Finite Group Databasebeta

← #10814893

Subgroups — #10814893

Order 768.

710 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2925C211yesmaximal, characteristic, complemented
2C4×#950C211yesmaximal, characteristic
3#79000C211yesmaximal, characteristic, complemented
4#71646C211yesmaximal, characteristic, complemented
5#70529C211yesmaximal, characteristic, complemented
6#70518C211yesmaximal, characteristic, complemented
7#70466C211yesmaximal, characteristic, complemented
8#70000C212yesmaximal
9#69996C212yesmaximal, complemented
10#69900C212yesmaximal, complemented
11#69890C212yesmaximal, complemented
12#22803—31nomaximal, Sylow
13C3×C4×#41C2211yescharacteristic
14C2×#950C2211yescharacteristic
15C4×#235C2211yescharacteristic
16C3×#433—22no—
17C3×#432—21no—
18C3×#431—21no—
19C3×#423C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C3×#408C2212yes—
22C3×#405C2212yes—
23C2×#837C2212yes—
24#5233C2211yescharacteristic
25C2×#825C2212yes—
26C2×#822C2212yes—
27C2×#822C2212yes—
28#5110C2212yes—
29#5101C2212yes—
30C2×#815C2212yes—
31C4×#216C411yescharacteristic, complemented
32C4×#216C411yescharacteristic, complemented
33C4×#215C411yescharacteristic, complemented
34C4×#215C411yescharacteristic, complemented
35C4×#214C411yescharacteristic, complemented
36C4×#214C411yescharacteristic, complemented
37C4×#213C411yescharacteristic, complemented
38C4×#213C411yescharacteristic, complemented
39#5002C2211yescharacteristic
40#4998C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42#4924C2211yescharacteristic
43C2×C4×C3⋊C8C2211yescharacteristic
44#4840—22no—
45#4835—21no—
46#4833—21no—
47#4555C2212yes—
48#4550C2212yes—
49#4532C2212yes—
50C4×#135—31no—
51#2925—31no—
52#2925D311yescharacteristic, complemented
53#1975—32no—
54#1929—32no—
55#1929—32no—
56#1848—31no—
57#1806—31no—
58#1806—31no—
59#1773—32no—
60#1770—31no—
61C2×C3×#41C2311yescharacteristic
62C2×#235C2311yescharacteristic
63C2×C3×M16C2312yes—
64C22×C3×C8C2312yes—
65C3×C4×Q8C2×C411yescharacteristic
66C3×C4×Q8C2×C411yescharacteristic
67C3×C4×D4—21no—
68C3×C4×D4—21no—
69C3×C4×D4C2×C411yescharacteristic
70C3×C4×D4C2×C411yescharacteristic
71C3×#117—21no—
72C3×#117C2311yescharacteristic
73C2×C3×C42—21no—
74C2×C3×C42C2311yescharacteristic
75#950—22no—
76#950—22no—
77C2×#216C2×C411yescharacteristic
78C2×#215C2×C411yescharacteristic
79C2×#214C2×C411yescharacteristic
80C2×#213C2×C411yescharacteristic
81C4×#67—21no—
82C4×#67—21no—
83#926C2311yescharacteristic
84C2×C4⋊Dic3—21no—
85C2×C4×Dic3C2312yes—
86C2×#208C2312yes—
87C22×C3⋊C8C2311yescharacteristic
88C22×C3⋊C8C2311yescharacteristic
89C4×D12C2×C411yescharacteristic
90C4×D12C2×C411yescharacteristic
91#872—21no—
92C42×D3—21no—
93C4×Dic6C2×C411yescharacteristic
94C4×Dic6C2×C411yescharacteristic
95C3×C4⋊C8—21no—
96C3×C4⋊C8—21no—
97C3×C4⋊C8D411yescharacteristic
98C3×C4⋊C8D411yescharacteristic
99C3×C4⋊C8D412yes—
100C3×C22⋊C8—22no—
101C3×C22⋊C8—22no—
102C3×#97—22no—
103C3×C4×C8—22no—
104#837—42no—
105#835C2×C411yescharacteristic
106#835C2×C411yescharacteristic
107#832C2×C411yescharacteristic
108#832C2×C411yescharacteristic
109C3⋊#95—21no—
110#825—42no—
111#822—24no—
112#822—42no—
113(C2×D3)⋊C8—22no—
114(C2×D3)⋊C8—22no—
115#815—22no—
116#815—24no—
117C8×Dic3—22no—
118#810C2×C411yescharacteristic
119#810C2×C411yescharacteristic
120#809C2×C411yescharacteristic
121#809C2×C411yescharacteristic
122#808C2×C411yescharacteristic
123#808C2×C411yescharacteristic
124#807C2×C411yescharacteristic
125#807C2×C411yescharacteristic
126C4⋊(C3⋊C8)—21no—
127C4⋊(C3⋊C8)—21no—
128C4×C3⋊C8C2×C411yescharacteristic
129C4×C3⋊C8C2×C411yescharacteristic
130C4×C3⋊C8C2×C411yescharacteristic
131C4×C3⋊C8C2×C411yescharacteristic
132C2×#135—31no—
133C4×#41—31no—
134C4×#41C2×D311yescharacteristic
135C4×Q16—31no—
136C4×Q16—31no—
137C4×SD16—31no—
138C4×SD16—31no—
139C4×SD16—31no—
140C4×SD16—31no—
141C4×D8—31no—
142C4×D8—31no—
143#433—22no—
144#433—62no—
145#432—21no—
146#432—61no—
147#431—21no—
148#431—61no—
149C2×#108—32no—
150C2×#108—32no—
151#427—31no—
152#423—31no—
153#423C2×D311yescharacteristic
154C2×C4⋊C8C2×D311yescharacteristic
155C2×#104—32no—
156C2×#104—32no—
157#416—31no—
158#416—31no—
159#408—32no—
160#408C2×D312yes—
161#405—32no—
162#405C2×D312yes—
163C2×#97—32no—
164C2×C4×C8—31no—
165C4×C4⋊C4—31no—
166#339—32no—
167#339—32no—
168#336—32no—
169C3×#41—21no—
170C3×#41—21no—
171C3×#41—21no—
172C3×#41—21no—
173C2×C3×Q8C22×C411yescharacteristic
174C2×C3×D4—21no—
175C2×C3×D4C22×C411yescharacteristic
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4#4111yescharacteristic
180C22×C3×C4#4111yescharacteristic
181C22×C3×C4C2411yescharacteristic
182C2×#67—21no—
183C22×Dic3—22no—
184#235—21no—
185#235—21no—
186#235—21no—
187#235—21no—
188C2×D12C22×C411yescharacteristic
189C2×C4×D3—21no—
190C2×C4×D3—21no—
191C2×C4×D3—21no—
192C2×Dic6C22×C411yescharacteristic
193C3×M16—22no—
194C3×M16—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8C2×D412yes—
199C2×C3×C8C2×D412yes—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4C22×C411yescharacteristic
203C3×C4⋊C4C22×C411yescharacteristic
204C3×C22⋊C4—21no—
205C3×C22⋊C4—21no—
206C3×C22⋊C4—21no—
207C3×C42—21no—
208C3×C42—21no—
209C3×C42—21no—
210C3×C42C22×C411yescharacteristic
211C3×C42C22×C411yescharacteristic
212C3×C42C2×D411yescharacteristic
213C3×C42C2×D411yescharacteristic
214#217—21no—
215#216—22no—
216#215—22no—
217#214—22no—
218#213—22no—
219(C2×D3)⋊C4—21no—
220(C2×D3)⋊C4—21no—
221C4⋊Dic3—42no—
222C4⋊Dic3C22×C411yescharacteristic
223C4⋊Dic3C22×C411yescharacteristic
224Dic3⋊C4—21no—
225Dic3⋊C4—21no—
226C4×Dic3—21no—
227C4×Dic3—21no—
228C4×Dic3—21no—
229C4×Dic3—22no—
230C4×Dic3—22no—
231#208—22no—
232#208—22no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—22no—
236C2×C3⋊C8C22×C411yescharacteristic
237C2×C3⋊C8C22×C411yescharacteristic
238C2×C3⋊C8C22×C411yescharacteristic
239C2×C3⋊C8C22×C411yescharacteristic
240C2×C3⋊C8#4112yes—
241C2×C3⋊C8#4112yes—
242C2×#41—31no—
243C2×#41C22×D311yescharacteristic
244#135—62no—
245#135—62no—
246C2×Q16—31no—
247C2×SD16—31no—
248C2×SD16—31no—
249C2×D8—31no—
250C2×M16—32no—
251C2×M16C22×D312yes—
252C22×C8—31no—
253C22×C8—31no—
254C22×C8C22×D312yes—
255C4×Q8—31no—
256C4×Q8—31no—
257C4×Q8C4×D311yescharacteristic
258C4×Q8C4×D311yescharacteristic
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—31no—
262C4×D4—31no—
263C4×D4—61no—
264C4×D4—61no—
265C4×D4C4×D311yescharacteristic
266C4×D4C4×D311yescharacteristic
267#117—21no—
268#117—31no—
269#117—61no—
270#117C22×D311yescharacteristic
271C2×C4⋊C4—61no—
272C2×C42—21no—
273C2×C42—32no—
274C2×C42—61no—
275C2×C42C22×D311yescharacteristic
276#108—64no—
277#108—122no—
278#107—31no—
279#107—31no—
280#106—31no—
281#106—31no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—61no—
285C4⋊C8—61no—
286C4⋊C8#6711yescharacteristic
287C4⋊C8#6711yescharacteristic
288C4⋊C8#6712yes—
289#104—122no—
290#104—122no—
291#103—31no—
292#103—31no—
293#103—31no—
294#103—31no—
295#102—31no—
296#102—31no—
297#102—31no—
298#102—31no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—62no—
302C22⋊C8—62no—
303#97—22no—
304#97—64no—
305#97—62no—
306C4×C8—22no—
307C4×C8—31no—
308C4×C8—31no—
309C4×C8—31no—
310C4×C8—31no—
311C4×C8—62no—
312#95—61no—
313C23×C3—21no—
314C23×C3C2×#4111yescharacteristic
315C22×D3—21no—
316C3×Q8—21no—
317C3×Q8—21no—
318C3×D4—21no—
319C3×D4—21no—
320C3×D4—41no—
321C3×D4—41no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—42no—
330C2×C3×C4—81no—
331C2×C3×C4C4×D411yescharacteristic
332C2×C3×C4C4×D411yescharacteristic
333C2×C3×C4C4×D411yescharacteristic
334C2×C3×C4C4×D411yescharacteristic
335C2×C3×C4C23×C411yescharacteristic
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4H(5,𝔽2)11yescharacteristic
340C2×C3×C42−1+411yescharacteristic
341#67—41no—
342#67—41no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349D12—21no—
350D12—21no—
351C4×D3—41no—
352C4×D3—41no—
353C4×D3—42no—
354C4×D3—81no—
355Dic6—21no—
356Dic6—21no—
357C3×C8—22no—
358C3×C8—22no—
359C3×C8—42no—
360C3⋊C8—21no—
361C3⋊C8—21no—
362C3⋊C8—42no—
363C3⋊C8#11712yes—
364C3⋊C8#11712yes—
365#41—21no—
366#41—21no—
367#41—21no—
368#41—21no—
369#41—61no—
370#41—61no—
371#41—61no—
372#41—61no—
373C2×Q8—31no—
374C2×Q8C2×C4×D311yescharacteristic
375C2×D4—21no—
376C2×D4—31no—
377C2×D4—61no—
378C2×D4C2×C4×D311yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—61no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—62no—
386C22×C4#23511yescharacteristic
387C22×C4#23511yescharacteristic
388C22×C4C23×D311yescharacteristic
389Q16—62no—
390SD16—62no—
391SD16—62no—
392D8—62no—
393M16—22no—
394M16—22no—
395M16—62no—
396M16—62no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—31no—
401C2×C8—31no—
402C2×C8—32no—
403C2×C8—32no—
404C2×C8—31no—
405C2×C8—31no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—62no—
409C2×C8C2×#6712yes—
410C2×C8C2×#6712yes—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—31no—
414C4⋊C4—31no—
415C4⋊C4—61no—
416C4⋊C4—61no—
417C4⋊C4—122no—
418C4⋊C4C2×C4×D311yescharacteristic
419C4⋊C4C2×C4×D311yescharacteristic
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—21no—
423C22⋊C4—61no—
424C22⋊C4—61no—
425C22⋊C4—61no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—61no—
430C42—61no—
431C42—61no—
432C42—62no—
433C42—62no—
434C42C2×C4×D311yescharacteristic
435C42C2×C4×D311yescharacteristic
436C42C2×#6711yescharacteristic
437C42C2×#6711yescharacteristic
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—41no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3C2×C4×D411yescharacteristic
444C22×C3#55511yescharacteristic
445C22×C3#55611yescharacteristic
446C2×D3—41no—
447C2×D3—41no—
448C2×D3—41no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4#44412yes—
456C3×C4C2×#11711yescharacteristic
457C3×C4#52011yescharacteristic
458C3×C4#54711yescharacteristic
459C3×C4#55011yescharacteristic
460Dic3—41no—
461Dic3—41no—
462Dic3—41no—
463Dic3—82no—
464C23—21no—
465C23—61no—
466C23C2×#23511yescharacteristic
467Q8—21no—
468Q8—21no—
469Q8—61no—
470Q8—61no—
471D4—21no—
472D4—21no—
473D4—41no—
474D4—41no—
475D4—61no—
476D4—61no—
477D4—121no—
478D4—121no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—61no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—81no—
491C2×C4—122no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—241no—
498C2×C4C4×#6711yescharacteristic
499C2×C4C4×#6711yescharacteristic
500C2×C4C4×#6711yescharacteristic
501C2×C4C4×#6711yescharacteristic
502C2×C4C22×C4×D311yescharacteristic
503C2×C4D3×#4111yescharacteristic
504C2×C4D3×#4111yescharacteristic
505C2×C4#100911yescharacteristic
506C2×C4#101011yescharacteristic
507C2×C4C22×#6711yescharacteristic
508C8—22no—
509C8—22no—
510C8—32no—
511C8—32no—
512C8—42no—
513C8—61no—
514C8—61no—
515C8—122no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—81no—
519C2×C3#235411yescharacteristic
520C2×C3C2×#44411yescharacteristic
521C2×C3#295711yescharacteristic
522D3—81no—
523C22—21no—
524C22—21no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—121no—
529C22—121no—
530C22—121no—
531C22#553211yescharacteristic
532C22#553311yescharacteristic
533C22C2×C4×#6711yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—121no—
541C4—121no—
542C4—121no—
543C4—242no—
544C4#514512yes—
545C4D3×#11711yescharacteristic
546C4#552311yescharacteristic
547C4#582411yescharacteristic
548C4#583411yescharacteristic
549C3#2280311yescharacteristic, complemented, Sylow
550C2—21no—
551C2—21no—
552C2—81no—
553C2—241no—
554C2#8224111yescharacteristic
555C2C2×#514511yescharacteristic
556C2#8381211yescharacteristic
subgroupG/H
centerC2×C4C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2925C2
socleC22×C3C2×C4×D4
prime radicalC2×#950C22
cosocleC22×C3×C4C24
hypercenter#2925D3
hypocenterC3#22803
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ #2925

Complements

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

NH ≅ G/Ntype
1. C3×#2925C2semidirect
3. #79000C2semidirect
4. #71646C2semidirect
5. #70529C2semidirect
6. #70518C2semidirect
7. #70466C2semidirect
9. #69996C2semidirect
10. #69900C2semidirect
11. #69890C2semidirect
31. C4×#216C4semidirect
32. C4×#216C4semidirect
33. C4×#215C4semidirect
34. C4×#215C4semidirect
35. C4×#214C4semidirect
36. C4×#214C4semidirect
37. C4×#213C4semidirect
38. C4×#213C4semidirect
52. #2925D3semidirect
549. C3#22803semidirect

Maximal subgroups

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

subgroupindex
C3×#29252
C4×#9502
#790002
#716462
#705292
#705182
#704662
#700002
#699962
#699002
#698902
#228033