Finite Group Databasebeta

← #10692946

Subgroups — #10692946

Order 768.

738 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C3×#2972C211yesmaximal, characteristic
2#82349C211yesmaximal, characteristic
3#71619C211yesmaximal, characteristic
4#69963C212yesmaximal, complemented
5#67676C212yesmaximal, complemented
6#9467—31nomaximal, Sylow
7C23×Dic6C2211yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C2×#924—21no—
10C3×C8×D4—24no—
11C3×#431—21no—
12C2×C3×C22⋊C8—21no—
13C2×C3×C22⋊C8C2212yes—
14#5220—21no—
15C2×#834—21no—
16#5212C2212yes—
17#5212C2211yescharacteristic
18C2×#816—24no—
19#4979C412yescomplemented
20#4947C412yescomplemented
21#4946—21no—
22#4929—21no—
23#4526—21no—
24#2972D311yescharacteristic
25#2409—31no—
26#1799—31no—
27#1787—32no—
28#1289—32no—
29C22×C3×D4D411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4C2311yescharacteristic
32C23×Dic3—21no—
33C22×Dic6—22no—
34C22×Dic6—21no—
35C22×Dic6—22no—
36C22×Dic6—44no—
37C22×Dic6C2×C412yes—
38C22×C3×C8—22no—
39C22×C3×C8D414yes—
40C3×C4×D4—22no—
41C3×C4×D4—22no—
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C22⋊C4C2×C411yescharacteristic
44C2×C3×C22⋊C4D411yescharacteristic
45C2×C3×C42D411yescharacteristic
46C2×#217—22no—
47C2×#217—21no—
48C2×#217—22no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—22no—
51#924—44no—
52C2×Dic3⋊C4—22no—
53C3×C4⋊C8—22no—
54C3×C22⋊C8—22no—
55C3×C22⋊C8—22no—
56C3×C22⋊C8—22no—
57C3×C4×C8—22no—
58#834—42no—
59C3⋊#95—22no—
60C3⋊#95—22no—
61C3⋊#95—22no—
62C3⋊#95C2×C412yes—
63#816—48no—
64C23×Q8—31no—
65C2×C22⋊Q8—61no—
66C2×C4×D4C2×D311yescharacteristic
67C8×D4—24no—
68#431—21no—
69C2×#103—64no—
70C2×#99—61no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8C2×D312yes—
73#395—32no—
74#393—32no—
75#391—61no—
76#386—61no—
77#385—61no—
78#380—31no—
79#380—32no—
80#342—61no—
81C24×C3C22⋊C411yescharacteristic
82C24×C3C2×D411yescharacteristic
83C2×C3×D4—22no—
84C2×C3×D4—22no—
85C2×C3×D4SD1612yes—
86C2×C3×D4Q1612yes—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—42no—
90C22×C3×C4—42no—
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D412yes—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4#4112yes—
98C22×Dic3—22no—
99C22×Dic3—22no—
100C22×Dic3—44no—
101C22×Dic3—42no—
102C22×Dic3—42no—
103C2×Dic6—28no—
104C2×Dic6—24no—
105C2×Dic6—24no—
106C2×Dic6—44no—
107C2×Dic6—48no—
108C2×Dic6—44no—
109C2×Dic6—42no—
110C2×Dic6—41no—
111C2×Dic6—41no—
112C2×C3×C8—24no—
113C2×C3×C8—44no—
114C2×C3×C8—82no—
115C3×C4⋊C4—22no—
116C3×C22⋊C4—22no—
117C3×C22⋊C4—22no—
118C3×C42—21no—
119C3×C42—21no—
120#217—44no—
121#217—42no—
122#217—42no—
123#217—42no—
124C4⋊Dic3—42no—
125C4⋊Dic3—44no—
126Dic3⋊C4—44no—
127C22×Q8—32no—
128C22×Q8—61no—
129C22×Q8—62no—
130C22×Q8—62no—
131C22×Q8—124no—
132C22×D4D1211yescharacteristic
133C23×C4—61no—
134C23×C4#6711yescharacteristic
135C23×C4C22×D311yescharacteristic
136C22×C8—22no—
137C22×C8#6714yes—
138C22⋊Q8—124no—
139C4×D4—22no—
140C4×D4—22no—
141C2×C4⋊C4—62no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—62no—
144C2×C4⋊C4C4×D311yescharacteristic
145C2×C22⋊C4—62no—
146C2×C22⋊C4—61no—
147C2×C22⋊C4—62no—
148C2×C22⋊C4C4×D311yescharacteristic
149C2×C22⋊C4D1211yescharacteristic
150C2×C42#6711yescharacteristic
151C4⋊C8—22no—
152#103—128no—
153#99—122no—
154C22⋊C8—22no—
155C22⋊C8—22no—
156C22⋊C8—22no—
157C4×C8—22no—
158#95—32no—
159#95—62no—
160#95—62no—
161#95—62no—
162C23×C3—21no—
163C23×C3—21no—
164C23×C3—41no—
165C23×C3—41no—
166C23×C3—41no—
167C23×C3—41no—
168C23×C3#10312yescomplemented
169C23×C3C2×C22⋊C411yescharacteristic
170C23×C3#12112yes—
171C23×C3C2×SD1611yescharacteristic
172C23×C3C2×Q1611yescharacteristic
173C23×C3#13512yes—
174C3×D4—24no—
175C3×D4—24no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—24no—
179C2×C3×C4—22no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—41no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—81no—
188C2×C3×C4#10312yescomplemented
189C2×C3×C4C4×D412yes—
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12311yescharacteristic
192C2×Dic3—24no—
193C2×Dic3—48no—
194C2×Dic3—42no—
195C2×Dic3—42no—
196C2×Dic3—42no—
197C2×Dic3—42no—
198C2×Dic3—42no—
199C2×Dic3—82no—
200C2×Dic3—82no—
201C2×Dic3—82no—
202Dic6—28no—
203Dic6—44no—
204Dic6—44no—
205Dic6—44no—
206C3×C8—44no—
207C24(C2×D3)⋊C411yescharacteristic
208C24C2×D1211yescharacteristic
209C2×Q8—64no—
210C2×Q8—68no—
211C2×Q8—64no—
212C2×Q8—124no—
213C2×Q8—122no—
214C2×Q8—124no—
215C2×Q8—128no—
216C2×Q8—121no—
217C2×Q8—121no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4#20412yes—
221C2×D4Dic1212yes—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—62no—
227C22×C4—62no—
228C22×C4—124no—
229C22×C4—122no—
230C22×C4—122no—
231C22×C4(C2×D3)⋊C411yescharacteristic
232C22×C4(C2×D3)⋊C411yescharacteristic
233C22×C4(C2×D3)⋊C411yescharacteristic
234C22×C4C2×C4×D311yescharacteristic
235C22×C4#23512yes—
236C22×C4C2×#6712yes—
237C22×C4C2×#6711yescharacteristic
238C2×C8—24no—
239C2×C8—44no—
240C2×C8—82no—
241C4⋊C4—22no—
242C4⋊C4—122no—
243C4⋊C4—124no—
244C4⋊C4—124no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—124no—
248C22⋊C4—122no—
249C22⋊C4—122no—
250C22⋊C4—122no—
251C42—21no—
252C42—21no—
253C22×C3—21no—
254C22×C3—22no—
255C22×C3—21no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—42no—
261C22×C3—42no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—81no—
265C22×C3#38611yescharacteristic
266C22×C3C2×#10311yescharacteristic
267C22×C3#41611yescharacteristic
268C22×C3#46512yes—
269C22×C3#46712yes—
270C3×C4—22no—
271C3×C4—21no—
272C3×C4—22no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4B(4,𝔽2)12yes—
276Dic3—44no—
277Dic3—82no—
278Dic3—162no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23#81612yescomplemented
286C23C2×#20411yescharacteristic
287C23#90412yes—
288C23C2×Dic1211yescharacteristic
289C23C2×(C2×D3)⋊C411yescharacteristic
290C23#93012yes—
291Q8—68no—
292Q8—124no—
293Q8—124no—
294Q8—124no—
295D4—24no—
296D4—24no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—64no—
309C2×C4—81no—
310C2×C4—122no—
311C2×C4—128no—
312C2×C4—122no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4—242no—
317C2×C4—242no—
318C2×C4—242no—
319C2×C4#81612yescomplemented
320C2×C4C4×#6712yes—
321C2×C4#92911yescharacteristic
322C2×C4#95311yescharacteristic
323C8—44no—
324C2×C3—21no—
325C2×C3—22no—
326C2×C3—21no—
327C2×C3—22no—
328C2×C3—42no—
329C2×C3#188911yescharacteristic
330C2×C3#189211yescharacteristic
331C2×C3#190911yescharacteristic
332C22—21no—
333C22—22no—
334C22—21no—
335C22—22no—
336C22—22no—
337C22—41no—
338C22—42no—
339C22—41no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22C2×#81611yescharacteristic
345C22#511311yescharacteristic
346C22#511412yes—
347C22#511612yes—
348C22#521311yescharacteristic
349C4—22no—
350C4—21no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—124no—
355C4—242no—
356C4—482no—
357C4#524412yes—
358C3#946711yescharacteristic, complemented, Sylow
359C2—21no—
360C2—22no—
361C2—21no—
362C2—22no—
363C2—42no—
364C2#7165411yescharacteristic
365C2#7197211yescharacteristic
366C2#7202411yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2972C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC23×C3×C4C23
hypercenter#2972D3
hypocenterC3#9467
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
4. #69963C2semidirect
5. #67676C2semidirect
19. #4979C4semidirect
20. #4947C4semidirect
168. C23×C3#103semidirect
188. C2×C3×C4#103semidirect
285. C23#816semidirect
319. C2×C4#816semidirect
358. C3#9467semidirect

Maximal subgroups

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

subgroupindex
C3×#29722
#823492
#716192
#699632
#676762
#94673