Finite Group Databasebeta

← #10799573

Subgroups — #10799573

Order 768.

770 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79070C211yesmaximal, characteristic, complemented
2#78818C211yesmaximal, characteristic
3#78648C211yesmaximal, characteristic, complemented
4C3×#1937C211yesmaximal, characteristic, complemented
5#70537C211yesmaximal, characteristic, complemented
6#70373C211yesmaximal, characteristic
7Dic3×#104C211yesmaximal, characteristic, complemented
8#70287C211yesmaximal, characteristic, complemented
9#69999C211yesmaximal, characteristic
10#69793C211yesmaximal, characteristic, complemented
11C4×C8×Dic3C211yesmaximal, characteristic, complemented
12#68918C211yesmaximal, characteristic, complemented
13#68916C211yesmaximal, characteristic, complemented
14#68747C211yesmaximal, characteristic, complemented
15#68728C211yesmaximal, characteristic, complemented
16#21292—31nomaximal, Sylow
17Dic3×#41C2211yescharacteristic
18C2×#950C2211yescharacteristic
19#5522C2211yescharacteristic
20C3×#424C2211yescharacteristic
21C2×C3×#104C2211yescharacteristic
22C3×#416C2211yescharacteristic
23C3×#408C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C2×#837C2211yescharacteristic
26#5234C2211yescharacteristic
27C2×#825C2211yescharacteristic
28#5129C2211yescharacteristic
29C2×#822C2211yescharacteristic
30C2×#822C2211yescharacteristic
31#5119C2211yescharacteristic
32Dic3×M16C2211yescharacteristic
33Dic3×M16C2211yescharacteristic
34#5113C2211yescharacteristic
35#5110C2211yescharacteristic
36#5102C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39C2×C8×Dic3C2211yescharacteristic
40C2×C8×Dic3C2211yescharacteristic
41#4999C2211yescharacteristic
42#4998C2211yescharacteristic
43C42×Dic3C2211yescharacteristic
44C2×#805C2211yescharacteristic
45C2×C4×C3⋊C8C2211yescharacteristic
46#4840—22no—
47#4833—22no—
48#4790C412yescomplemented
49#4788C412yescomplemented
50#4758C412yescomplemented
51#4756C412yescomplemented
52#4720—22no—
53#4718—22no—
54C3×#341C2211yescharacteristic
55C3×#339C2211yescharacteristic
56#4555C2211yescharacteristic
57#4552C2211yescharacteristic
58#4532C2211yescharacteristic
59#4523C2211yescharacteristic
60C4×#135—31no—
61#2922—31no—
62#2922—31no—
63#1974—31no—
64#1974—31no—
65#1945—31no—
66#1945—31no—
67#1937—31no—
68#1937—31no—
69#1937—31no—
70#1937D311yescharacteristic, complemented
71C4×#108—31no—
72C4×#104—31no—
73C4×#104—31no—
74C42×C8—31no—
75C2×C3×#41C2311yescharacteristic
76C2×#235C2311yescharacteristic
77C2×C3×M16C2311yescharacteristic
78C2×C3×M16C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×#117C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83#950—22no—
84#950—22no—
85Q8×Dic3C2×C412yes—
86C2×#216C2×C411yescharacteristic
87C2×#215C2×C411yescharacteristic
88D4×Dic3—22no—
89D4×Dic3C2×C412yes—
90C2×#214C2×C411yescharacteristic
91C2×#213C2×C411yescharacteristic
92#926—21no—
93#926C2311yescharacteristic
94C2×C4×Dic3—22no—
95C2×C4×Dic3—21no—
96C2×C4×Dic3C2311yescharacteristic
97C2×C4×Dic3C2311yescharacteristic
98C2×#208C2311yescharacteristic
99C2×#208C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2311yescharacteristic
102#893C2×C412yes—
103#887C2×C412yes—
104#67⋊C4—22no—
105#872—21no—
106C42×D3—21no—
107C3×C4⋊C8—22no—
108C3×#104—22no—
109C3×#103C2×C412yes—
110C3×#102C2×C412yes—
111C3×C22⋊C8—22no—
112C3×C4×C8—22no—
113#837—22no—
114#830—22no—
115#825—22no—
116#822—22no—
117#822—22no—
118#821C2×C412yes—
119(C2×D3)⋊C8—22no—
120(C2×D3)⋊C8—22no—
121#816C2×C412yes—
122#815—21no—
123#815—21no—
124Dic3⋊C8—22no—
125Dic3⋊C8—22no—
126C8×Dic3—21no—
127C8×Dic3—21no—
128C8×Dic3—22no—
129C8×Dic3—22no—
130C8×Dic3C2×C412yes—
131C8×Dic3C2×C412yes—
132C8×Dic3D412yes—
133C8×Dic3D412yes—
134#808C2×C412yes—
135#807C2×C412yes—
136#805—22no—
137C4⋊(C3⋊C8)—22no—
138C4×C3⋊C8—22no—
139C2×#135—31no—
140C4×#41—31no—
141C4×#41—31no—
142C4×Q16—32no—
143C4×SD16—32no—
144C4×SD16—32no—
145C4×D8—32no—
146#433—62no—
147#433—62no—
148#431—62no—
149#431—62no—
150C2×#108—31no—
151C2×#108—31no—
152#427—31no—
153#424—31no—
154#424—31no—
155#424—31no—
156#424C2×D311yescharacteristic
157C2×#104—31no—
158C2×#104—31no—
159C2×#104—31no—
160C2×#104C2×D311yescharacteristic
161#416—31no—
162#416C2×D311yescharacteristic
163#408—31no—
164#408—31no—
165#408—31no—
166#408C2×D311yescharacteristic
167C4×M16—31no—
168C4×M16—31no—
169C2×C4×C8—31no—
170C2×C4×C8—31no—
171C2×C4×C8—31no—
172C2×C4×C8—31no—
173C2×C4×C8—31no—
174C2×C4×C8C2×D311yescharacteristic
175C43—31no—
176#341—31no—
177#341C2×D311yescharacteristic
178#339—31no—
179#339—31no—
180#339—31no—
181#339C2×D311yescharacteristic
182C3×#41—21no—
183C3×#41—21no—
184C3×#41—21no—
185C3×#41—21no—
186C2×C3×Q8C22×C411yescharacteristic
187C2×C3×D4—21no—
188C2×C3×D4C22×C411yescharacteristic
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4C2411yescharacteristic
192C2×#67—21no—
193C22×Dic3—21no—
194C22×Dic3—22no—
195C22×Dic3#4112yes—
196#235—21no—
197#235—21no—
198#235—21no—
199#235—21no—
200C2×D12C22×C411yescharacteristic
201C2×C4×D3—21no—
202C2×C4×D3—22no—
203C2×Dic6C22×C411yescharacteristic
204C3×M16—21no—
205C3×M16—21no—
206C3×M16—21no—
207C3×M16—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8C22×C412yes—
215C2×C3×C8C2×D412yes—
216C3×C4⋊C4C22×C412yes—
217C3×C22⋊C4—21no—
218C3×C42—21no—
219C3×C42—21no—
220C3×C42#4112yes—
221#217—21no—
222#217—22no—
223#216—22no—
224#215—22no—
225#214—22no—
226#213—22no—
227(C2×D3)⋊C4—22no—
228C4⋊Dic3—22no—
229C4⋊Dic3C22×C412yes—
230Dic3⋊C4—22no—
231C4×Dic3—28no—
232C4×Dic3—21no—
233C4×Dic3—21no—
234C4×Dic3—22no—
235C4×Dic3—22no—
236C4×Dic3C22×C412yes—
237C4×Dic3C2×D411yescharacteristic
238C4×Dic3C2×D411yescharacteristic
239C4×Dic3#4112yes—
240#208—21no—
241#208—21no—
242#208—21no—
243#208—21no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8C22×C411yescharacteristic
249C2×C3⋊C8C22×C411yescharacteristic
250C2×C3⋊C8C2×D412yes—
251C2×C3⋊C8#4112yes—
252C2×C3⋊C8#4112yes—
253C2×#41—31no—
254C2×#41C22×D311yescharacteristic
255#135—62no—
256#135—62no—
257C2×Q16—31no—
258C2×SD16—31no—
259C2×SD16—31no—
260C2×D8—31no—
261C2×M16—31no—
262C2×M16—31no—
263C2×M16C22×D311yescharacteristic
264C2×M16C22×D311yescharacteristic
265C22×C8—31no—
266C22×C8—31no—
267C22×C8C22×D311yescharacteristic
268C22×C8C22×D311yescharacteristic
269C4×Q8—32no—
270C4×Q8—32no—
271C4×D4—32no—
272C4×D4—32no—
273C4×D4—62no—
274C4×D4—62no—
275#117—31no—
276#117—61no—
277#117—61no—
278#117C22×D311yescharacteristic
279C2×C42—31no—
280C2×C42—31no—
281C2×C42—61no—
282C2×C42—61no—
283C2×C42—62no—
284C2×C42C22×D311yescharacteristic
285#108—62no—
286#108—62no—
287#107—32no—
288#106—32no—
289C4⋊C8—22no—
290C4⋊C8—62no—
291C4⋊C8—62no—
292C4⋊C8—62no—
293#104—22no—
294#104—62no—
295#104—62no—
296#104—62no—
297#103—32no—
298#103C4×D312yes—
299#102—32no—
300#102C4×D312yes—
301C22⋊C8—22no—
302C22⋊C8—62no—
303C22⋊C8—62no—
304C22⋊C8—62no—
305#97—61no—
306#97—61no—
307C4×C8—22no—
308C4×C8—32no—
309C4×C8—32no—
310C4×C8—32no—
311C4×C8—32no—
312C4×C8—62no—
313C4×C8—62no—
314C4×C8—61no—
315C4×C8—61no—
316C4×C8—62no—
317C23×C3—21no—
318C23×C3C2×#4111yescharacteristic
319C22×D3—21no—
320C3×Q8—21no—
321C3×Q8—21no—
322C3×D4—21no—
323C3×D4—21no—
324C3×D4—41no—
325C3×D4—41no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4C23×C411yescharacteristic
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C2×#4111yescharacteristic
335C2×C3×C4C2×#4111yescharacteristic
336C2×C3×C4C2×#4111yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338#67—41no—
339#67—41no—
340C2×Dic3—24no—
341C2×Dic3—22no—
342C2×Dic3—24no—
343C2×Dic3—21no—
344C2×Dic3—22no—
345C2×Dic3—44no—
346C2×Dic3C4×D412yes—
347C2×Dic3C4×D412yes—
348D12—21no—
349D12—21no—
350C4×D3—41no—
351C4×D3—41no—
352C4×D3—44no—
353Dic6—21no—
354Dic6—21no—
355C3×C8—22no—
356C3×C8—22no—
357C3×C8—41no—
358C3×C8—41no—
359C3⋊C8—22no—
360C3⋊C8—41no—
361C3⋊C8—41no—
362C3⋊C8#11712yes—
363C3⋊C8#11712yes—
364#41—21no—
365#41—21no—
366#41—21no—
367#41—21no—
368#41—61no—
369#41—61no—
370#41—61no—
371#41—61no—
372C2×Q8—31no—
373C2×Q8C2×C4×D311yescharacteristic
374C2×D4—21no—
375C2×D4—31no—
376C2×D4—61no—
377C2×D4C2×C4×D311yescharacteristic
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—32no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4C23×D311yescharacteristic
386Q16—62no—
387SD16—62no—
388SD16—62no—
389D8—62no—
390M16—21no—
391M16—21no—
392M16—21no—
393M16—21no—
394M16—61no—
395M16—61no—
396M16—61no—
397M16—61no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—31no—
405C2×C8—31no—
406C2×C8—32no—
407C2×C8—32no—
408C2×C8—32no—
409C2×C8—61no—
410C2×C8—61no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8C2×C4×D312yes—
414C2×C8D3×D412yes—
415C4⋊C4—32no—
416C4⋊C4—62no—
417C4⋊C4—62no—
418C4⋊C4C2×C4×D312yes—
419C22⋊C4—21no—
420C22⋊C4—62no—
421C22⋊C4—62no—
422C22⋊C4—61no—
423C42—21no—
424C42—21no—
425C42—32no—
426C42—31no—
427C42—31no—
428C42—32no—
429C42—62no—
430C42—62no—
431C42—68no—
432C42—61no—
433C42—61no—
434C42#23712yes—
435C22×C3—22no—
436C22×C3—42no—
437C22×C3—41no—
438C22×C3C2×C4×D411yescharacteristic
439C22×C3#52911yescharacteristic
440C22×C3#52911yescharacteristic
441C2×D3—42no—
442C2×D3—41no—
443C3×C4—24no—
444C3×C4—42no—
445C3×C4—41no—
446C3×C4C2×#11711yescharacteristic
447C3×C4C4×#4111yescharacteristic
448C3×C4#53211yescharacteristic
449C3×C4#53211yescharacteristic
450Dic3—24no—
451Dic3—22no—
452Dic3—42no—
453Dic3—41no—
454Dic3#44314yes—
455C23—21no—
456C23—61no—
457C23D3×#4111yescharacteristic
458Q8—21no—
459Q8—21no—
460Q8—61no—
461Q8—61no—
462D4—21no—
463D4—21no—
464D4—41no—
465D4—41no—
466D4—61no—
467D4—61no—
468D4—121no—
469D4—121no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—32no—
475C2×C4—32no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—64no—
479C2×C4—64no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—61no—
483C2×C4—124no—
484C2×C4—124no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4C22×C4×D311yescharacteristic
488C2×C4C2×D3×D411yescharacteristic
489C2×C4C2×#23711yescharacteristic
490C2×C4D3×#4111yescharacteristic
491C2×C4D3×#4111yescharacteristic
492C2×C4D3×#4111yescharacteristic
493C8—22no—
494C8—22no—
495C8—32no—
496C8—32no—
497C8—41no—
498C8—41no—
499C8—62no—
500C8—121no—
501C8—121no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—81no—
505C2×C3#229311yescharacteristic
506C2×C3C2×#44312yes—
507D3—81no—
508C22—22no—
509C22—42no—
510C22—41no—
511C22—122no—
512C22—121no—
513C22C4×D3×D411yescharacteristic
514C22#564911yescharacteristic
515C22#570611yescharacteristic
516C4—24no—
517C4—34no—
518C4—42no—
519C4—41no—
520C4—64no—
521C4—62no—
522C4—122no—
523C4—121no—
524C4D3×#11711yescharacteristic
525C4C4×#23711yescharacteristic
526C4#559611yescharacteristic
527C4#566811yescharacteristic
528C3#2129211yescharacteristic, complemented, Sylow
529C2—21no—
530C2—21no—
531C2—81no—
532C2—241no—
533C2#7545011yescharacteristic
534C2D3×#44312yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1937C2
socleC22×C3C2×C4×D4
prime radicalC2×#950C22
cosocleC22×C3×C4C24
hypercenter#1937D3
hypocenterC3#21292
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1937

Complements

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

NH ≅ G/Ntype
1. #79070C2semidirect
3. #78648C2semidirect
4. C3×#1937C2semidirect
5. #70537C2semidirect
7. Dic3×#104C2semidirect
8. #70287C2semidirect
10. #69793C2semidirect
11. C4×C8×Dic3C2semidirect
12. #68918C2semidirect
13. #68916C2semidirect
14. #68747C2semidirect
15. #68728C2semidirect
48. #4790C4semidirect
49. #4788C4semidirect
50. #4758C4semidirect
51. #4756C4semidirect
70. #1937D3semidirect
528. C3#21292semidirect

Maximal subgroups

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

subgroupindex
#790702
#788182
#786482
C3×#19372
#705372
#703732
Dic3×#1042
#702872
#699992
#697932
C4×C8×Dic32
#689182
#689162
#687472
#687282
#212923