Finite Group Databasebeta

← #10897671

Subgroups — #10897671

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82514C211yesmaximal, characteristic, complemented
2#78980C211yesmaximal, characteristic, complemented
3C3×#1906C211yesmaximal, characteristic, complemented
4#72121C211yesmaximal, characteristic
5#72116C211yesmaximal, characteristic, complemented
6#71645C211yesmaximal, characteristic, complemented
7#70416C211yesmaximal, characteristic, complemented
8#70405C211yesmaximal, characteristic, complemented
9#70372C211yesmaximal, characteristic, complemented
10#70044C211yesmaximal, characteristic, complemented
11#70043C211yesmaximal, characteristic, complemented
12#69034C211yesmaximal, characteristic, complemented
13#69016C211yesmaximal, characteristic, complemented
14#68749C211yesmaximal, characteristic, complemented
15#68746C211yesmaximal, characteristic, complemented
16#29387—31nomaximal, Sylow
17C3×#575C2211yescharacteristic
18C3×C4×#41C2211yescharacteristic
19#5822C2211yescharacteristic
20#5822C2211yescharacteristic
21#5819C2211yescharacteristic
22#5745C2211yescharacteristic
23C3×#421C2211yescharacteristic
24C2×C3×#104C2211yescharacteristic
25C3×#410C2211yescharacteristic
26#5236C2211yescharacteristic
27#5234C2211yescharacteristic
28#5234C2211yescharacteristic
29#5234C2211yescharacteristic
30#5233C2211yescharacteristic
31#5129C2211yescharacteristic
32#5124C2211yescharacteristic
33#5119C2211yescharacteristic
34#5006C2211yescharacteristic
35#5002C2211yescharacteristic
36#5000—22no—
37#4999C2211yescharacteristic
38#4999C2211yescharacteristic
39#4999C2211yescharacteristic
40C2×#805C2211yescharacteristic
41#4926—22no—
42C4×#208C2211yescharacteristic
43C2×#803C2211yescharacteristic
44#4824—22no—
45#4823—22no—
46C3×#344C2211yescharacteristic
47C3×#339C2211yescharacteristic
48C3⋊#338C2211yescharacteristic
49#4559C2211yescharacteristic
50#4558C2211yescharacteristic
51#4558C2211yescharacteristic
52#4533C2211yescharacteristic
53#4533C2211yescharacteristic
54#4531C2211yescharacteristic
55#4523C2211yescharacteristic
56#2956—31no—
57#2923—31no—
58#2103—31no—
59#2098—31no—
60#2053—31no—
61#2017—31no—
62#1938—31no—
63#1938—31no—
64#1906—31no—
65#1906D311yescharacteristic, complemented
66#1881—31no—
67#1881—31no—
68#1868—31no—
69#1868—31no—
70#1841—31no—
71C2×C3×#41C2311yescharacteristic
72C2×C3×#41C2311yescharacteristic
73C2×#235C2311yescharacteristic
74C3×#137—21no—
75C3×#137—21no—
76C3×AGL(1,ℤ/8)—21no—
77C3×AGL(1,ℤ/8)—21no—
78C3×#135—41no—
79C2×C3×M16C2311yescharacteristic
80C2×C3×M16C2311yescharacteristic
81C3×C4×Q8—21no—
82C3×C4×D4—21no—
83C3×C4×D4—22no—
84C3×#117—21no—
85C3×#117C2311yescharacteristic
86C3×#117C2311yescharacteristic
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
89#951—21no—
90#951—21no—
91#950—22no—
92#950—41no—
93#949—21no—
94#949—21no—
95#948—22no—
96#948—41no—
97#942—21no—
98#942—21no—
99#932—21no—
100#932—21no—
101#926C2311yescharacteristic
102C2×#208—41no—
103C2×#208C2311yescharacteristic
104C2×#208C2311yescharacteristic
105C2×#208C2311yescharacteristic
106C2×#208C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#907—22no—
110D3×M16—41no—
111#901—41no—
112C3×#104—21no—
113C3×#104—21no—
114C3×#104—41no—
115C3×#99—22no—
116#837—41no—
117#835—21no—
118#832—21no—
119#830—22no—
120#830—22no—
121#830—41no—
122#822—22no—
123(C2×D3)⋊C8—41no—
124Dic3⋊C8—41no—
125#810—21no—
126#809—21no—
127#806—22no—
128#805—21no—
129#805—21no—
130C4⋊(C3⋊C8)—22no—
131C4⋊(C3⋊C8)—22no—
132C4⋊(C3⋊C8)—41no—
133#803—21no—
134#803—21no—
135#803D412yes—
136C4×C3⋊C8D412yes—
137#575—31no—
138#575—31no—
139#575C2×D311yescharacteristic
140#568—31no—
141#568—31no—
142C4×#41C2×D311yescharacteristic
143#444—62no—
144#443—62no—
145#433—62no—
146#432—62no—
147#430—31no—
148#424—31no—
149#424—31no—
150#424—31no—
151#424—31no—
152#421—31no—
153#421C2×D311yescharacteristic
154C2×#104—31no—
155C2×#104C2×D311yescharacteristic
156#416—31no—
157#416—31no—
158#410—31no—
159#410C2×D311yescharacteristic
160#408—31no—
161#408—31no—
162#408—31no—
163#408—31no—
164C4×M16—31no—
165C2×#97—31no—
166#344—31no—
167#344—31no—
168#344—31no—
169#344C2×D311yescharacteristic
170#341—31no—
171#341—31no—
172#339—31no—
173#339C2×D311yescharacteristic
174#338—31no—
175H(3,ℤ/4)—31no—
176C3×#41—21no—
177C3×#41—21no—
178C3×#41—21no—
179C3×#41—21no—
180C3×#41—21no—
181C3×#41—21no—
182C3×#41—21no—
183C3×#41—21no—
184C2×C3×Q8#4111yescharacteristic
185C2×C3×Q8#4111yescharacteristic
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4#4111yescharacteristic
189C2×C3×D4#4111yescharacteristic
190C22×C3×C4—22no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4C2411yescharacteristic
195C2×#67—21no—
196#235—21no—
197#235—21no—
198#235—21no—
199#235—21no—
200C2×D12#4111yescharacteristic
201C2×C4×D3—21no—
202C2×Dic6#4111yescharacteristic
203C3×Q16—41no—
204C3×SD16—41no—
205C3×SD16—41no—
206C3×D8—41no—
207C3×M16—21no—
208C3×M16—21no—
209C3×M16—21no—
210C3×M16—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C22⋊C4—22no—
217C3×C22⋊C4—21no—
218C3×C22⋊C4#4112yes—
219C3×C42—22no—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42C2×D411yescharacteristic
224C3×C42C2×D411yescharacteristic
225#217—21no—
226#216—22no—
227#216—41no—
228#215—22no—
229#215—41no—
230#214—22no—
231#214—41no—
232#213—22no—
233#213—41no—
234C4⋊Dic3—21no—
235C4×Dic3—21no—
236#208—21no—
237#208—21no—
238#208—21no—
239#208—21no—
240#208—21no—
241#208—21no—
242#208—21no—
243#208—21no—
244#208—42no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—42no—
254C2×C3⋊C8C2×D412yes—
255C2×C3⋊C8C2×D412yes—
256ΓL(1,𝔽25)—42no—
257D3×C8—42no—
258C2×#41—31no—
259C2×#41C22×D311yescharacteristic
260C2×#41C22×D311yescharacteristic
261#137—21no—
262#137—21no—
263#137—61no—
264#137—61no—
265#137—61no—
266#137—61no—
267AGL(1,ℤ/8)—21no—
268AGL(1,ℤ/8)—21no—
269AGL(1,ℤ/8)—61no—
270AGL(1,ℤ/8)—61no—
271AGL(1,ℤ/8)—61no—
272AGL(1,ℤ/8)—61no—
273#135—41no—
274#135—62no—
275#135—121no—
276#131—62no—
277#131—62no—
278#131—121no—
279#131—121no—
280C2×M16—31no—
281C2×M16—31no—
282C2×M16—31no—
283C2×M16—31no—
284C2×M16—121no—
285C2×M16—121no—
286C2×M16C22×D311yescharacteristic
287C2×M16C22×D311yescharacteristic
288C22×C8—31no—
289C22×C8—31no—
290C4×Q8—21no—
291C4×D4—21no—
292C4×D4—22no—
293#117—21no—
294#117—31no—
295#117C22×D311yescharacteristic
296#117C22×D311yescharacteristic
297C2×C42—21no—
298C2×C42C22×D311yescharacteristic
299#108—62no—
300C4⋊C8—62no—
301C4⋊C8—62no—
302C4⋊C8—121no—
303C4⋊C8—121no—
304#104—21no—
305#104—21no—
306#104—41no—
307#104—61no—
308#104—61no—
309#104—121no—
310#103—61no—
311#103—61no—
312#102—61no—
313#102—61no—
314#99—22no—
315#99—62no—
316C22⋊C8—62no—
317C22⋊C8—62no—
318C22⋊C8—121no—
319C22⋊C8—121no—
320#97—32no—
321#97—61no—
322#97—61no—
323C4×C8—32no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3C2×#4111yescharacteristic
327C22×D3—21no—
328C3×Q8—21no—
329C3×Q8—21no—
330C3×Q8—21no—
331C3×Q8—21no—
332C3×D4—21no—
333C3×D4—21no—
334C3×D4—21no—
335C3×D4—21no—
336C3×D4—41no—
337C3×D4—41no—
338C3×D4—41no—
339C3×D4—41no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—81no—
351C2×C3×C4—82no—
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4H(5,𝔽2)11yescharacteristic
357C2×C3×C4H(5,𝔽2)11yescharacteristic
358#67—41no—
359#67—41no—
360C2×Dic3—21no—
361C2×Dic3—41no—
362D12—21no—
363D12—21no—
364C4×D3—41no—
365C4×D3—41no—
366Dic6—21no—
367Dic6—21no—
368C3×C8—41no—
369C3×C8—41no—
370C3⋊C8—22no—
371C3⋊C8—22no—
372C3⋊C8—41no—
373C3⋊C8—41no—
374C3⋊C8—41no—
375C3⋊C8—41no—
376#41—21no—
377#41—21no—
378#41—21no—
379#41—21no—
380#41—21no—
381#41—21no—
382#41—21no—
383#41—21no—
384#41—61no—
385#41—61no—
386#41—61no—
387#41—61no—
388C2×Q8—31no—
389C2×Q8#23511yescharacteristic
390C2×Q8#23711yescharacteristic
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—31no—
394C2×D4—61no—
395C2×D4#23511yescharacteristic
396C2×D4#23711yescharacteristic
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—61no—
402C22×C4C23×D311yescharacteristic
403Q16—41no—
404Q16—62no—
405Q16—121no—
406SD16—41no—
407SD16—41no—
408SD16—62no—
409SD16—62no—
410SD16—121no—
411SD16—121no—
412D8—41no—
413D8—62no—
414D8—121no—
415M16—21no—
416M16—21no—
417M16—21no—
418M16—21no—
419M16—61no—
420M16—61no—
421M16—61no—
422M16—61no—
423M16—61no—
424M16—61no—
425M16—61no—
426M16—61no—
427M16—122no—
428M16—122no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—32no—
432C2×C8—32no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—122no—
442C2×C8—122no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—61no—
447C22⋊C4—22no—
448C22⋊C4—21no—
449C22⋊C4—61no—
450C22⋊C4#23712yes—
451C42—22no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—61no—
456C42D3×D411yescharacteristic
457C42D3×D411yescharacteristic
458C22×C3—41no—
459C22×C3—42no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—81no—
463C22×C3#52511yescharacteristic
464C22×C3#55311yescharacteristic
465C22×C3#55311yescharacteristic
466C2×D3—41no—
467C2×D3—81no—
468C3×C4—42no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—41no—
472C3×C4—41no—
473C3×C4—81no—
474C3×C4#52511yescharacteristic
475C3×C4#53511yescharacteristic
476C3×C4#55111yescharacteristic
477C3×C4#55111yescharacteristic
478Dic3—41no—
479Dic3—81no—
480C23—21no—
481C23—21no—
482C23—61no—
483C23C2×#23711yescharacteristic
484Q8—21no—
485Q8—21no—
486Q8—21no—
487Q8—21no—
488Q8—61no—
489Q8—61no—
490D4—21no—
491D4—21no—
492D4—21no—
493D4—21no—
494D4—41no—
495D4—41no—
496D4—41no—
497D4—41no—
498D4—61no—
499D4—61no—
500D4—121no—
501D4—121no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—61no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4C2×#23511yescharacteristic
519C2×C4C2×D3×D411yescharacteristic
520C2×C4C2×#23711yescharacteristic
521C2×C4#100411yescharacteristic
522C2×C4D3×#4111yescharacteristic
523C2×C4#100911yescharacteristic
524C8—41no—
525C8—41no—
526C8—62no—
527C8—62no—
528C8—121no—
529C8—121no—
530C8—121no—
531C8—121no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—81no—
536C2×C3—81no—
537C2×C3#259811yescharacteristic
538D3—81no—
539C22—41no—
540C22—42no—
541C22—41no—
542C22—41no—
543C22—81no—
544C22—121no—
545C22—241no—
546C22#556511yescharacteristic
547C22#558611yescharacteristic
548C22#561411yescharacteristic
549C4—41no—
550C4—42no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—81no—
555C4—121no—
556C4—241no—
557C4#549511yescharacteristic
558C4#554711yescharacteristic
559C4#560711yescharacteristic
560C4#565811yescharacteristic
561C3#2938711yescharacteristic, complemented, Sylow
562C2—21no—
563C2—21no—
564C2—21no—
565C2—81no—
566C2—81no—
567C2—241no—
568C2#7477511yescharacteristic
subgroupG/H
centerC4#5607
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1906C2
socleC2×C3#2598
prime radical#72121C2
hypercenter#1906D3
hypocenterC3#29387
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1906

Complements

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

NH ≅ G/Ntype
1. #82514C2semidirect
2. #78980C2semidirect
3. C3×#1906C2semidirect
5. #72116C2semidirect
6. #71645C2semidirect
7. #70416C2semidirect
8. #70405C2semidirect
9. #70372C2semidirect
10. #70044C2semidirect
11. #70043C2semidirect
12. #69034C2semidirect
13. #69016C2semidirect
14. #68749C2semidirect
15. #68746C2semidirect
65. #1906D3semidirect
561. C3#29387semidirect

Maximal subgroups

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

subgroupindex
#825142
#789802
C3×#19062
#721212
#721162
#716452
#704162
#704052
#703722
#700442
#700432
#690342
#690162
#687492
#687462
#293873