Finite Group Databasebeta

← #10692883

Subgroups — #10692883

Order 768.

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

normal subgroups only

This group has 554 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#83259C211yesmaximal, characteristic
2#79335C211yesmaximal, characteristic
3C3×#1556C211yesmaximal, characteristic, complemented
4#71776C211yesmaximal, characteristic, complemented
5#70827C211yesmaximal, characteristic
6#67750C211yesmaximal, characteristic, complemented
7#67749C211yesmaximal, characteristic, complemented
8#9455—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#946—21no—
11C2×#946—21no—
12C2×C4×D12C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C3×#432C2211yescharacteristic
15#4993—21no—
16#4993—21no—
17#4988C412yes—
18#4975C412yes—
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#2542—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
48#946—44no—
49#946—44no—
50C2×(C2×D3)⋊C4—22no—
51C2×(C2×D3)⋊C4—22no—
52C2×(C2×D3)⋊C4D411yescharacteristic
53C2×(C2×D3)⋊C4D411yescharacteristic
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3C2×C411yescharacteristic
56C2×Dic3⋊C4—22no—
57C2×Dic3⋊C4—22no—
58C2×C4×Dic3C2×C412yes—
59C2×ΓL(1,𝔽25)—22no—
60C2×D3×C8—22no—
61C4×D12—21no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8D412yes—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#97—21no—
70C3×#97—21no—
71C3⋊#95—21no—
72#830—21no—
73#830—21no—
74(C2×D3)⋊C8—22no—
75(C2×D3)⋊C8—22no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D412yes—
78#803—21no—
79#803—21no—
80C2×C4⋊Q8C2×D311yescharacteristic
81C2×C22⋊Q8—61no—
82C2×C22⋊Q8—61no—
83C2×C4×D4—31no—
84#435—62no—
85#435—62no—
86#432—31no—
87#432—31no—
88#432C2×D311yescharacteristic
89#432C2×D311yescharacteristic
90#407—61no—
91C2×C22⋊C8—61no—
92#399—61no—
93#399—61no—
94#390—32no—
95#384—32no—
96C4×C4⋊C4—31no—
97#332—22no—
98#332—62no—
99#323—61no—
100#323—61no—
101C23×D3C22⋊C411yescharacteristic
102C23×D3C22⋊C411yescharacteristic
103C2×C3×Q8—42no—
104C2×C3×Q8—42no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C22×C3×C4—22no—
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×Dic3—22no—
112C22×Dic3C22⋊C411yescharacteristic
113C22×Dic3C22⋊C411yescharacteristic
114C2×D12—22no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—41no—
118C2×C4×D3—21no—
119C2×C4×D3—21no—
120C2×C4×D3—21no—
121C2×C4×D3—21no—
122C2×C4×D3—42no—
123C2×C4×D3—42no—
124C2×C3×C8—22no—
125C2×C3×C8—41no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133C3×C42#4111yescharacteristic
134(C2×D3)⋊C4—21no—
135(C2×D3)⋊C4—21no—
136(C2×D3)⋊C4—21no—
137(C2×D3)⋊C4—21no—
138(C2×D3)⋊C4—44no—
139(C2×D3)⋊C4—44no—
140C4⋊Dic3—24no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143Dic3⋊C4—44no—
144Dic3⋊C4—44no—
145C4×Dic3—42no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—41no—
148ΓL(1,𝔽25)—82no—
149D3×C8—82no—
150C22×Q8—21no—
151C22×D4—31no—
152C23×C4—31no—
153C23×C4—31no—
154C2×M16—62no—
155C22×C8—62no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C22⋊Q8—124no—
161C22⋊Q8—124no—
162C4×D4—61no—
163C4×D4—61no—
164C4×D4—61no—
165C4×D4—61no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—62no—
170C2×C4⋊C4—62no—
171C2×C4⋊C4C4×D312yes—
172C2×C22⋊C4—31no—
173C2×C22⋊C4—31no—
174C2×C22⋊C4—62no—
175C2×C22⋊C4—62no—
176C2×C42—32no—
177C2×C42C22×D311yescharacteristic
178C4⋊C8—21no—
179C4⋊C8—32no—
180C4⋊C8—61no—
181C4⋊C8#6712yes—
182C22⋊C8—21no—
183C22⋊C8—21no—
184C22⋊C8—62no—
185C22⋊C8—62no—
186C22⋊C8—61no—
187C22⋊C8—61no—
188#97—21no—
189#97—21no—
190#97—61no—
191#97—61no—
192#95—61no—
193C23×C3C2×C22⋊C411yescharacteristic
194C22×D3—21no—
195C22×D3—21no—
196C22×D3—21no—
197C22×D3—21no—
198C22×D3—21no—
199C22×D3—41no—
200C22×D3—41no—
201C22×D3—41no—
202C22×D3—41no—
203C22×D3#10112yes—
204C3×Q8—82no—
205C3×Q8—82no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—22no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4C4×D411yescharacteristic
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12311yescharacteristic
219C2×Dic3—24no—
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226D12—42no—
227D12—42no—
228C4×D3—42no—
229C4×D3—42no—
230C4×D3—81no—
231C4×D3—81no—
232C3×C8—42no—
233C3×C8—81no—
234C3⋊C8—42no—
235C3⋊C8—81no—
236C24—31no—
237C24—31no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×D4—62no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—121no—
246C22×C4—22no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—62no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—122no—
255C22×C4—122no—
256C22×C4C2×C4×D311yescharacteristic
257C22×C4D3×D411yescharacteristic
258C22×C4D3×D411yescharacteristic
259M16—242no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—62no—
263C2×C8—121no—
264C2×C8—242no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—64no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—124no—
273C4⋊C4—124no—
274C22⋊C4—61no—
275C22⋊C4—61no—
276C22⋊C4—61no—
277C22⋊C4—61no—
278C22⋊C4—124no—
279C22⋊C4—124no—
280C42—122no—
281C42#23511yescharacteristic
282C42D3×D411yescharacteristic
283C42#23711yescharacteristic
284C42C2×#6711yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3#38611yescharacteristic
288C22×C3C2×#10111yescharacteristic
289C22×C3#41311yescharacteristic
290C2×D3—22no—
291C2×D3—41no—
292C2×D3—41no—
293C2×D3—42no—
294C2×D3—42no—
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—41no—
298C2×D3—81no—
299C2×D3—81no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—82no—
306C3×C4#45312yes—
307Dic3—44no—
308Dic3—41no—
309Dic3—41no—
310C23—32no—
311C23—61no—
312C23—61no—
313C23—61no—
314C23—61no—
315C23—61no—
316C23—121no—
317C23—121no—
318C23—121no—
319C23—121no—
320C23D3×C22⋊C411yescharacteristic
321Q8—82no—
322Q8—82no—
323D4—122no—
324D4—122no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—64no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—122no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—241no—
342C2×C4—241no—
343C2×C4#67⋊C411yescharacteristic
344C2×C4#88311yescharacteristic
345C2×C4#88411yescharacteristic
346C2×C4C4×#6711yescharacteristic
347C2×C4#93711yescharacteristic
348C2×C4#93811yescharacteristic
349C8—42no—
350C8—81no—
351C8—122no—
352C8—241no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3#188711yescharacteristic
356C2×C3#190111yescharacteristic
357C2×C3#190911yescharacteristic
358D3—42no—
359D3—81no—
360C22—21no—
361C22—21no—
362C22—62no—
363C22—122no—
364C22—122no—
365C22—121no—
366C22—121no—
367C22—121no—
368C22—121no—
369C22—121no—
370C22—241no—
371C22—241no—
372C22D3×#10111yescharacteristic
373C22#475111yescharacteristic
374C22#495611yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—82no—
381C4—124no—
382C4—121no—
383C4—121no—
384C4#519812yes—
385C3#945511yescharacteristic, complemented, Sylow
386C2—21no—
387C2—21no—
388C2—122no—
389C2—241no—
390C2#7011511yescharacteristic
391C2#7032311yescharacteristic
392C2#7035011yescharacteristic
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#9455
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
385. C3#9455semidirect

Maximal subgroups

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

subgroupindex
#832592
#793352
C3×#15562
#717762
#708272
#677502
#677492
#94553