Finite Group Databasebeta

← #10692898

Subgroups — #10692898

Order 768.

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

normal subgroups only

This group has 536 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#83179C211yesmaximal, characteristic
2#79314C211yesmaximal, characteristic, complemented
3C3×#1555C211yesmaximal, characteristic
4#71760C211yesmaximal, characteristic
5#70830C211yesmaximal, characteristic, complemented
6#67750C211yesmaximal, characteristic, complemented
7#67749C211yesmaximal, characteristic, complemented
8#9458—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C2×#936C412yes—
11C2×C4×Dic6C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14#5222—21no—
15#5222—21no—
16#5221—21no—
17#5221—21no—
18#5221C412yes—
19C4×C4⋊Dic3C2211yescharacteristic
20#4926C2211yescharacteristic
21#4926C2211yescharacteristic
22#4828—22no—
23#4719—21no—
24#4718—21no—
25#4701—22no—
26C3×#331—22no—
27#4544—22no—
28#4469—21no—
29#4469—21no—
30#2983—31no—
31#2548—31no—
32#1555—31no—
33#1555D311yescharacteristic
34#1467—31no—
35#1285—31no—
36#1285—31no—
37C22×C3×D4—21no—
38C22×C3×D4—21no—
39C22×C3×D4C2×C412yes—
40C22×Dic6C2×C411yescharacteristic
41C3×#127—21no—
42C3×#127—21no—
43C3×#127—21no—
44C3×#127—21no—
45C2×C3×C42C2311yescharacteristic
46C2×#217—22no—
47C2×#217—22no—
48C2×#217—22no—
49C2×#217—22no—
50#936—44no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3C2×C411yescharacteristic
53C2×Dic3⋊C4D411yescharacteristic
54C2×Dic3⋊C4D411yescharacteristic
55C2×C4×Dic3C2×C412yes—
56C2×C4×Dic3D411yescharacteristic
57C2×C4×Dic3D411yescharacteristic
58C4×Dic6—21no—
59C4×Dic6—21no—
60C4×Dic6—21no—
61C4×Dic6—21no—
62C3×C4⋊C8—21no—
63C3×C4⋊C8D412yes—
64C3×C22⋊C8—21no—
65C3×C22⋊C8—21no—
66C3×#97—21no—
67C3×#97—21no—
68C3⋊#95—21no—
69#830—21no—
70#830—21no—
71#815—22no—
72Dic3⋊C8—22no—
73Dic3⋊C8—22no—
74C8×Dic3—22no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D412yes—
77#803—21no—
78#803—21no—
79C2×#127C2×D311yescharacteristic
80C2×#124—32no—
81C2×C4×Q8—31no—
82#446—62no—
83#446—62no—
84#432—31no—
85#432—31no—
86#432—61no—
87#432C2×D311yescharacteristic
88#432C2×D311yescharacteristic
89#431—61no—
90#394—61no—
91#394—61no—
92#390—32no—
93#390—61no—
94#390—61no—
95C4×C4⋊C4—31no—
96#331—22no—
97#331—62no—
98#324—61no—
99#324—61no—
100C24×C3—22no—
101C2×C3×D4—22no—
102C2×C3×D4—22no—
103C2×C3×D4—42no—
104C2×C3×D4—42no—
105C2×C3×D4—42no—
106C2×C3×D4—42no—
107C2×C3×D4—41no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—82no—
112C22×C3×C4C22×C411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—22no—
116C22×Dic3C22⋊C411yescharacteristic
117C22×Dic3C22⋊C411yescharacteristic
118C22×Dic3C22⋊C411yescharacteristic
119C22×Dic3C22⋊C411yescharacteristic
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—22no—
123C2×Dic6—41no—
124C2×C3×C8—22no—
125C2×C3×C8—41no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129C3×C42#4111yescharacteristic
130#217—44no—
131#217—44no—
132#217—82no—
133#217—82no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—42no—
137Dic3⋊C4—21no—
138Dic3⋊C4—21no—
139Dic3⋊C4—21no—
140Dic3⋊C4—21no—
141C4×Dic3—21no—
142C4×Dic3—21no—
143C4×Dic3—21no—
144C4×Dic3—21no—
145C4×Dic3—42no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—41no—
148C22×Q8—31no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4C4×D312yes—
152#127—21no—
153#127—21no—
154#127—21no—
155#127—21no—
156#124—124no—
157C4×Q8—61no—
158C4×Q8—61no—
159C4×Q8—61no—
160C4×Q8—61no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—61no—
165C2×C22⋊C4—62no—
166C2×C22⋊C4—62no—
167C2×C22⋊C4—62no—
168C2×C22⋊C4—62no—
169C2×C42—31no—
170C2×C42—31no—
171C2×C42—32no—
172C2×C42C22×D311yescharacteristic
173C4⋊C8—21no—
174C4⋊C8—32no—
175C4⋊C8—62no—
176C4⋊C8—62no—
177C4⋊C8—61no—
178C4⋊C8#6712yes—
179C22⋊C8—21no—
180C22⋊C8—21no—
181C22⋊C8—61no—
182C22⋊C8—61no—
183#97—21no—
184#97—21no—
185#97—62no—
186#97—61no—
187#97—61no—
188C4×C8—62no—
189#95—61no—
190C23×C3—42no—
191C23×C3—42no—
192C23×C3—42no—
193C23×C3—42no—
194C23×C3—42no—
195C23×C3—42no—
196C23×C3—42no—
197C23×C3C2×C22⋊C411yescharacteristic
198C3×D4—82no—
199C3×D4—82no—
200C3×D4—82no—
201C3×D4—82no—
202C3×D4—82no—
203C3×D4—82no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—22no—
207C2×C3×C4—41no—
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12311yescharacteristic
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226C2×Dic3#10012yes—
227Dic6—42no—
228Dic6—42no—
229C3×C8—42no—
230C3×C8—81no—
231C3⋊C8—42no—
232C3⋊C8—81no—
233C24—22no—
234C2×Q8—62no—
235C2×Q8—61no—
236C2×Q8—61no—
237C2×Q8—121no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—82no—
249C22×C4—31no—
250C22×C4—31no—
251C22×C4—31no—
252C22×C4—31no—
253C22×C4—62no—
254C22×C4C2×C4×D311yescharacteristic
255C22×C4D3×D411yescharacteristic
256C22×C4D3×D411yescharacteristic
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—62no—
260C2×C8—121no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—122no—
268C22⋊C4—124no—
269C22⋊C4—124no—
270C22⋊C4—242no—
271C22⋊C4—242no—
272C42—61no—
273C42—61no—
274C42—61no—
275C42—61no—
276C42—122no—
277C42#23511yescharacteristic
278C42D3×D411yescharacteristic
279C42#23711yescharacteristic
280C42C2×#6711yescharacteristic
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—82no—
284C22×C3—82no—
285C22×C3—82no—
286C22×C3—82no—
287C22×C3—82no—
288C22×C3—82no—
289C22×C3—82no—
290C22×C3#38611yescharacteristic
291C22×C3C2×#10011yescharacteristic
292C22×C3#41311yescharacteristic
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4#45412yes—
299Dic3—22no—
300Dic3—41no—
301Dic3—41no—
302Dic3—41no—
303Dic3—82no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23D3×C22⋊C411yescharacteristic
312Q8—122no—
313Q8—122no—
314D4—82no—
315D4—82no—
316D4—82no—
317D4—82no—
318D4—82no—
319D4—82no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—32no—
324C2×C4—41no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4—122no—
337C2×C4#67⋊C411yescharacteristic
338C2×C4#88311yescharacteristic
339C2×C4#88411yescharacteristic
340C2×C4C4×#6711yescharacteristic
341C2×C4#93711yescharacteristic
342C2×C4#93811yescharacteristic
343C8—42no—
344C8—81no—
345C8—122no—
346C8—241no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—162no—
350C2×C3#188811yescharacteristic
351C2×C3#190011yescharacteristic
352C2×C3#191011yescharacteristic
353C22—21no—
354C22—21no—
355C22—82no—
356C22—82no—
357C22—82no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22D3×#10011yescharacteristic
363C22#474511yescharacteristic
364C22#495611yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—62no—
371C4—121no—
372C4—121no—
373C4—121no—
374C4—242no—
375C4#520112yes—
376C3#945811yescharacteristic, complemented, Sylow
377C2—21no—
378C2—21no—
379C2—162no—
380C2#7008111yescharacteristic
381C2#7032511yescharacteristic
382C2#7033011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1555C2
socleC22×C3#386
prime radicalC2×C3×#127C22
cosocleC2×C3×C42C23
hypercenter#1555D3
hypocenterC3#9458
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1555

Complements

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

NH ≅ G/Ntype
2. #79314C2semidirect
5. #70830C2semidirect
6. #67750C2semidirect
7. #67749C2semidirect
376. C3#9458semidirect

Maximal subgroups

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

subgroupindex
#831792
#793142
C3×#15552
#717602
#708302
#677502
#677492
#94583