—731no—
356C2×#107—731no—
357C2×#107—731no—
358C2×#106—731no—
359C2×#106C2×D7311yescharacteristic
360C2×C4⋊C8—731no—
361C2×#103—731no—
362C2×#103—731no—
363C2×#103—731no—
364C2×#102—731no—
365C2×#102C2×D7311yescharacteristic
366C2×#102C2×D7311yescharacteristic
367C2×#97—731no—
368C2×#97—731no—
369C2×C4×C8—731no—
370#389—731no—
371#389—731no—
372#389—731no—
373#389—1461no—
374#389—1461no—
375#389C2×D7311yescharacteristic
376#385—1461no—
377#385—1461no—
378#384—731no—
379#382—731no—
380#382—1461no—
381#380—1461no—
382C4×C4⋊C4—731no—
383C4×C4⋊C4—1461no—
384#340—731no—
385#340—731no—
386#340—731no—
387#340C2×D7311yescharacteristic
388#336—731no—
389#336C2×D7311yescharacteristic
390C22×Q8—731no—
391C22×D4C22×D7311yescharacteristic
392C23×C4—21no—
393C23×C4—1461no—
394C23×C4—1461no—
395C2×SD16—734no—
396C2×SD16—734no—
397C2×SD16—734no—
398C22×C8—731no—
399C22×C8—731no—
400C22×C8C22×D7311yescharacteristic
401C22×C8C22×D7311yescharacteristic
402C4×Q8—2921no—
403C4×Q8—2921no—
404C4×Q8—734no—
405C4×Q8—1462no—
406C4×Q8—1462no—
407C4×Q8—1462no—
408C4×D4—22no—
409C4×D4—41no—
410C4×D4—2921no—
411C4×D4—734no—
412C4×D4—1462no—
413C4×D4—1462no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—2924no—
416C2×C4⋊C4—731no—
417C2×C4⋊C4—731no—
418C2×C4⋊C4—731no—
419C2×C4⋊C4—731no—
420C2×C4⋊C4—731no—
421C2×C4⋊C4—1461no—
422C2×C4⋊C4—1461no—
423C2×C4⋊C4—1461no—
424C2×C4⋊C4—1461no—
425C2×C4⋊C4—1461no—
426C2×C4⋊C4—1461no—
427C2×C4⋊C4—1461no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4C22×D7311yescharacteristic
430C2×C4⋊C4C22×D7311yescharacteristic
431C2×C4⋊C4C22×D7311yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—1461no—
434C2×C22⋊C4—1461no—
435C2×C42—731no—
436C2×C42—731no—
437C2×C42—1461no—
438C2×C42—1461no—
439C2×C42—1461no—
440C2×C42C22×D7311yescharacteristic
441#107—2921no—
442#107—734no—
443#107—1462no—
444#106—41no—
445#106—1462no—
446C4⋊C8—1462no—
447#103—734no—
448#103—1462no—
449#103—1462no—
450#102—22no—
451#102—1462no—
452#102C4×D7314yes—
453#97—734no—
454#97—1462no—
455C4×C8—1462no—
456#95—21no—
457#95—731no—
458#95—731no—
459#95—1461no—
460#95—1461no—
461#95—1461no—
462#95—1461no—
463#95—1461no—
464C24—21no—
465C2×Q8—732no—
466C2×Q8—732no—
467C2×Q8—732no—
468C2×Q8—1461no—
469C2×Q8—1461no—
470C2×Q8—1461no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4C2×C4×D7312yes—
475C2×D4D4×D7312yes—
476C2×D4Q8×D7312yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—81no—
483C22×C4—2922no—
484C22×C4—2922no—
485C22×C4—2922no—
486C22×C4—2922no—
487C22×C4—5841no—
488C22×C4—731no—
489C22×C4—731no—
490C22×C4—731no—
491C22×C4—731no—
492C22×C4—1461no—
493C22×C4—1461no—
494C22×C4—1464no—
495C22×C4—1461no—
496C22×C4—1461no—
497C22×C4D4×D7311yescharacteristic
498C22×C4D4×D7311yescharacteristic
499C22×C4C23×D7311yescharacteristic
500SD16—7316no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—732no—
507C2×C8—732no—
508C2×C8—732no—
509C2×C8—1461no—
510C2×C8—1461no—
511C2×C8—1461no—
512C2×C8C2×C4×D7312yes—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—42no—
519C4⋊C4—2922no—
520C4⋊C4—2922no—
521C4⋊C4—2921no—
522C4⋊C4—2922no—
523C4⋊C4—2922no—
524C4⋊C4—2922no—
525C4⋊C4—2922no—
526C4⋊C4—5841no—
527C4⋊C4—732no—
528C4⋊C4—1462no—
529C4⋊C4—1464no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4—1461no—
534C4⋊C4—1461no—
535C4⋊C4C2×C4×D7312yes—
536C22⋊C4—42no—
537C22⋊C4—2922no—
538C22⋊C4—1464no—
539C42—21no—
540C42—21no—
541C42—2922no—
542C42—2922no—
543C42—732no—
544C42—732no—
545C42—1464no—
546C42—1461no—
547C42—1461no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23C2×D4×D7311yescharacteristic
556Q8—734no—
557Q8—1462no—
558Q8—1462no—
559Q8—1462no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4C4⋊C4×D7314yes—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—84no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2928no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—2921no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—2921no—
589C2×C4—5844no—
590C2×C4—732no—
591C2×C4—732no—
592C2×C4—732no—
593C2×C4—1461no—
594C2×C4—1461no—
595C2×C4—1461no—
596C2×C4—1462no—
597C2×C4—1461no—
598C2×C4—1461no—
599C2×C4—1461no—
600C2×C4—1461no—
601C2×C4—1461no—
602C2×C4—1462no—
603C2×C4—1462no—
604C2×C4—1462no—
605C2×C4—1461no—
606C2×C4C22×C4×D7311yescharacteristic
607C2×C4C2×D4×D7311yescharacteristic
608C2×C4C2×Q8×D7311yescharacteristic
609C2×C4#41×D7311yescharacteristic
610C2×C4#1455663611yescharacteristic
611C2×C4#1455663811yescharacteristic
612C8—22no—
613C8—41no—
614C8—2921no—
615C8—734no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22C4×D4×D7311yescharacteristic
624C22#1798921911yescharacteristic
625C22#1798925011yescharacteristic
626C22AGL(1,ℤ/8)×D7311yescharacteristic
627C22#1798938311yescharacteristic
628C22#1798939311yescharacteristic
629C22#1798940111yescharacteristic
630C4—41no—
631C4—42no—
632C4—41no—
633C4—81no—
634C4—81no—
635C4—2921no—
636C4—2922no—
637C4—5841no—
638C4—5841no—
639C4—1464no—
640C4—1462no—
641C4—1462no—
642C4—1462no—
643C4C2×C4⋊C4×D7311yescharacteristic
644C4#1798906611yescharacteristic
645C4#1798923511yescharacteristic
646C4#1798929111yescharacteristic
647C2—44no—
648C2#2261282911yescharacteristic
649C2#2261874211yescharacteristic
650C2#2261876711yescharacteristic
651C2#2262142411yescharacteristic
652C2#2262148411yescharacteristic
653C2#2262284711yescharacteristic
654C2#2262286911yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1864C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#1864D73
hypocenterC73#22677
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
4. #22612850C2semidirect
6. #22610666C2semidirect
7. #22610663C2semidirect
8. #22610196C2semidirect
9. #22610188C2semidirect
11. #22610096C2semidirect
13. #22609935C2semidirect
14. #22609804C2semidirect
24. #17988684C4semidirect
75. #14556556C2×C4semidirect
103. #14556524C2×C4semidirect
107. #14556516C2×C4semidirect
111. #14556501C2×C4semidirect
156. #11924016C4⋊C4semidirect
283. Dic146C2×C4⋊C4semidirect
287. #10596831C2×C4⋊C4semidirect
343. C73#22677semidirect

Maximal subgroups

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

subgroupindex
C2×#179886872
C2×#179886842
C2×#179886782
#226128502
D4×#119240072
#226106662
#226106632
#226101962
#226101882
#226101802
#226100962
#226099392
#226099352
#226098042
C73×#18642
#2267773