Finite Group Databasebeta

← #10692365

Subgroups — #10692365

Order 768.

500 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 93 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
1#82727C211yesmaximal, characteristic
2#73881C211yesmaximal, characteristic
3C3×#1996C211yesmaximal, characteristic
4#71688C211yesmaximal, characteristic
5#69442C211yesmaximal, characteristic
6#67626C211yesmaximal, characteristic
7#67618C211yesmaximal, characteristic
8#9363—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C3×#380C2211yescharacteristic
12C3×C4×C22⋊C4C2211yescharacteristic
13#5225C412yescomplemented
14C4×#217C2211yescharacteristic
15#5207C2211yescharacteristic
16#5207C2211yescharacteristic
17#5045—22no—
18#5044—21no—
19#5043—21no—
20Dic3×C4⋊C4—21no—
21#4972—21no—
22#4964C412yescomplemented
23#4945—22no—
24#4650—21no—
25#4648—21no—
26C3×#342—22no—
27#4538—21no—
28#4538—21no—
29#4523—22no—
30#3095—31no—
31#2346—31no—
32#1996D311yescharacteristic
33#1826—31no—
34#1508—31no—
35#1305—31no—
36#1297—31no—
37C22×C3×Q8C2×C411yescharacteristic
38C23×C3×C4C2311yescharacteristic
39C3×C22⋊Q8—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4D412yes—
49C2×C3×C42—21no—
50C2×#217—21no—
51C2×#217—21no—
52#926—21no—
53#926—21no—
54#926C2×C412yes—
55#926D412yes—
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4⋊Dic3D411yescharacteristic
58C2×Dic3⋊C4—22no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3—22no—
61C2×C4×Dic3—21no—
62C2×#208—21no—
63C2×#208C2×C412yes—
64C3×#95—21no—
65C3×#95—21no—
66#835—22no—
67#835—22no—
68C3⋊#95—22no—
69C3⋊#95—22no—
70C3⋊#95D411yescharacteristic
71C3⋊#95D411yescharacteristic
72#830—21no—
73#830—21no—
74#830—21no—
75#830—21no—
76#808—22no—
77#807—22no—
78C2×C22⋊Q8C2×D311yescharacteristic
79C2×#117—31no—
80#485—62no—
81#484—61no—
82#483—61no—
83#428—32no—
84#419—32no—
85#407—31no—
86#407—31no—
87#389—61no—
88#388—62no—
89#382—61no—
90#380C2×D311yescharacteristic
91C4×C4⋊C4—61no—
92C4×C22⋊C4—31no—
93C4×C22⋊C4C2×D311yescharacteristic
94#376—61no—
95#342—22no—
96#339—62no—
97#328—61no—
98#328—61no—
99C24×C3C2×D411yescharacteristic
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—42no—
103C22×C3×C4—22no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—42no—
107C22×C3×C4—41no—
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—42no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—41no—
125C3×C22⋊C4—41no—
126C3×C22⋊C4—41no—
127C3×C22⋊C4—42no—
128C3×C42—42no—
129#217—22no—
130#217—22no—
131#217—42no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3C22⋊C412yes—
135C4⋊Dic3C22⋊C412yes—
136Dic3⋊C4—82no—
137C4×Dic3—22no—
138C4×Dic3—22no—
139C4×Dic3—44no—
140C4×Dic3—42no—
141#208—42no—
142#208—42no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—41no—
145C22×Q8C4×D311yescharacteristic
146C23×C4C22×D311yescharacteristic
147C2×M16—32no—
148C2×M16—61no—
149C22⋊Q8—21no—
150C22⋊Q8—21no—
151C22⋊Q8—21no—
152C22⋊Q8—21no—
153#117—32no—
154#117—32no—
155#117—61no—
156#117—61no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—62no—
162C2×C4⋊C4C4×D311yescharacteristic
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4#6712yes—
168C2×C42—21no—
169C2×C42—61no—
170C2×C42—62no—
171C2×C42—61no—
172#107—62no—
173#106—62no—
174#103—62no—
175#103—62no—
176C22⋊C8—61no—
177C22⋊C8—61no—
178C22⋊C8—61no—
179C22⋊C8—61no—
180#95—21no—
181#95—21no—
182#95—31no—
183#95—31no—
184#95—62no—
185#95—62no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3C4×D411yescharacteristic
191C23×C3#12011yescharacteristic
192C23×C3#12111yescharacteristic
193C3×Q8—42no—
194C3×Q8—81no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4—42no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4—81no—
210C2×C3×C4—81no—
211C2×C3×C4C2×C22⋊C411yescharacteristic
212C2×C3×C4C4×D411yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12311yescharacteristic
215C2×Dic3—22no—
216C2×Dic3—22no—
217C2×Dic3—22no—
218C2×Dic3—41no—
219C2×Dic3—42no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C3⋊C8—42no—
227C3⋊C8—81no—
228C24C2×#6711yescharacteristic
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—42no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4C2×C4×D311yescharacteristic
242C22×C4#23511yescharacteristic
243C22×C4D3×D411yescharacteristic
244C22×C4D3×D411yescharacteristic
245C22×C4D3×D411yescharacteristic
246C22×C4#23711yescharacteristic
247M16—122no—
248M16—122no—
249C2×C8—62no—
250C2×C8—121no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—32no—
254C4⋊C4—32no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—242no—
261C22⋊C4—22no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—42no—
266C22⋊C4—62no—
267C22⋊C4—62no—
268C22⋊C4—122no—
269C42—42no—
270C42—62no—
271C42—62no—
272C42—122no—
273C42—124no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—81no—
283C22×C3—81no—
284C22×C3#38611yescharacteristic
285C22×C3#45511yescharacteristic
286C22×C3#45611yescharacteristic
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—82no—
291C3×C4—82no—
292C3×C4—81no—
293C3×C4—81no—
294C3×C4#41012yes—
295Dic3—42no—
296Dic3—42no—
297Dic3—42no—
298Dic3—81no—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23C4×#6711yescharacteristic
304C23#93711yescharacteristic
305C23#93811yescharacteristic
306Q8—42no—
307Q8—81no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—62no—
321C2×C4—62no—
322C2×C4—62no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—121no—
328C2×C4—122no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4D3×C22⋊C411yescharacteristic
336C2×C4#67⋊C411yescharacteristic
337C2×C4#88311yescharacteristic
338C2×C4#88411yescharacteristic
339C8—122no—
340C8—241no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3#188211yescharacteristic
346C2×C3#189911yescharacteristic
347C2×C3#190611yescharacteristic
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—81no—
357C22—81no—
358C22#495611yescharacteristic
359C22#506111yescharacteristic
360C22#509511yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—82no—
365C4—82no—
366C4—81no—
367C4—81no—
368C4—122no—
369C4—122no—
370C4—122no—
371C4—241no—
372C4#474012yes—
373C3#936311yescharacteristic, complemented, Sylow
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#6969311yescharacteristic
379C2#7002911yescharacteristic
380C2#7005511yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1996C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1996D3
hypocenterC3#9363
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
13. #5225C4semidirect
22. #4964C4semidirect
373. C3#9363semidirect

Maximal subgroups

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

subgroupindex
#827272
#738812
C3×#19962
#716882
#694422
#676262
#676182
#93633