Finite Group Databasebeta

← #10705474

Subgroups — #10705474

Order 768.

448 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 83 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#76810C211yesmaximal, characteristic
2#68951C211yesmaximal, characteristic
3#68100C211yesmaximal, characteristic
4C3×#1351C211yesmaximal, characteristic
5#67646C211yesmaximal, characteristic
6#67490C211yesmaximal, characteristic
7#67449C211yesmaximal, characteristic
8#11717—31nomaximal, Sylow
9C2×#944C411yescharacteristic, complemented
10C2×#877—21no—
11C2×#877C2211yescharacteristic
12C3×#393C2211yescharacteristic
13#5230C411yescharacteristic, complemented
14#4949—21no—
15#4947C2211yescharacteristic
16#4942—21no—
17#4718—21no—
18#4658—21no—
19#4656C411yescharacteristic, complemented
20#4650C411yescharacteristic, complemented
21#4647—21no—
22C3×#342C2211yescharacteristic
23C3×#323C2211yescharacteristic
24#4527—21no—
25#4526C2211yescharacteristic
26#4525C2211yescharacteristic
27#4469—21no—
28#2628—31no—
29#1740—31no—
30#1458—31no—
31#1351—31no—
32#1351—31no—
33#1351—31no—
34#1351D311yescharacteristic
35C22×C3×Q8C2×C411yescharacteristic
36C23×Dic3D411yescharacteristic
37C22×Dic6C2×C411yescharacteristic
38C2×C3×C22⋊C4—21no—
39C2×C3×C22⋊C4—21no—
40C2×C3×C22⋊C4C2311yescharacteristic
41C2×#217—21no—
42C2×#217—21no—
43C2×#217D411yescharacteristic
44#944—41no—
45#944—41no—
46C2×C4⋊Dic3—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—41no—
52C2×Dic3⋊C4C2×C411yescharacteristic
53C2×Dic3⋊C4D411yescharacteristic
54C2×C4×Dic3—41no—
55C2×C4×Dic3C2×C411yescharacteristic
56C2×C4×Dic3D411yescharacteristic
57#877—24no—
58#877—41no—
59#877—41no—
60#877—42no—
61C3×C22⋊C8—21no—
62C3×#95—21no—
63C3×#95C2×C411yescharacteristic
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95C2×C411yescharacteristic
69#830—21no—
70Dic3⋊C8—41no—
71C8×Dic3—41no—
72C2×C4⋊Q8—31no—
73C2×#117—31no—
74C2×#117—61no—
75#431—61no—
76#400—61no—
77#399—61no—
78#395—31no—
79#395—31no—
80#393—31no—
81#393C2×D311yescharacteristic
82#389—61no—
83#382—31no—
84#380—61no—
85#342—31no—
86#342—61no—
87#342C2×D311yescharacteristic
88#324—61no—
89#323—31no—
90#323C2×D311yescharacteristic
91C24×C3C2×D411yescharacteristic
92C2×C3×Q8—41no—
93C2×C3×Q8—81no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—41no—
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—22no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104C22×Dic3—41no—
105C22×Dic3C22⋊C411yescharacteristic
106C22×Dic3C22⋊C411yescharacteristic
107C22×Dic3C22⋊C411yescharacteristic
108C22×Dic3C22⋊C411yescharacteristic
109C22×Dic3SD1612yes—
110C22×Dic3Q1612yes—
111C2×Dic6—41no—
112C2×Dic6—81no—
113C2×C3×C8—41no—
114C3×C22⋊C4—22no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—41no—
117C3×C22⋊C4—81no—
118#217—22no—
119#217—41no—
120#217—41no—
121#217—81no—
122C4⋊Dic3—81no—
123Dic3⋊C4—41no—
124Dic3⋊C4—41no—
125Dic3⋊C4—44no—
126Dic3⋊C4—42no—
127Dic3⋊C4—41no—
128Dic3⋊C4—41no—
129Dic3⋊C4—42no—
130Dic3⋊C4—81no—
131C4×Dic3—21no—
132C4×Dic3—21no—
133C4×Dic3—21no—
134C4×Dic3—21no—
135C4×Dic3—44no—
136C2×C3⋊C8—41no—
137C22×Q8—31no—
138C22×Q8C4×D311yescharacteristic
139C23×C4—31no—
140C4⋊Q8—121no—
141C4⋊Q8—121no—
142#117—64no—
143#117—121no—
144#117—121no—
145#117—122no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—121no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—31no—
157C2×C22⋊C4—61no—
158C2×C22⋊C4—61no—
159C2×C22⋊C4C22×D311yescharacteristic
160C2×C42—31no—
161C2×C42—31no—
162C2×C42—121no—
163C4⋊C8—121no—
164C22⋊C8—21no—
165C22⋊C8—61no—
166C4×C8—121no—
167#95—21no—
168#95—31no—
169#95—61no—
170#95—61no—
171#95—61no—
172#95—61no—
173#95C4×D311yescharacteristic
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—41no—
177C23×C3—41no—
178C23×C3C2×C22⋊C411yescharacteristic
179C23×C3C2×SD1611yescharacteristic
180C23×C3C2×Q1611yescharacteristic
181C3×Q8—82no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—42no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—81no—
194C2×C3×C4—81no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—22no—
198C2×Dic3—22no—
199C2×Dic3—22no—
200C2×Dic3—22no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—42no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—81no—
213C2×Dic3—81no—
214C2×Dic3#9911yescharacteristic
215C2×Dic3#9911yescharacteristic
216C2×Dic3#10312yes—
217C2×Dic3#10312yes—
218C2×Dic3#10412yes—
219Dic6—82no—
220C3×C8—81no—
221C3⋊C8—81no—
222C24D3×D411yescharacteristic
223C2×Q8—41no—
224C2×Q8—81no—
225C2×Q8—121no—
226C2×Q8—241no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—32no—
232C22×C4—32no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—41no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—62no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—121no—
242C22×C4C2×C4×D311yescharacteristic
243C22×C4D3×D411yescharacteristic
244C2×C8—41no—
245C2×C8—121no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C4⋊C4—122no—
251C4⋊C4—122no—
252C4⋊C4—124no—
253C4⋊C4—241no—
254C4⋊C4—241no—
255C22⋊C4—22no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—62no—
259C22⋊C4—81no—
260C22⋊C4—121no—
261C22⋊C4—121no—
262C22⋊C4—241no—
263C42—61no—
264C42—61no—
265C42—61no—
266C42—61no—
267C42—124no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—81no—
277C22×C3C2×#9911yescharacteristic
278C22×C3C2×#10311yescharacteristic
279C22×C3C2×#10411yescharacteristic
280C3×C4—41no—
281C3×C4—41no—
282C3×C4—81no—
283C3×C4—82no—
284C3×C4—81no—
285Dic3—22no—
286Dic3—41no—
287Dic3—41no—
288Dic3—81no—
289Dic3—82no—
290Dic3—81no—
291Dic3#32812yes—
292Dic3#32812yes—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23D3×C22⋊C411yescharacteristic
298C23D3×SD1611yescharacteristic
299C23D3×Q1611yescharacteristic
300Q8—82no—
301Q8—242no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—32no—
307C2×C4—32no—
308C2×C4—32no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—62no—
319C2×C4—62no—
320C2×C4—62no—
321C2×C4—62no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—122no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—241no—
338C2×C4—241no—
339C8—81no—
340C8—241no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3C2×#32811yescharacteristic
346C2×C3#212211yescharacteristic
347C2×C3#212911yescharacteristic
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—81no—
357C22D3×#9911yescharacteristic
358C22D3×#10311yescharacteristic
359C22D3×#10411yescharacteristic
360C4—32no—
361C4—32no—
362C4—41no—
363C4—41no—
364C4—62no—
365C4—81no—
366C4—82no—
367C4—81no—
368C4—121no—
369C4—121no—
370C4—241no—
371C4—242no—
372C4—241no—
373C3#1171711yescharacteristic, complemented, Sylow
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2D3×#32811yescharacteristic
379C2#6913711yescharacteristic
380C2#6915911yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1351C2
socleC22×C3C2×#99
prime radicalC24×C3C2×D4
cosocleC2×C3×C22⋊C4C23
hypercenter#1351D3
hypocenterC3#11717
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1351

Complements

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

NH ≅ G/Ntype
9. C2×#944C4semidirect
13. #5230C4semidirect
19. #4656C4semidirect
20. #4650C4semidirect
373. C3#11717semidirect

Maximal subgroups

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

subgroupindex
#768102
#689512
#681002
C3×#13512
#676462
#674902
#674492
#117173