Finite Group Databasebeta

← #10798405

Subgroups — #10798405

Order 768.

800 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2922C211yesmaximal, characteristic, complemented
2C4×#950C211yesmaximal, characteristic
3C4×#901C211yesmaximal, characteristic, complemented
4C4×#837C211yesmaximal, characteristic, complemented
5#70545C212yesmaximal, complemented
6#70534C212yesmaximal, complemented
7#70011C212yesmaximal
8#69996C212yesmaximal, complemented
9C4×#822C211yesmaximal, characteristic, complemented
10C4×#825C211yesmaximal, characteristic, complemented
11#69750C211yesmaximal, characteristic, complemented
12#21137—31nomaximal, Sylow
13C3×C4×#41C2211yescharacteristic
14C2×#950C2211yescharacteristic
15C2×#901C2212yes—
16C4×#235C2211yescharacteristic
17C3×#433—22no—
18C3×#431—21no—
19C3×#431—21no—
20C3×#424C2212yes—
21C3×#408C2212yes—
22C3×C4×M16C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24C2×#837C2212yes—
25#5233C2211yescharacteristic
26C2×#825C2212yes—
27C2×#822C2212yes—
28#5101C2212yes—
29C4×#216C411yescharacteristic, complemented
30C4×#216C411yescharacteristic, complemented
31C4×#215C411yescharacteristic, complemented
32C4×#215C411yescharacteristic, complemented
33C4×#214C411yescharacteristic, complemented
34C4×#214C411yescharacteristic, complemented
35C4×#213C411yescharacteristic, complemented
36C4×#213C411yescharacteristic, complemented
37#5002C2211yescharacteristic
38#4998C2211yescharacteristic
39C42×Dic3C2211yescharacteristic
40C4×#208C2211yescharacteristic
41C2×C4×C3⋊C8C2211yescharacteristic
42#4689—22no—
43C4×ΓL(1,𝔽25)—21no—
44C4×D3×C8—21no—
45#4558C2212yes—
46#4555C2212yes—
47#4550C2212yes—
48#4550C2212yes—
49#4532C2212yes—
50C4×#135—31no—
51#2922D311yescharacteristic, complemented
52C4×#131—31no—
53#1929—32no—
54#1922—32no—
55#1876—32no—
56#1866—32no—
57C4×#108—31no—
58C4×#104—31no—
59C4×#104—31no—
60#1744—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—24no—
76C2×#216C2×C411yescharacteristic
77C2×#215C2×C411yescharacteristic
78C2×#214C2×C411yescharacteristic
79C2×#213C2×C411yescharacteristic
80C4×#67—21no—
81C4×#67—21no—
82#926C2311yescharacteristic
83C2×C4×Dic3—21no—
84C2×C4×Dic3—21no—
85C2×C4×Dic3C2312yes—
86C2×#208C2312yes—
87C22×C3⋊C8C2311yescharacteristic
88C22×C3⋊C8C2311yescharacteristic
89#901—24no—
90#901—24no—
91C2×ΓL(1,𝔽25)—22no—
92C2×D3×C8—22no—
93C4×D12C2×C411yescharacteristic
94C4×D12C2×C411yescharacteristic
95#872—21no—
96C42×D3—21no—
97C4×Dic6C2×C411yescharacteristic
98C4×Dic6C2×C411yescharacteristic
99C3×C4⋊C8—22no—
100C3×C4⋊C8—22no—
101C3×C22⋊C8—22no—
102C3×C22⋊C8—22no—
103C3×#97—21no—
104C3×C4×C8—21no—
105C3×C4×C8D411yescharacteristic
106C3×C4×C8D411yescharacteristic
107C3×C4×C8D412yes—
108#837—24no—
109#835C2×C411yescharacteristic
110#835C2×C411yescharacteristic
111#832C2×C411yescharacteristic
112#832C2×C411yescharacteristic
113#825—24no—
114#822—24no—
115#815—22no—
116C8×Dic3—22no—
117#810C2×C411yescharacteristic
118#810C2×C411yescharacteristic
119#809C2×C411yescharacteristic
120#809C2×C411yescharacteristic
121#808C2×C411yescharacteristic
122#808C2×C411yescharacteristic
123#807C2×C411yescharacteristic
124#807C2×C411yescharacteristic
125#803—21no—
126C4×C3⋊C8—21no—
127C4×C3⋊C8C2×C411yescharacteristic
128C4×C3⋊C8C2×C411yescharacteristic
129C4×C3⋊C8C2×C411yescharacteristic
130C4×C3⋊C8C2×C411yescharacteristic
131C2×#135—31no—
132C2×#131—32no—
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#431—21no—
145#431—21no—
146C2×#108—32no—
147#427—31no—
148#424C2×D312yes—
149C2×#104—32no—
150C2×#104—32no—
151#416—31no—
152#416—31no—
153#408C2×D312yes—
154#405—32no—
155#405—62no—
156C4×M16—31no—
157C4×M16—61no—
158C4×M16C2×D311yescharacteristic
159C2×C4×C8—31no—
160C2×C4×C8—61no—
161C2×C4×C8C2×D311yescharacteristic
162C43—31no—
163#341—32no—
164#339—32no—
165#339—32no—
166#336—32no—
167#336—32no—
168C3×#41—22no—
169C3×#41—22no—
170C2×C3×Q8C22×C411yescharacteristic
171C2×C3×D4—21no—
172C2×C3×D4C22×C411yescharacteristic
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4#4111yescharacteristic
177C22×C3×C4#4111yescharacteristic
178C22×C3×C4C2411yescharacteristic
179C2×#67—21no—
180C22×Dic3—22no—
181#235—22no—
182#235#4114yes—
183C2×D12C22×C411yescharacteristic
184C2×C4×D3—21no—
185C2×C4×D3—21no—
186C2×C4×D3—21no—
187C2×Dic6C22×C411yescharacteristic
188C3×M16—22no—
189C3×M16—22no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8C2×D412yes—
194C2×C3×C8C2×D412yes—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4C22×C411yescharacteristic
198C3×C4⋊C4C22×C411yescharacteristic
199C3×C22⋊C4—21no—
200C3×C22⋊C4—21no—
201C3×C22⋊C4—21no—
202C3×C42—21no—
203C3×C42—21no—
204C3×C42—21no—
205C3×C42C22×C411yescharacteristic
206C3×C42C22×C411yescharacteristic
207C3×C42C2×D411yescharacteristic
208C3×C42C2×D411yescharacteristic
209#217—21no—
210#216—22no—
211#215—22no—
212#214—22no—
213#213—22no—
214(C2×D3)⋊C4—21no—
215(C2×D3)⋊C4—21no—
216C4⋊Dic3C22×C411yescharacteristic
217C4⋊Dic3C22×C411yescharacteristic
218Dic3⋊C4—21no—
219Dic3⋊C4—21no—
220C4×Dic3—24no—
221C4×Dic3—21no—
222C4×Dic3—21no—
223C4×Dic3—21no—
224C4×Dic3—22no—
225C4×Dic3—22no—
226C4×Dic3—42no—
227#208—22no—
228#208#4114yes—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8C22×C411yescharacteristic
233C2×C3⋊C8C22×C411yescharacteristic
234C2×C3⋊C8C22×C411yescharacteristic
235C2×C3⋊C8C22×C411yescharacteristic
236ΓL(1,𝔽25)—28no—
237D3×C8—28no—
238C2×#41—31no—
239C2×#41C22×D311yescharacteristic
240#135—64no—
241C2×Q16—31no—
242C2×SD16—31no—
243C2×SD16—31no—
244C2×D8—31no—
245#131—64no—
246#131—64no—
247C2×M16—32no—
248C2×M16—62no—
249C2×M16C22×D312yes—
250C22×C8—31no—
251C22×C8—31no—
252C22×C8—62no—
253C22×C8C22×D312yes—
254C4×Q8—31no—
255C4×Q8—31no—
256C4×Q8C4×D311yescharacteristic
257C4×Q8C4×D311yescharacteristic
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—31no—
261C4×D4—31no—
262C4×D4—61no—
263C4×D4—61no—
264C4×D4C4×D311yescharacteristic
265C4×D4C4×D311yescharacteristic
266#117—21no—
267#117—31no—
268#117—61no—
269#117C22×D311yescharacteristic
270C2×C42—21no—
271C2×C42—32no—
272C2×C42—61no—
273C2×C42—61no—
274C2×C42—61no—
275C2×C42C22×D311yescharacteristic
276#108—64no—
277#107—31no—
278#107—31no—
279#106—31no—
280#106—31no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283#104—64no—
284#104—64no—
285#103—31no—
286#103—31no—
287#103—31no—
288#103—31no—
289#102—31no—
290#102—31no—
291#102—31no—
292#102—31no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295#97—21no—
296#97—62no—
297#97—61no—
298C4×C8—21no—
299C4×C8—31no—
300C4×C8—31no—
301C4×C8—31no—
302C4×C8—31no—
303C4×C8—62no—
304C4×C8—61no—
305C4×C8#6712yes—
306C4×C8#6711yescharacteristic
307C4×C8#6711yescharacteristic
308C23×C3—21no—
309C23×C3C2×#4111yescharacteristic
310C22×D3—21no—
311C3×Q8—21no—
312C3×Q8—21no—
313C3×D4—21no—
314C3×D4—21no—
315C3×D4—42no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4C4×D411yescharacteristic
325C2×C3×C4C4×D411yescharacteristic
326C2×C3×C4C4×D411yescharacteristic
327C2×C3×C4C4×D411yescharacteristic
328C2×C3×C4C23×C411yescharacteristic
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C2×#4112yes—
331C2×C3×C4C2×#4112yes—
332#67—24no—
333C2×Dic3—21no—
334C2×Dic3—24no—
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339D12—21no—
340D12#11712yes—
341C4×D3—24no—
342C4×D3—42no—
343C4×D3—42no—
344Dic6—21no—
345Dic6#11712yes—
346C3×C8—22no—
347C3×C8—22no—
348C3×C8—42no—
349C3⋊C8—21no—
350C3⋊C8—21no—
351C3⋊C8—21no—
352C3⋊C8—21no—
353C3⋊C8—24no—
354#41—22no—
355#41—22no—
356#41—34no—
357#41—62no—
358C2×Q8—31no—
359C2×Q8C2×C4×D311yescharacteristic
360C2×D4—21no—
361C2×D4—31no—
362C2×D4—61no—
363C2×D4C2×C4×D311yescharacteristic
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—61no—
368C22×C4—62no—
369C22×C4—61no—
370C22×C4—61no—
371C22×C4#23511yescharacteristic
372C22×C4#23511yescharacteristic
373C22×C4C23×D311yescharacteristic
374Q16—62no—
375SD16—62no—
376SD16—62no—
377D8—62no—
378M16—22no—
379M16—22no—
380M16—34no—
381M16—68no—
382M16—62no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—31no—
387C2×C8—31no—
388C2×C8—31no—
389C2×C8—31no—
390C2×C8—68no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8C2×#6712yes—
395C2×C8C2×#6712yes—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—31no—
399C4⋊C4—31no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C4⋊C4C2×C4×D311yescharacteristic
403C4⋊C4C2×C4×D311yescharacteristic
404C22⋊C4—21no—
405C22⋊C4—21no—
406C22⋊C4—21no—
407C22⋊C4—61no—
408C22⋊C4—61no—
409C22⋊C4—61no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—64no—
414C42—61no—
415C42—61no—
416C42—61no—
417C42—62no—
418C42—62no—
419C42—122no—
420C42C2×C4×D311yescharacteristic
421C42C2×C4×D311yescharacteristic
422C42C2×#6711yescharacteristic
423C42C2×#6711yescharacteristic
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3C2×C4×D411yescharacteristic
430C22×C3#52912yes—
431C2×D3—22no—
432C2×D3—41no—
433C2×D3—41no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—41no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4#44312yes—
440C3×C4#44412yes—
441C3×C4C2×#11711yescharacteristic
442C3×C4C4×#4111yescharacteristic
443C3×C4#52512yes—
444Dic3—22no—
445Dic3—44no—
446Dic3—41no—
447Dic3—41no—
448C23—21no—
449C23—61no—
450C23C2×#23511yescharacteristic
451Q8—21no—
452Q8—21no—
453Q8—32no—
454Q8—61no—
455D4—21no—
456D4—21no—
457D4—32no—
458D4—42no—
459D4—64no—
460D4—61no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—61no—
470C2×C4—64no—
471C2×C4—64no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—122no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4C4×#6711yescharacteristic
479C2×C4C4×#6711yescharacteristic
480C2×C4C4×#6711yescharacteristic
481C2×C4C4×#6711yescharacteristic
482C2×C4C22×C4×D311yescharacteristic
483C2×C4D3×#4112yes—
484C2×C4D3×#4112yes—
485C2×C4C22×#6711yescharacteristic
486C8—22no—
487C8—22no—
488C8—42no—
489C8—61no—
490C8—61no—
491C8—61no—
492C8—61no—
493C8—64no—
494C2×C3—22no—
495C2×C3—81no—
496C2×C3#229011yescharacteristic
497C2×C3C2×#44311yescharacteristic
498C2×C3C2×#44411yescharacteristic
499D3—42no—
500C22—21no—
501C22—21no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—62no—
506C22—121no—
507C22—121no—
508C22#552812yes—
509C22C2×C4×#6711yescharacteristic
510C4—21no—
511C4—21no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—62no—
516C4—124no—
517C4—121no—
518C4—121no—
519C4#514412yes—
520C4#514512yes—
521C4D3×#11711yescharacteristic
522C4#552211yescharacteristic
523C4#582812yes—
524C3#2113711yescharacteristic, complemented, Sylow
525C2—22no—
526C2—81no—
527C2—122no—
528C2#8226811yescharacteristic
529C2C2×#514411yescharacteristic
530C2C2×#514511yescharacteristic
subgroupG/H
centerC42C2×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2922C2
socleC22×C3C2×C4×D4
prime radicalC2×#950C22
cosocleC22×C3×C4C24
hypercenter#2922D3
hypocenterC3#21137
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C42 ◁ C2×C4×C8 ◁ #2922

Complements

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

NH ≅ G/Ntype
1. C3×#2922C2semidirect
3. C4×#901C2semidirect
4. C4×#837C2semidirect
5. #70545C2semidirect
6. #70534C2semidirect
8. #69996C2semidirect
9. C4×#822C2semidirect
10. C4×#825C2semidirect
11. #69750C2semidirect
29. C4×#216C4semidirect
30. C4×#216C4semidirect
31. C4×#215C4semidirect
32. C4×#215C4semidirect
33. C4×#214C4semidirect
34. C4×#214C4semidirect
35. C4×#213C4semidirect
36. C4×#213C4semidirect
51. #2922D3semidirect
524. C3#21137semidirect

Maximal subgroups

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

subgroupindex
C3×#29222
C4×#9502
C4×#9012
C4×#8372
#705452
#705342
#700112
#699962
C4×#8222
C4×#8252
#697502
#211373