Finite Group Databasebeta

← #10692259

Subgroups — #10692259

Order 768.

470 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3094C211yesmaximal, characteristic
2#82210C211yesmaximal, characteristic, complemented
3#71676C211yesmaximal, characteristic
4#71675C211yesmaximal, characteristic
5#71575C211yesmaximal, characteristic, complemented
6#67688C211yesmaximal, characteristic, complemented
7#67684C211yesmaximal, characteristic, complemented
8#9348—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C3×#482—21no—
12C3×#482—21no—
13C3×#481—21no—
14C3×#480—21no—
15C3×#428C411yescharacteristic, complemented
16C3×#428C411yescharacteristic, complemented
17C3×#418C411yescharacteristic, complemented
18C3×#418C411yescharacteristic, complemented
19C3×#407C2211yescharacteristic
20#5212C2211yescharacteristic
21#5207C2211yescharacteristic
22C2×#830C2211yescharacteristic
23C2×#830C2211yescharacteristic
24#5000—22no—
25#4999—21no—
26#4997—21no—
27#4926—21no—
28#4925—21no—
29#4537—21no—
30#4537—21no—
31#4529—21no—
32#4529—21no—
33#4525—22no—
34#3094D311yescharacteristic
35#2924—31no—
36#1825—31no—
37#1519—31no—
38#1471—31no—
39#1305—31no—
40#1298—31no—
41C22×C3×D4—21no—
42C22×C3×D4C2×C411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C2×C3×M16C2×C411yescharacteristic
45C2×C3×M16C2×C411yescharacteristic
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#117—21no—
51C3×#117—21no—
52C3×#117C2×C411yescharacteristic
53C3×#117C2×C411yescharacteristic
54C3×#117D411yescharacteristic
55C3×#117D411yescharacteristic
56C2×C3×C4⋊C4C2×C411yescharacteristic
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C42—21no—
62C2×#217—21no—
63C2×C4⋊Dic3—21no—
64C2×#208—21no—
65C22×C3⋊C8—21no—
66C22×C3⋊C8—21no—
67C3×#107—21no—
68C3×#107—21no—
69C3×#106—21no—
70C3×#106—21no—
71C3×#102—21no—
72C3×#102—21no—
73C3×#102—21no—
74C3×#102—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3⋊#95—21no—
78#830—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830D411yescharacteristic
83#830D411yescharacteristic
84#830D412yes—
85C4⋊(C3⋊C8)—22no—
86C4⋊(C3⋊C8)—22no—
87#803—22no—
88C4×C3⋊C8—22no—
89C2×#121C2×D311yescharacteristic
90C2×#117C2×D311yescharacteristic
91#482—21no—
92#482—21no—
93#481—21no—
94#480—21no—
95#433—62no—
96#432—61no—
97#431—61no—
98#428Dic311yescharacteristic, complemented
99#428Dic311yescharacteristic, complemented
100#424—61no—
101#418Dic311yescharacteristic, complemented
102#418Dic311yescharacteristic, complemented
103#407—31no—
104#407C2×D311yescharacteristic
105C2×C22⋊C8—31no—
106C2×C22⋊C8—31no—
107#405—61no—
108#380—31no—
109#340—61no—
110#340—61no—
111#327—61no—
112#327—61no—
113#323—62no—
114C24×C3—21no—
115C24×C3#4111yescharacteristic
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—81no—
123C2×C3×D4—81no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4C22×C411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C22×Dic3—41no—
136C3×M16—41no—
137C3×M16—41no—
138C2×C3×C8—21no—
139C2×C3×C8—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4C22⋊C411yescharacteristic
145C3×C4⋊C4C22⋊C411yescharacteristic
146C3×C4⋊C4C22⋊C411yescharacteristic
147C3×C4⋊C4C22⋊C411yescharacteristic
148C3×C22⋊C4—21no—
149C3×C22⋊C4—21no—
150C3×C22⋊C4—21no—
151C3×C22⋊C4—21no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C42—21no—
155C3×C42—21no—
156C3×C42—21no—
157C3×C42—21no—
158#217—81no—
159C4⋊Dic3—41no—
160C4⋊Dic3—41no—
161#208—41no—
162#208—41no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—41no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171C22×D4—21no—
172C22×D4C2×Dic311yescharacteristic
173C23×C4C22×D311yescharacteristic
174C2×M16—61no—
175C2×M16C2×Dic311yescharacteristic
176C2×M16C2×Dic311yescharacteristic
177C22×C8—61no—
178C22×C8—61no—
179#121—21no—
180#121—21no—
181#121—21no—
182#121—21no—
183#117—21no—
184#117—21no—
185#117C2×Dic311yescharacteristic
186#117C2×Dic311yescharacteristic
187#117#6711yescharacteristic
188#117#6711yescharacteristic
189C2×C4⋊C4—61no—
190C2×C4⋊C4C2×Dic311yescharacteristic
191C2×C4⋊C4#6711yescharacteristic
192C2×C4⋊C4#6711yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—61no—
196C2×C42—21no—
197#107—21no—
198#107—21no—
199#106—21no—
200#106—21no—
201C4⋊C8—62no—
202C4⋊C8—62no—
203#102—21no—
204#102—21no—
205#102—21no—
206#102—21no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—32no—
210C22⋊C8—31no—
211C22⋊C8—31no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216#97—62no—
217C4×C8—62no—
218#95—61no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—42no—
224C23×C3—41no—
225C23×C3—81no—
226C23×C3—81no—
227C23×C3C4×D411yescharacteristic
228C23×C3#12111yescharacteristic
229C23×C3#12311yescharacteristic
230C3×D4—41no—
231C3×D4—41no—
232C3×D4—81no—
233C3×D4—82no—
234C3×D4—81no—
235C3×D4—81no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—81no—
253C2×C3×C4—81no—
254C2×C3×C4C2×C22⋊C411yescharacteristic
255C2×C3×C4C4×D411yescharacteristic
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×Dic3—81no—
259C2×Dic3—81no—
260C2×Dic3—81no—
261C3×C8—41no—
262C3×C8—41no—
263C3⋊C8—41no—
264C3⋊C8—41no—
265C3⋊C8—42no—
266C3⋊C8—81no—
267C24—21no—
268C24#23711yescharacteristic
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—81no—
276C2×D4—81no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—121no—
283C22×C4D3×D411yescharacteristic
284C22×C4D3×D411yescharacteristic
285C22×C4#23711yescharacteristic
286C22×C4C22×Dic311yescharacteristic
287C22×C4C2×#6711yescharacteristic
288C22×C4C2×#6711yescharacteristic
289M16—41no—
290M16—41no—
291M16—121no—
292M16—121no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—61no—
296C2×C8—62no—
297C2×C8—61no—
298C2×C8—121no—
299C2×C8—121no—
300C2×C8—121no—
301C2×C8—121no—
302C2×C8—121no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—121no—
308C4⋊C4—121no—
309C4⋊C4#21711yescharacteristic
310C4⋊C4#21711yescharacteristic
311C4⋊C4#21711yescharacteristic
312C4⋊C4#21711yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—241no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3—162no—
338C22×C3#38611yescharacteristic
339C22×C3#47111yescharacteristic
340C22×C3#47211yescharacteristic
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—41no—
345C3×C4—41no—
346C3×C4—41no—
347C3×C4—41no—
348C3×C4—81no—
349C3×C4#41011yescharacteristic
350C3×C4#41011yescharacteristic
351Dic3—161no—
352C23—21no—
353C23—21no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—41no—
358C23—81no—
359C23—81no—
360C23D4×Dic311yescharacteristic
361C23#93511yescharacteristic
362C23#93911yescharacteristic
363D4—41no—
364D4—41no—
365D4—81no—
366D4—82no—
367D4—81no—
368D4—81no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—241no—
388C2×C4—241no—
389C2×C4—241no—
390C2×C4D4×Dic311yescharacteristic
391C2×C4#93711yescharacteristic
392C2×C4#93911yescharacteristic
393C2×C4C2×#21711yescharacteristic
394C8—41no—
395C8—41no—
396C8—121no—
397C8—121no—
398C8—122no—
399C8—241no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—41no—
403C2×C3—41no—
404C2×C3—161no—
405C2×C3#188111yescharacteristic
406C2×C3#189911yescharacteristic
407C2×C3#190711yescharacteristic
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—162no—
422C22#519911yescharacteristic
423C22#520011yescharacteristic
424C22#522011yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—81no—
433C4—481no—
434C4#524011yescharacteristic
435C4#524011yescharacteristic
436C3#934811yescharacteristic, complemented, Sylow
437C2—21no—
438C2—21no—
439C2—41no—
440C2—41no—
441C2—161no—
442C2#7170511yescharacteristic
443C2#7209111yescharacteristic
444C2#7211611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#3094C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#3094D3
hypocenterC3#9348
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3094

Complements

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

NH ≅ G/Ntype
2. #82210C2semidirect
5. #71575C2semidirect
6. #67688C2semidirect
7. #67684C2semidirect
15. C3×#428C4semidirect
16. C3×#428C4semidirect
17. C3×#418C4semidirect
18. C3×#418C4semidirect
98. #428Dic3semidirect
99. #428Dic3semidirect
101. #418Dic3semidirect
102. #418Dic3semidirect
436. C3#9348semidirect

Maximal subgroups

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

subgroupindex
C3×#30942
#822102
#716762
#716752
#715752
#676882
#676842
#93483