Finite Group Databasebeta

← #10687356

Subgroups — #10687356

Order 768.

682 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3507C211yesmaximal, characteristic
2#71831C211yesmaximal, characteristic, complemented
3#71822C211yesmaximal, characteristic, complemented
4#71807C211yesmaximal, characteristic
5#71798C211yesmaximal, characteristic
6#71635C211yesmaximal, characteristic, complemented
7#71625C211yesmaximal, characteristic, complemented
8#8461—31nomaximal, Sylow
9C3×#559—21no—
10C3×#557—21no—
11C3×#556—21no—
12C3×#556—22no—
13C3×#554—21no—
14C3×#552—22no—
15C3×#539—22no—
16C3×#536C412yes—
17C3×#533C2211yescharacteristic
18C3×#532C2211yescharacteristic
19C3×#529—22no—
20C3×#520C412yes—
21C2×C3×C4×Q8C2211yescharacteristic
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×#803C2211yescharacteristic
26#4547—21no—
27#4547—21no—
28#4546—21no—
29#4544—21no—
30#4543—22no—
31#4540—22no—
32#4528—22no—
33#4524—22no—
34#3507D311yescharacteristic
35#1985—31no—
36#1552—31no—
37#1551—31no—
38#1531—31no—
39#1530—31no—
40#1467—31no—
41C2×C3×#41C2×C412yes—
42C22×C3×Q8C2×C411yescharacteristic
43C3×C4⋊Q8—22no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#127D411yescharacteristic
48C3×#126—24no—
49C3×#126—24no—
50C3×#125—22no—
51C3×#125—24no—
52C3×#125D412yes—
53C3×#124—22no—
54C3×#124—22no—
55C3×#124D412yes—
56C3×#123—24no—
57C3×#123—24no—
58C3×C22⋊Q8—24no—
59C3×C22⋊Q8—21no—
60C3×C22⋊Q8—21no—
61C3×C22⋊Q8—22no—
62C3×C22⋊Q8—22no—
63C3×#121—21no—
64C3×#121—21no—
65C3×C4×Q8—22no—
66C3×C4×Q8—22no—
67C3×C4×Q8—21no—
68C3×C4×Q8—21no—
69C3×C4×Q8—21no—
70C3×C4×Q8—21no—
71C3×C4×Q8—21no—
72C3×C4×Q8—21no—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C3×C4×D4—24no—
76C3×C4×D4—21no—
77C3×C4×D4—21no—
78C3×#117—24no—
79C3×#117—22no—
80C3×#117C2×C412yes—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4C2×C411yescharacteristic
83C2×C3×C4⋊C4D412yes—
84C2×C3×C42D412yes—
85C2×C3×C42C2311yescharacteristic
86C22×C3⋊C8—21no—
87C22×C3⋊C8—21no—
88#830—21no—
89#830—21no—
90C4⋊(C3⋊C8)—21no—
91C4⋊(C3⋊C8)—21no—
92C4⋊(C3⋊C8)—21no—
93C4⋊(C3⋊C8)—21no—
94#803—21no—
95#803—21no—
96#803—21no—
97#803—21no—
98#559—21no—
99#557—21no—
100#556—21no—
101#556—22no—
102#554—21no—
103#552—22no—
104#539—22no—
105#536Dic312yes—
106#533C2×D311yescharacteristic
107#532C2×D311yescharacteristic
108#529—22no—
109#520Dic312yes—
110C2×C4×Q8C2×D311yescharacteristic
111#432—31no—
112#432—31no—
113C2×C4⋊C8—31no—
114C2×#97—31no—
115#336—62no—
116#333—61no—
117#332—61no—
118#332—61no—
119#331—61no—
120#330—62no—
121#329—62no—
122#324—62no—
123C3×#41—44no—
124C2×C3×Q8—21no—
125C2×C3×Q8—21no—
126C2×C3×Q8—21no—
127C2×C3×Q8—21no—
128C2×C3×Q8—42no—
129C2×C3×Q8C22⋊C412yes—
130C2×C3×D4—22no—
131C2×C3×D4—21no—
132C2×C3×D4—21no—
133C2×C3×D4C22⋊C412yes—
134C22×C3×C4—24no—
135C22×C3×C4—22no—
136C22×C3×C4C22⋊C412yes—
137C22×C3×C4C22⋊C412yes—
138C22×C3×C4C22×C411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C3×C4⋊C4—24no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4—24no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4C22⋊C412yes—
155C3×C4⋊C4C22⋊C412yes—
156C3×C22⋊C4—24no—
157C3×C22⋊C4—24no—
158C3×C22⋊C4—24no—
159C3×C22⋊C4—22no—
160C3×C22⋊C4—22no—
161C3×C22⋊C4—22no—
162C3×C42—24no—
163C3×C42—22no—
164C3×C42—22no—
165C3×C42—22no—
166C3×C42C2×D411yescharacteristic
167C3×C42C2×D411yescharacteristic
168C3×C42C2×D411yescharacteristic
169C3×C42C2×D411yescharacteristic
170C2×C3⋊C8—22no—
171C2×C3⋊C8—41no—
172C2×C3⋊C8—41no—
173C2×C3⋊C8—42no—
174C2×C3⋊C8—41no—
175C2×C3⋊C8—41no—
176C2×C3⋊C8—41no—
177C2×#41C2×Dic312yes—
178C22×Q8C2×Dic311yescharacteristic
179C22×C8—61no—
180C22×C8—61no—
181C4⋊Q8—22no—
182C4⋊Q8#6711yescharacteristic
183C4⋊Q8#6711yescharacteristic
184C4⋊Q8#6711yescharacteristic
185#127#6711yescharacteristic
186#126—24no—
187#126—24no—
188#125—22no—
189#125—24no—
190#125#6712yes—
191#124—22no—
192#124—22no—
193#124#6712yes—
194#123—24no—
195#123—24no—
196C22⋊Q8—24no—
197C22⋊Q8—21no—
198C22⋊Q8—21no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201#121—21no—
202#121—21no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—24no—
214C4×D4—21no—
215C4×D4—21no—
216#117—24no—
217#117—22no—
218#117C2×Dic312yes—
219C2×C4⋊C4—22no—
220C2×C4⋊C4C2×Dic311yescharacteristic
221C2×C4⋊C4#6712yes—
222C2×C42#6712yes—
223C2×C42C22×D311yescharacteristic
224C4⋊C8—61no—
225C4⋊C8—61no—
226C4⋊C8—61no—
227C4⋊C8—61no—
228C22⋊C8—61no—
229C22⋊C8—61no—
230#97—61no—
231#97—61no—
232#97—61no—
233#97—61no—
234C23×C3—22no—
235C23×C3C2×C22⋊C411yescharacteristic
236C3×Q8—42no—
237C3×Q8—42no—
238C3×Q8—42no—
239C3×Q8—41no—
240C3×Q8—41no—
241C3×Q8—81no—
242C3×D4—42no—
243C3×D4—44no—
244C3×D4—41no—
245C3×D4—41no—
246C3×D4—81no—
247C2×C3×C4—22no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—24no—
256C2×C3×C4—41no—
257C2×C3×C4—44no—
258C2×C3×C4—42no—
259C2×C3×C4—84no—
260C2×C3×C4C2×C22⋊C411yescharacteristic
261C2×C3×C4C2×C22⋊C411yescharacteristic
262C2×C3×C4#12011yescharacteristic
263C2×C3×C4#12011yescharacteristic
264C2×C3×C4#12011yescharacteristic
265C2×C3×C4#12011yescharacteristic
266C3⋊C8—42no—
267C3⋊C8—81no—
268C3⋊C8—81no—
269C3⋊C8—81no—
270#41—44no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—42no—
276C2×Q8#21712yes—
277C2×D4—22no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4#21712yes—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4#21712yes—
284C22×C4#21712yes—
285C22×C4C22×Dic311yescharacteristic
286C22×C4C2×#6711yescharacteristic
287C22×C4C2×#6711yescharacteristic
288C2×C8—62no—
289C2×C8—121no—
290C2×C8—121no—
291C2×C8—122no—
292C2×C8—121no—
293C2×C8—121no—
294C2×C8—121no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4#21712yes—
309C4⋊C4#21712yes—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C42—24no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42C2×#6711yescharacteristic
321C42C2×#6711yescharacteristic
322C42C2×#6711yescharacteristic
323C42C2×#6711yescharacteristic
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3—42no—
329C22×C3#37911yescharacteristic
330C22×C3#41312yes—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—42no—
337C3×C4—42no—
338C3×C4—42no—
339C3×C4—42no—
340C3×C4—42no—
341C3×C4#45411yescharacteristic
342C3×C4#45611yescharacteristic
343C23—22no—
344C23C2×#21711yescharacteristic
345Q8—42no—
346Q8—42no—
347Q8—42no—
348Q8—41no—
349Q8—41no—
350Q8—81no—
351D4—42no—
352D4—44no—
353D4—41no—
354D4—41no—
355D4—81no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—84no—
369C2×C4C2×#21711yescharacteristic
370C2×C4C2×#21711yescharacteristic
371C2×C4#95311yescharacteristic
372C2×C4#95311yescharacteristic
373C2×C4#95311yescharacteristic
374C2×C4#95311yescharacteristic
375C8—122no—
376C8—241no—
377C8—241no—
378C8—241no—
379C2×C3—22no—
380C2×C3—82no—
381C2×C3#178511yescharacteristic
382C2×C3#179311yescharacteristic
383C2×C3#179411yescharacteristic
384C22—21no—
385C22—21no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22#523712yes—
390C22#524711yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4#524511yescharacteristic
402C4#524611yescharacteristic
403C3#846111yescharacteristic, complemented, Sylow
404C2—22no—
405C2—82no—
406C2#7213711yescharacteristic
407C2#7214311yescharacteristic
408C2#7214611yescharacteristic
subgroupG/H
centerC22#5247
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3507C2
socleC22×C3#379
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3507D3
hypocenterC3#8461
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3507

Complements

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

NH ≅ G/Ntype
2. #71831C2semidirect
3. #71822C2semidirect
6. #71635C2semidirect
7. #71625C2semidirect
403. C3#8461semidirect

Maximal subgroups

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

subgroupindex
C3×#35072
#718312
#718222
#718072
#717982
#716352
#716252
#84613