Finite Group Databasebeta

← #10687635

Subgroups — #10687635

Order 768.

736 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×#5672C211yesmaximal, characteristic
2C2×C3×#329C211yesmaximal, characteristic
3#71726C211yesmaximal, characteristic
4#69827C212yesmaximal, complemented
5C2×#4465C212yesmaximal, complemented
6#8507—31nomaximal, Sylow
7C2×C3×#124C2211yescharacteristic
8C2×#944—21no—
9C2×#935—21no—
10#5672—22no—
11#5672—24no—
12#5672C414yescomplemented
13C2×#879—22no—
14C2×#878—22no—
15C2×#870C2212yes—
16C2×C3×#97C2212yes—
17C2×#835—21no—
18C2×#832—21no—
19#4935C2211yescharacteristic
20C2×#803C2211yescharacteristic
21C3×#329—22no—
22#4551—22no—
23#4551—22no—
24#4465—24no—
25C2×#563—31no—
26#1992—32no—
27#1959—31no—
28C2×#330—32no—
29C2×#329D311yescharacteristic
30C22×C3×Q8D411yescharacteristic
31C22×C3×D4D411yescharacteristic
32C23×Dic3—21no—
33C22×Dic6—21no—
34C22×C3×C8—22no—
35C3×#124—22no—
36C3×#124C2×C412yes—
37C3×#124D412yes—
38C2×C3×C22⋊C4—22no—
39C2×C3×C42C2311yescharacteristic
40C2×#217—22no—
41C2×#217—21no—
42#944—24no—
43#944—24no—
44#935—24no—
45#935—24no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—22no—
48C2×C4⋊Dic3D412yes—
49C2×C4⋊Dic3D412yes—
50C2×Dic3⋊C4—22no—
51C2×Dic3⋊C4—22no—
52C2×Dic3⋊C4—22no—
53C2×C4×Dic3—21no—
54C22×C3⋊C8—21no—
55#879—28no—
56#879—28no—
57#878—28no—
58#878—28no—
59#870—22no—
60#870—22no—
61#870C2×C414yes—
62#870D414yes—
63C3×#97—22no—
64C3×#97—22no—
65#835—42no—
66#832—42no—
67#803—22no—
68#563—34no—
69#563—64no—
70#563—62no—
71C2×C4⋊Q8—61no—
72C2×#126—62no—
73C2×#125—32no—
74C2×#124C2×D311yescharacteristic
75C2×#123—61no—
76C2×C22⋊Q8—62no—
77C2×#103—61no—
78C2×#102—61no—
79C2×#97—31no—
80C2×#97C2×D312yes—
81#384—31no—
82#340—62no—
83#340—62no—
84#330—64no—
85#329—22no—
86C24×C3—21no—
87C2×C3×Q8—21no—
88C2×C3×Q8—41no—
89C2×C3×Q8—41no—
90C2×C3×Q8C22⋊C412yes—
91C2×C3×D4—21no—
92C2×C3×D4—41no—
93C2×C3×D4—41no—
94C2×C3×D4C22⋊C412yes—
95C22×C3×C4—21no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D412yes—
98C22×Dic3—22no—
99C22×Dic3—22no—
100C22×Dic3—24no—
101C22×Dic3—41no—
102C22×Dic3—81no—
103C22×Dic3—81no—
104C2×Dic6—24no—
105C2×Dic6—81no—
106C2×Dic6—81no—
107C2×C3×C8—24no—
108C2×C3×C8—42no—
109C2×C3×C8—42no—
110C3×C22⋊C4—24no—
111C3×C22⋊C4—24no—
112C3×C42C22×C411yescharacteristic
113C3×C42C2×D412yes—
114C3×C42C2×D411yescharacteristic
115#217—28no—
116#217—24no—
117C4⋊Dic3—22no—
118C4⋊Dic3—22no—
119C4⋊Dic3—41no—
120C4⋊Dic3—41no—
121C4⋊Dic3—44no—
122C4⋊Dic3C22⋊C414yes—
123C4⋊Dic3C22⋊C414yes—
124Dic3⋊C4—28no—
125Dic3⋊C4—24no—
126Dic3⋊C4—24no—
127Dic3⋊C4—24no—
128Dic3⋊C4—24no—
129C4×Dic3—24no—
130C2×C3⋊C8—22no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8—41no—
133C22×Q8—61no—
134C22×Q8D1211yescharacteristic
135C22×D4D1211yescharacteristic
136C23×C4—61no—
137C22×C8—22no—
138C22×C8—61no—
139C4⋊Q8—64no—
140C4⋊Q8—64no—
141#126—68no—
142#126—68no—
143#125—34no—
144#125—34no—
145#125—62no—
146#125—62no—
147#124—22no—
148#124C4×D312yes—
149#124#6712yes—
150#123—64no—
151#123—64no—
152C22⋊Q8—68no—
153C22⋊Q8—68no—
154C2×C4⋊C4—32no—
155C2×C4⋊C4—32no—
156C2×C4⋊C4—62no—
157C2×C4⋊C4—62no—
158C2×C4⋊C4—62no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—62no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—62no—
163C2×C22⋊C4—61no—
164C2×C42—61no—
165C2×C42C22×D311yescharacteristic
166#103—122no—
167#102—122no—
168#97—22no—
169#97—22no—
170#97—62no—
171C23×C3—22no—
172C23×C3—41no—
173C23×C3—42no—
174C23×C3—41no—
175C23×C3—42no—
176C23×C3#12011yescharacteristic
177C3×Q8—42no—
178C3×Q8—81no—
179C3×D4—42no—
180C3×D4—81no—
181C2×C3×C4—21no—
182C2×C3×C4—22no—
183C2×C3×C4—21no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4C2×C22⋊C412yes—
189C2×C3×C4C2×C22⋊C411yescharacteristic
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12012yes—
192C2×Dic3—24no—
193C2×Dic3—24no—
194C2×Dic3—42no—
195C2×Dic3—42no—
196C2×Dic3—42no—
197C2×Dic3—42no—
198C2×Dic3—84no—
199C2×Dic3—81no—
200C2×Dic3—82no—
201Dic6—84no—
202C3×C8—44no—
203C3⋊C8—42no—
204C24—21no—
205C2×Q8—21no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—64no—
209C2×Q8—241no—
210C2×Q8—241no—
211C2×Q8(C2×D3)⋊C412yes—
212C2×D4—21no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4(C2×D3)⋊C412yes—
216C22×C4—21no—
217C22×C4—62no—
218C22×C4—62no—
219C22×C4—64no—
220C22×C4—121no—
221C22×C4—241no—
222C22×C4—241no—
223C22×C4C2×D1211yescharacteristic
224C22×C4D3×D412yes—
225C2×C8—24no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—62no—
229C2×C8—121no—
230C2×C8—121no—
231C4⋊C4—34no—
232C4⋊C4—34no—
233C4⋊C4—68no—
234C4⋊C4—64no—
235C4⋊C4—64no—
236C4⋊C4—64no—
237C4⋊C4—62no—
238C4⋊C4—62no—
239C4⋊C4—64no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C4⋊C4—124no—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—68no—
246C22⋊C4—64no—
247C42—64no—
248C42C2×C4×D311yescharacteristic
249C42D3×D412yes—
250C42C2×#6711yescharacteristic
251C22×C3—42no—
252C22×C3—44no—
253C22×C3—81no—
254C22×C3—81no—
255C22×C3—82no—
256C22×C3#37911yescharacteristic
257C22×C3#45311yescharacteristic
258C22×C3#45412yes—
259C22×C3#45511yescharacteristic
260C22×C3#45612yes—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—22no—
264C3×C4—22no—
265C3×C4—42no—
266Dic3—44no—
267Dic3—44no—
268Dic3—161no—
269C23—22no—
270C23—41no—
271C23—41no—
272C23—42no—
273C23—42no—
274C23#88211yescharacteristic
275Q8—42no—
276Q8—81no—
277Q8—244no—
278D4—42no—
279D4—81no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—64no—
288C2×C4—64no—
289C2×C4—122no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4—122no—
293C2×C4—244no—
294C2×C4—241no—
295C2×C4—242no—
296C2×C4D3×C22⋊C412yes—
297C2×C4#88211yescharacteristic
298C2×C4C2×(C2×D3)⋊C411yescharacteristic
299C2×C4#93712yes—
300C8—44no—
301C8—122no—
302C2×C3—82no—
303C2×C3#165711yescharacteristic
304C2×C3#166011yescharacteristic
305C2×C3#166412yes—
306C2×C3#179211yescharacteristic
307C2×C3#179312yes—
308C22—42no—
309C22—44no—
310C22—81no—
311C22—81no—
312C22—82no—
313C22#474611yescharacteristic
314C22#474711yescharacteristic
315C22#482612yes—
316C22#482712yes—
317C22#495511yescharacteristic
318C4—21no—
319C4—21no—
320C4—22no—
321C4—22no—
322C4—42no—
323C4—124no—
324C4—124no—
325C4—481no—
326C3#850711yescharacteristic, complemented, Sylow
327C2—82no—
328C2#6835411yescharacteristic
329C2#6835511yescharacteristic
330C2#6835912yes—
331C2#7008611yescharacteristic
332C2#7038712yes—
subgroupG/H
centerC23#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#329C2
socleC23×C3#120
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenterC2×#329D3
hypocenterC3#8507
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
4. #69827C2semidirect
5. C2×#4465C2semidirect
12. #5672C4semidirect
326. C3#8507semidirect

Maximal subgroups

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

subgroupindex
C2×#56722
C2×C3×#3292
#717262
#698272
C2×#44652
#85073