Finite Group Databasebeta

← #10692217

Subgroups — #10692217

Order 768.

478 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3195C211yesmaximal, characteristic
2#82750C211yesmaximal, characteristic
3#71701C211yesmaximal, characteristic
4#71689C211yesmaximal, characteristic
5#71688C211yesmaximal, characteristic
6#67685C211yesmaximal, characteristic
7#67682C211yesmaximal, characteristic
8#9342—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C3×#483—21no—
12C3×#483C412yescomplemented
13C3×#470—22no—
14C3×#451—21no—
15C3×#451C412yescomplemented
16C3×#407C2211yescharacteristic
17#5229—21no—
18#5212C2211yescharacteristic
19#5211C2211yescharacteristic
20C4×#217C2211yescharacteristic
21#5207C2211yescharacteristic
22#4984—21no—
23#4980—21no—
24#4980—21no—
25#4951—21no—
26#4951—21no—
27#4538—22no—
28#4538—21no—
29#4538—21no—
30#4526—21no—
31#4526—21no—
32#3195D311yescharacteristic
33#2378—31no—
34#1996—31no—
35#1511—31no—
36#1508—31no—
37#1305—31no—
38#1284—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C22×C3×Q8D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C2×C3×Q16—22no—
43C2×C3×M16—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8C2×C412yes—
51C3×C22⋊Q8D412yes—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4C2×C411yescharacteristic
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×#217—21no—
59C2×#217—21no—
60C2×#217D411yescharacteristic
61C2×#217D411yescharacteristic
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C2×C4×Dic3—21no—
66C2×C4×Dic3—21no—
67C2×C4×Dic3—21no—
68C2×#208—21no—
69C3×#106—22no—
70C3×#103—22no—
71C3×#103—22no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8C2×C412yes—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95D411yescharacteristic
82C3⋊#95D411yescharacteristic
83#830—21no—
84#830—21no—
85C2×C22⋊Q8C2×D311yescharacteristic
86C2×C22⋊Q8C2×D311yescharacteristic
87#483—21no—
88#483Dic312yescomplemented
89#470—22no—
90#451—21no—
91#451Dic312yescomplemented
92#407—31no—
93#407C2×D311yescharacteristic
94#391—61no—
95#389—61no—
96#388—61no—
97#388—61no—
98#387—61no—
99#387—61no—
100#381—31no—
101#380—31no—
102C4×C22⋊C4—31no—
103#342—61no—
104#342—61no—
105#328—61no—
106#328—61no—
107#328—62no—
108C24×C3C2×D411yescharacteristic
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—81no—
113C2×C3×Q8—81no—
114C2×C3×Q8C22⋊C412yes—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—81no—
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—41no—
131C3×Q16—82no—
132C3×M16—81no—
133C2×C3×C8—22no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4C22⋊C412yes—
142C3×C22⋊C4—22no—
143C3×C22⋊C4—22no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146#217—22no—
147#217—41no—
148#217—41no—
149#217—41no—
150#217—81no—
151C4⋊Dic3—81no—
152C4⋊Dic3—81no—
153C4⋊Dic3—81no—
154C4×Dic3—81no—
155C4×Dic3—81no—
156C4×Dic3—81no—
157#208—81no—
158C2×C3⋊C8—41no—
159C22×Q8C2×Dic311yescharacteristic
160C22×Q8#6711yescharacteristic
161C23×C4C22×D311yescharacteristic
162C2×Q16—22no—
163C2×M16—21no—
164C2×M16—61no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8C2×Dic312yes—
172C22⋊Q8#6712yes—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4C2×Dic311yescharacteristic
179C2×C4⋊C4#6711yescharacteristic
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—31no—
183C2×C22⋊C4—31no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C42—61no—
187C2×C42—61no—
188C2×C42—61no—
189#106—22no—
190#103—22no—
191#103—22no—
192C22⋊C8—21no—
193C22⋊C8—61no—
194C22⋊C8—61no—
195C22⋊C8C2×Dic312yes—
196#95—31no—
197#95—31no—
198#95—61no—
199#95—61no—
200#95—61no—
201#95—61no—
202#95—61no—
203#95—61no—
204#95—61no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3C4×D411yescharacteristic
210C23×C3#12011yescharacteristic
211C23×C3#12111yescharacteristic
212C3×Q8—42no—
213C3×Q8—42no—
214C3×Q8—81no—
215C3×Q8—81no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—81no—
231C2×C3×C4—81no—
232C2×C3×C4C2×C22⋊C411yescharacteristic
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12311yescharacteristic
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—42no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—81no—
250C2×Dic3—81no—
251C3×C8—42no—
252C3⋊C8—81no—
253C24D3×D411yescharacteristic
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—81no—
258C2×Q8—81no—
259C2×Q8#21712yes—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—81no—
269C22×C4—121no—
270C22×C4D3×D411yescharacteristic
271C22×C4#23711yescharacteristic
272C22×C4#23711yescharacteristic
273C22×C4C22×Dic311yescharacteristic
274C22×C4C2×#6711yescharacteristic
275C22×C4C2×#6711yescharacteristic
276Q16—82no—
277M16—81no—
278M16—241no—
279C2×C8—22no—
280C2×C8—121no—
281C4⋊C4—22no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—241no—
289C4⋊C4—241no—
290C4⋊C4—241no—
291C4⋊C4#21712yes—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—62no—
297C22⋊C4—121no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C22⋊C4—241no—
301C42—241no—
302C42—241no—
303C42—241no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—81no—
313C22×C3—81no—
314C22×C3#38611yescharacteristic
315C22×C3#45511yescharacteristic
316C22×C3#45611yescharacteristic
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—42no—
322C3×C4—42no—
323C3×C4—81no—
324C3×C4—81no—
325Dic3—81no—
326Dic3—81no—
327Dic3—81no—
328Dic3—81no—
329Dic3—82no—
330C23—21no—
331C23—21no—
332C23—41no—
333C23—41no—
334C23D4×Dic311yescharacteristic
335C23#93711yescharacteristic
336C23#93911yescharacteristic
337Q8—42no—
338Q8—42no—
339Q8—81no—
340Q8—81no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—122no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—241no—
371C2×C4—241no—
372C2×C4D4×Dic311yescharacteristic
373C2×C4#93511yescharacteristic
374C2×C4#93911yescharacteristic
375C2×C4C2×#21711yescharacteristic
376C8—42no—
377C8—241no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—41no—
381C2×C3—41no—
382C2×C3#188211yescharacteristic
383C2×C3#190511yescharacteristic
384C2×C3#191311yescharacteristic
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—81no—
394C22—81no—
395C22#520211yescharacteristic
396C22#520511yescharacteristic
397C22#522011yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—42no—
404C4—81no—
405C4—81no—
406C4—241no—
407C4—241no—
408C4—241no—
409C4—241no—
410C4—242no—
411C3#934211yescharacteristic, complemented, Sylow
412C2—21no—
413C2—21no—
414C2—41no—
415C2—41no—
416C2#7184811yescharacteristic
417C2#7210011yescharacteristic
418C2#7210111yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#3195C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#3195D3
hypocenterC3#9342
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3195

Complements

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

NH ≅ G/Ntype
12. C3×#483C4semidirect
15. C3×#451C4semidirect
88. #483Dic3semidirect
91. #451Dic3semidirect
411. C3#9342semidirect

Maximal subgroups

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

subgroupindex
C3×#31952
#827502
#717012
#716892
#716882
#676852
#676822
#93423