Finite Group Databasebeta

← #10691192

Subgroups — #10691192

Order 768.

458 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 105 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#83262C211yesmaximal, characteristic
2#79039C211yesmaximal, characteristic
3C3×#1548C211yesmaximal, characteristic, complemented
4#71774C211yesmaximal, characteristic, complemented
5#70782C211yesmaximal, characteristic
6#67738C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9192—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#872C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13#5231C412yes—
14#5230C412yes—
15#5126—21no—
16Dic3×M16—21no—
17#4993—21no—
18#4987—21no—
19#4985—21no—
20#4972—21no—
21#4936C2211yescharacteristic
22#4925C2211yescharacteristic
23#4924C2211yescharacteristic
24#4840—22no—
25#4840—22no—
26C3×#333—22no—
27#4546—22no—
28#4469—21no—
29#4468—21no—
30#2921—31no—
31#2628—31no—
32#1548—31no—
33#1548D311yescharacteristic, complemented
34#1458—31no—
35#1286—31no—
36#1286—31no—
37C22×C3×Q8C2×C412yes—
38C22×C4×D3D411yescharacteristic
39C2×C3×M16—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C42C2311yescharacteristic
47C2×(C2×D3)⋊C4—22no—
48C2×(C2×D3)⋊C4C2×C411yescharacteristic
49C2×(C2×D3)⋊C4D411yescharacteristic
50C2×C4⋊Dic3—21no—
51C2×Dic3⋊C4—22no—
52C2×Dic3⋊C4C2×C411yescharacteristic
53C2×Dic3⋊C4D411yescharacteristic
54C2×C4×Dic3D411yescharacteristic
55C2×#208—21no—
56#872—21no—
57#872—21no—
58#872—21no—
59#872—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×C4⋊C8D412yes—
63C3×C22⋊C8—21no—
64C3×C4×C8—21no—
65C3⋊#95—21no—
66C3⋊#95—22no—
67C3⋊#95—22no—
68C3⋊#95C2×C412yes—
69#830—21no—
70(C2×D3)⋊C8—24no—
71#815—22no—
72C8×Dic3—22no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)D412yes—
76C4×C3⋊C8—21no—
77C2×C4⋊Q8C2×D311yescharacteristic
78C2×#117—31no—
79#433—62no—
80#433—62no—
81#431—31no—
82#431C2×D311yescharacteristic
83#423—31no—
84#423C2×D311yescharacteristic
85#407—61no—
86C4×M16—61no—
87#400—61no—
88#399—61no—
89#395—32no—
90#393—32no—
91#389—61no—
92#382—31no—
93#380—61no—
94#333—22no—
95#333—62no—
96#324—61no—
97#323—61no—
98C23×D3C22⋊C411yescharacteristic
99C2×C3×Q8—22no—
100C2×C3×Q8—22no—
101C2×C3×Q8—82no—
102C22×C3×C4—22no—
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×Dic3—22no—
107C22×Dic3C22⋊C411yescharacteristic
108C22×Dic3C22⋊C411yescharacteristic
109C22×Dic3C22⋊C411yescharacteristic
110C2×C4×D3—21no—
111C2×C4×D3—21no—
112C2×C4×D3—21no—
113C2×C4×D3—21no—
114C2×C4×D3—21no—
115C2×C4×D3—21no—
116C3×M16—41no—
117C3×M16—41no—
118C2×C3×C8—22no—
119C2×C3×C8—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42#4111yescharacteristic
127C3×C42#4111yescharacteristic
128(C2×D3)⋊C4—21no—
129(C2×D3)⋊C4—21no—
130(C2×D3)⋊C4—21no—
131(C2×D3)⋊C4—21no—
132(C2×D3)⋊C4—82no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135Dic3⋊C4—21no—
136Dic3⋊C4—21no—
137Dic3⋊C4—21no—
138Dic3⋊C4—21no—
139Dic3⋊C4—82no—
140C4×Dic3—21no—
141C4×Dic3—21no—
142#208—41no—
143#208—41no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—41no—
146C22×Q8C4×D312yes—
147C23×C4—31no—
148C2×M16—21no—
149C2×M16—61no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154#117—61no—
155#117—61no—
156#117—61no—
157#117—61no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—62no—
164C2×C22⋊C4—31no—
165C2×C22⋊C4—31no—
166C2×C22⋊C4—62no—
167C2×C42—31no—
168C2×C42C22×D311yescharacteristic
169C4⋊C8—21no—
170C4⋊C8—21no—
171C4⋊C8—32no—
172C4⋊C8—61no—
173C4⋊C8—61no—
174C4⋊C8#6712yes—
175C22⋊C8—21no—
176C22⋊C8—64no—
177C22⋊C8—61no—
178#97—62no—
179C4×C8—21no—
180C4×C8—62no—
181C4×C8—61no—
182#95—32no—
183#95—62no—
184#95—62no—
185#95—61no—
186C23×C3C2×C22⋊C411yescharacteristic
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—41no—
191C22×D3—41no—
192C3×Q8—82no—
193C3×Q8—82no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4C4×D411yescharacteristic
200C2×C3×C4C4×D411yescharacteristic
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12311yescharacteristic
205C2×C3×C4#13511yescharacteristic
206C2×C3×C4#13511yescharacteristic
207C2×C3×C4#13711yescharacteristic
208C2×C3×C4#13711yescharacteristic
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—42no—
217C2×Dic3—42no—
218C2×Dic3—42no—
219C4×D3—41no—
220C4×D3—41no—
221C4×D3—41no—
222C4×D3—41no—
223C3×C8—42no—
224C3×C8—81no—
225C3⋊C8—42no—
226C3⋊C8—81no—
227C24—31no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—82no—
231C22×C4—22no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—62no—
242C22×C4C2×C4×D311yescharacteristic
243C22×C4D3×D411yescharacteristic
244C22×C4D3×D411yescharacteristic
245M16—41no—
246M16—41no—
247M16—121no—
248M16—121no—
249C2×C8—22no—
250C2×C8—41no—
251C2×C8—62no—
252C2×C8—121no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—242no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—242no—
269C42—61no—
270C42—61no—
271C42#23511yescharacteristic
272C42D3×D411yescharacteristic
273C42#23711yescharacteristic
274C42C2×#6711yescharacteristic
275C22×C3—21no—
276C22×C3—21no—
277C22×C3#38611yescharacteristic
278C22×C3#41611yescharacteristic
279C22×C3#41911yescharacteristic
280C2×D3—42no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—81no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—82no—
290C3×C4#45211yescharacteristic
291C3×C4#45211yescharacteristic
292C3×C4#48511yescharacteristic
293C3×C4#48511yescharacteristic
294Dic3—41no—
295Dic3—41no—
296Dic3—41no—
297Dic3—82no—
298C23—61no—
299C23—61no—
300C23—61no—
301C23—121no—
302C23—121no—
303C23D3×C22⋊C411yescharacteristic
304Q8—82no—
305Q8—82no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4#67⋊C411yescharacteristic
326C2×C4#88311yescharacteristic
327C2×C4#88411yescharacteristic
328C2×C4#91511yescharacteristic
329C2×C4#91611yescharacteristic
330C2×C4#91811yescharacteristic
331C2×C4#91911yescharacteristic
332C2×C4C4×#6711yescharacteristic
333C2×C4#93711yescharacteristic
334C2×C4#93811yescharacteristic
335C8—42no—
336C8—81no—
337C8—122no—
338C8—241no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3#187311yescharacteristic
342C2×C3#190911yescharacteristic
343C2×C3#191411yescharacteristic
344D3—81no—
345C22—21no—
346C22—21no—
347C22—122no—
348C22—121no—
349C22—121no—
350C22—121no—
351C22—241no—
352C22#480211yescharacteristic
353C22#480411yescharacteristic
354C22#495611yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—82no—
360C4—121no—
361C4—121no—
362C4—121no—
363C4—242no—
364C4#481411yescharacteristic
365C4#481511yescharacteristic
366C4#517311yescharacteristic
367C4#518711yescharacteristic
368C3#919211yescharacteristic, complemented, Sylow
369C2—21no—
370C2—21no—
371C2—241no—
372C2#7008911yescharacteristic
373C2#7026911yescharacteristic
374C2#7035011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#386
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1548D3
hypocenterC3#9192
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
3. C3×#1548C2semidirect
4. #71774C2semidirect
6. #67738C2semidirect
7. #67713C2semidirect
33. #1548D3semidirect
368. C3#9192semidirect

Maximal subgroups

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

subgroupindex
#832622
#790392
C3×#15482
#717742
#707822
#677382
#677132
#91923