Finite Group Databasebeta

← #10692873

Subgroups — #10692873

Order 768.

498 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 95 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#83260C211yesmaximal, characteristic
2#79335C211yesmaximal, characteristic
3C3×#1556C211yesmaximal, characteristic, complemented
4#71776C211yesmaximal, characteristic, complemented
5#70827C211yesmaximal, characteristic
6#67750C211yesmaximal, characteristic, complemented
7#67749C211yesmaximal, characteristic, complemented
8#9453—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13#5231—21no—
14#5231—21no—
15#4988C412yes—
16#4987—21no—
17#4987—21no—
18#4977C412yes—
19C4×C4⋊Dic3C2211yescharacteristic
20#4926C2211yescharacteristic
21#4926C2211yescharacteristic
22#4836—22no—
23#4726—21no—
24D3×C22⋊C8—21no—
25#4706—22no—
26C3×#332—22no—
27#4547—22no—
28#4468—21no—
29#4468—21no—
30#2973—31no—
31#2563—31no—
32#1556—31no—
33#1556D311yescharacteristic, complemented
34#1466—31no—
35#1285—31no—
36#1285—31no—
37C22×C3×Q8—21no—
38C22×D12C2×C411yescharacteristic
39C22×C4×D3D411yescharacteristic
40C22×C4×D3D411yescharacteristic
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4C2×C412yes—
47C2×C3×C42C2311yescharacteristic
48C2×(C2×D3)⋊C4—22no—
49C2×(C2×D3)⋊C4—22no—
50C2×(C2×D3)⋊C4D411yescharacteristic
51C2×(C2×D3)⋊C4D411yescharacteristic
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3C2×C411yescharacteristic
54C2×C4×Dic3C2×C412yes—
55C2×ΓL(1,𝔽25)—22no—
56C2×D3×C8—22no—
57C4×D12—21no—
58C4×D12—21no—
59C4×D12—21no—
60C4×D12—21no—
61C3×C4⋊C8—21no—
62C3×C4⋊C8D412yes—
63C3×C22⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×#97—21no—
66C3×#97—21no—
67C3⋊#95—21no—
68C3⋊#95—22no—
69C3⋊#95—22no—
70#830—21no—
71#830—21no—
72(C2×D3)⋊C8—22no—
73(C2×D3)⋊C8—22no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)D412yes—
76#803—21no—
77#803—21no—
78C2×C4⋊Q8C2×D311yescharacteristic
79C2×C4×D4—31no—
80#435—62no—
81#435—62no—
82#432—31no—
83#432—31no—
84#432C2×D311yescharacteristic
85#432C2×D311yescharacteristic
86#407—61no—
87C2×C22⋊C8—61no—
88#393—61no—
89#393—61no—
90#390—32no—
91#382—32no—
92#380—61no—
93#380—61no—
94C4×C4⋊C4—31no—
95#332—22no—
96#332—62no—
97#323—61no—
98#323—61no—
99C23×D3C22⋊C411yescharacteristic
100C23×D3C22⋊C411yescharacteristic
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—82no—
104C22×C3×C4—22no—
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×Dic3—22no—
109C22×Dic3C22⋊C411yescharacteristic
110C22×Dic3C22⋊C411yescharacteristic
111C2×D12—22no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12—41no—
115C2×C4×D3—21no—
116C2×C4×D3—21no—
117C2×C4×D3—21no—
118C2×C4×D3—21no—
119C2×C4×D3—42no—
120C2×C4×D3—42no—
121C2×C3×C8—22no—
122C2×C3×C8—41no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130C3×C42#4111yescharacteristic
131(C2×D3)⋊C4—21no—
132(C2×D3)⋊C4—21no—
133(C2×D3)⋊C4—21no—
134(C2×D3)⋊C4—21no—
135(C2×D3)⋊C4—82no—
136(C2×D3)⋊C4—82no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—42no—
140C4×Dic3—42no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—41no—
143ΓL(1,𝔽25)—82no—
144D3×C8—82no—
145C22×Q8—21no—
146C22×D4—31no—
147C23×C4—31no—
148C23×C4—31no—
149C2×M16—62no—
150C22×C8—62no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4×D4—61no—
156C4×D4—61no—
157C4×D4—61no—
158C4×D4—61no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4C4×D312yes—
163C2×C22⋊C4—31no—
164C2×C22⋊C4—31no—
165C2×C22⋊C4—62no—
166C2×C22⋊C4—62no—
167C2×C42—32no—
168C2×C42C22×D311yescharacteristic
169C4⋊C8—21no—
170C4⋊C8—32no—
171C4⋊C8—61no—
172C4⋊C8#6712yes—
173C22⋊C8—21no—
174C22⋊C8—21no—
175C22⋊C8—62no—
176C22⋊C8—62no—
177C22⋊C8—61no—
178C22⋊C8—61no—
179#97—21no—
180#97—21no—
181#97—61no—
182#97—61no—
183#95—62no—
184#95—62no—
185#95—61no—
186C23×C3C2×C22⋊C411yescharacteristic
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—21no—
191C22×D3—21no—
192C22×D3—41no—
193C22×D3—41no—
194C22×D3—41no—
195C22×D3—41no—
196C22×D3#10112yes—
197C3×Q8—82no—
198C3×Q8—82no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—22no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12311yescharacteristic
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—42no—
217C2×Dic3—42no—
218C2×Dic3—42no—
219D12—42no—
220D12—42no—
221C4×D3—42no—
222C4×D3—42no—
223C4×D3—81no—
224C4×D3—81no—
225C3×C8—42no—
226C3×C8—81no—
227C3⋊C8—42no—
228C3⋊C8—81no—
229C24—31no—
230C24—31no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—82no—
234C2×D4—62no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—121no—
238C22×C4—22no—
239C22×C4—31no—
240C22×C4—31no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—62no—
246C22×C4—122no—
247C22×C4—122no—
248C22×C4C2×C4×D311yescharacteristic
249C22×C4D3×D411yescharacteristic
250C22×C4D3×D411yescharacteristic
251M16—242no—
252C2×C8—22no—
253C2×C8—41no—
254C2×C8—62no—
255C2×C8—121no—
256C2×C8—242no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—122no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—242no—
269C22⋊C4—242no—
270C42—122no—
271C42#23511yescharacteristic
272C42D3×D411yescharacteristic
273C42#23711yescharacteristic
274C42C2×#6711yescharacteristic
275C22×C3—21no—
276C22×C3—21no—
277C22×C3#38611yescharacteristic
278C22×C3C2×#10111yescharacteristic
279C22×C3#41311yescharacteristic
280C2×D3—22no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—42no—
284C2×D3—42no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—81no—
289C2×D3—81no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—82no—
296C3×C4#45512yes—
297Dic3—41no—
298Dic3—41no—
299Dic3—82no—
300C23—32no—
301C23—61no—
302C23—61no—
303C23—61no—
304C23—61no—
305C23—61no—
306C23—121no—
307C23—121no—
308C23—121no—
309C23—121no—
310C23D3×C22⋊C411yescharacteristic
311Q8—82no—
312Q8—82no—
313D4—122no—
314D4—122no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—122no—
325C2×C4—122no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—122no—
329C2×C4—122no—
330C2×C4—122no—
331C2×C4—241no—
332C2×C4—241no—
333C2×C4#67⋊C411yescharacteristic
334C2×C4#88311yescharacteristic
335C2×C4#88411yescharacteristic
336C2×C4C4×#6711yescharacteristic
337C2×C4#93711yescharacteristic
338C2×C4#93811yescharacteristic
339C8—42no—
340C8—81no—
341C8—122no—
342C8—241no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3#188511yescharacteristic
346C2×C3#190111yescharacteristic
347C2×C3#191011yescharacteristic
348D3—42no—
349D3—81no—
350C22—21no—
351C22—21no—
352C22—62no—
353C22—122no—
354C22—122no—
355C22—121no—
356C22—121no—
357C22—121no—
358C22—121no—
359C22—121no—
360C22—241no—
361C22—241no—
362C22D3×#10111yescharacteristic
363C22#475111yescharacteristic
364C22#495611yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—82no—
371C4—121no—
372C4—121no—
373C4—242no—
374C4#520212yes—
375C3#945311yescharacteristic, complemented, Sylow
376C2—21no—
377C2—21no—
378C2—122no—
379C2—241no—
380C2#7011511yescharacteristic
381C2#7033011yescharacteristic
382C2#7034311yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1556C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1556D3
hypocenterC3#9453
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
3. C3×#1556C2semidirect
4. #71776C2semidirect
6. #67750C2semidirect
7. #67749C2semidirect
33. #1556D3semidirect
375. C3#9453semidirect

Maximal subgroups

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

subgroupindex
#832602
#793352
C3×#15562
#717762
#708272
#677502
#677492
#94533