Finite Group Databasebeta

← #10692944

Subgroups — #10692944

Order 768.

738 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 131 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#82349C211yesmaximal, characteristic
2#82301C211yesmaximal, characteristic
3C3×#1799C211yesmaximal, characteristic
4#69525C212yesmaximal, complemented
5#67625C212yesmaximal, complemented
6#9467—31nomaximal, Sylow
7C23×Dic6C2211yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C2×#924—21no—
10C2×C3×#99—21no—
11C3×#380C2212yes—
12#5220—21no—
13#5212C2211yescharacteristic
14C2×#830—21no—
15C2×#830C2212yes—
16D4×C3⋊C8—24no—
17#4979C412yescomplemented
18C2×#810—24no—
19#4947C412yescomplemented
20#4946—21no—
21#4929—21no—
22#4925—21no—
23C3×#342—21no—
24#2972—31no—
25#2409—31no—
26#1799D311yescharacteristic
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—
38C3×C4×D4—22no—
39C3×C4×D4—22no—
40C2×C3×C4⋊C4—22no—
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4C2×C411yescharacteristic
44C2×C3×C22⋊C4D411yescharacteristic
45C2×C3×C42D411yescharacteristic
46C2×#217—22no—
47C2×#217—21no—
48C2×C4⋊Dic3—21no—
49#924—44no—
50C2×Dic3⋊C4—22no—
51C22×C3⋊C8—22no—
52C22×C3⋊C8D414yes—
53C3×#99—42no—
54C3×#95—22no—
55C3⋊#95—22no—
56C3⋊#95—22no—
57C3⋊#95C2×C412yes—
58#830—22no—
59#830—22no—
60#830—22no—
61#810—48no—
62C4⋊(C3⋊C8)—22no—
63C4×C3⋊C8—22no—
64C23×Q8—31no—
65C2×C22⋊Q8—61no—
66C2×C4×D4C2×D311yescharacteristic
67C8×D4—64no—
68#431—61no—
69C2×#103—64no—
70C2×#99—21no—
71C2×C22⋊C8—32no—
72C2×C22⋊C8—61no—
73#395—32no—
74#393—32no—
75#391—61no—
76#386—61no—
77#385—61no—
78#380—31no—
79#380C2×D312yes—
80#342—21no—
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×C4—42no—
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22×C411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D412yes—
98C22×C3×C4#4112yes—
99C22×Dic3—22no—
100C22×Dic3—22no—
101C22×Dic3—44no—
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—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—44no—
114C3×C22⋊C4—22no—
115C3×C22⋊C4—22no—
116C3×C22⋊C4—42no—
117C3×C22⋊C4—42no—
118C3×C42—21no—
119C3×C42—21no—
120#217—44no—
121#217—42no—
122C4⋊Dic3—42no—
123Dic3⋊C4—44no—
124C2×C3⋊C8—24no—
125C2×C3⋊C8—44no—
126C2×C3⋊C8—82no—
127C22×Q8—32no—
128C22×Q8—61no—
129C22×Q8—62no—
130C22×Q8—62no—
131C22×Q8—124no—
132C22×D4#6711yescharacteristic
133C23×C4—61no—
134C23×C4D1211yescharacteristic
135C23×C4C22×D311yescharacteristic
136C22×C8—34no—
137C22×C8—62no—
138C22⋊Q8—124no—
139C4×D4—22no—
140C4×D4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—62no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4C4×D311yescharacteristic
145C2×C22⋊C4—22no—
146C2×C22⋊C4—62no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4C4×D311yescharacteristic
149C2×C22⋊C4#6711yescharacteristic
150C2×C42D1211yescharacteristic
151C4⋊C8—62no—
152#103—128no—
153#99—42no—
154C22⋊C8—62no—
155C22⋊C8—62no—
156C22⋊C8—62no—
157C4×C8—62no—
158#95—22no—
159#95—32no—
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—82no—
189C2×C3×C4—82no—
190C2×C3×C4—82no—
191C2×C3×C4#10312yescomplemented
192C2×C3×C4C4×D412yes—
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12311yescharacteristic
195C2×Dic3—24no—
196C2×Dic3—48no—
197C2×Dic3—42no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—42no—
201C2×Dic3—42no—
202Dic6—28no—
203Dic6—44no—
204Dic6—44no—
205Dic6—44no—
206C3⋊C8—44no—
207C24(C2×D3)⋊C411yescharacteristic
208C24C2×#6711yescharacteristic
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#21412yes—
221C2×D4#21612yes—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—62no—
228C22×C4—62no—
229C22×C4—124no—
230C22×C4—122no—
231C22×C4(C2×D3)⋊C411yescharacteristic
232C22×C4(C2×D3)⋊C411yescharacteristic
233C22×C4(C2×D3)⋊C411yescharacteristic
234C22×C4C2×C4×D311yescharacteristic
235C22×C4C2×D1211yescharacteristic
236C22×C4D3×D412yes—
237C22×C4#23712yes—
238C2×C8—64no—
239C2×C8—124no—
240C2×C8—242no—
241C4⋊C4—22no—
242C4⋊C4—44no—
243C4⋊C4—122no—
244C4⋊C4—124no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—124no—
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×C4—162no—
276C3×C4B(4,𝔽2)12yes—
277Dic3—44no—
278Dic3—82no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23#81012yescomplemented
286C23C2×(C2×D3)⋊C411yescharacteristic
287C23C2×#21411yescharacteristic
288C23#93912yes—
289C23C2×#21611yescharacteristic
290C23#95012yes—
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—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—122no—
314C2×C4—128no—
315C2×C4—122no—
316C2×C4—122no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4#81012yescomplemented
320C2×C4#67⋊C412yes—
321C2×C4#88211yescharacteristic
322C2×C4#88611yescharacteristic
323C8—124no—
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—41no—
339C22—42no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22#495811yescharacteristic
345C22C2×#81011yescharacteristic
346C22#500211yescharacteristic
347C22#504112yes—
348C22#505112yes—
349C4—21no—
350C4—22no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—124no—
355C4—162no—
356C4—242no—
357C4#474112yes—
358C3#946711yescharacteristic, complemented, Sylow
359C2—22no—
360C2—21no—
361C2—21no—
362C2—22no—
363C2—42no—
364C2#6951411yescharacteristic
365C2#7002211yescharacteristic
366C2#7004611yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1799C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC23×C3×C4C23
hypercenter#1799D3
hypocenterC3#9467
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
4. #69525C2semidirect
5. #67625C2semidirect
17. #4979C4semidirect
19. #4947C4semidirect
168. C23×C3#103semidirect
191. C2×C3×C4#103semidirect
285. C23#810semidirect
319. C2×C4#810semidirect
358. C3#9467semidirect

Maximal subgroups

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

subgroupindex
#823492
#823012
C3×#17992
#695252
#676252
#94673