Finite Group Databasebeta

← #10692502

Subgroups — #10692502

Order 768.

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

normal subgroups only

This group has 570 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
1C3×#2972C211yesmaximal, characteristic
2#82385C211yesmaximal, characteristic
3#71619C211yesmaximal, characteristic
4#69946C212yesmaximal, complemented
5#67675C212yesmaximal, complemented
6#9383—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#924C2211yescharacteristic
9C3×C8×D4—24no—
10C3×#431—21no—
11C2×C3×C22⋊C8—21no—
12C2×C3×C22⋊C8C2212yes—
13#5220—21no—
14C2×#834—21no—
15#5212C2212yes—
16#5210—21no—
17C2×C3⋊#95C2211yescharacteristic
18#4986C411yescharacteristic, complemented
19#4979C411yescharacteristic, complemented
20#4949C411yescharacteristic, complemented
21#4947C411yescharacteristic, complemented
22#4942—21no—
23#4933—21no—
24#4526—21no—
25#4467—24no—
26#2972D311yescharacteristic
27#2564—31no—
28#1799—31no—
29#1520—32no—
30#1294—32no—
31C22×C3×D4D411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4C2311yescharacteristic
34C23×Dic3—21no—
35C22×Dic6C2×C411yescharacteristic
36C22×C3×C8—22no—
37C22×C3×C8—22no—
38C3×C4×D4—22no—
39C3×C4×D4—22no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C22⋊C4C2×C411yescharacteristic
42C2×C3×C22⋊C4D411yescharacteristic
43C2×C3×C42D411yescharacteristic
44C2×#217—21no—
45C2×#217—21no—
46C2×#217—22no—
47C2×#217—21no—
48C2×C4⋊Dic3—22no—
49C2×C4⋊Dic3C2×C411yescharacteristic
50#924—22no—
51#924—22no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C3×C4⋊C8—22no—
56C3×C22⋊C8—22no—
57C3×C22⋊C8—22no—
58C3×C22⋊C8D414yes—
59C3×C4×C8—22no—
60#834—42no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—22no—
67C3⋊#95C2×C411yescharacteristic
68C3⋊#95C2×C411yescharacteristic
69C2×C22⋊Q8—31no—
70C2×C4×D4C2×D311yescharacteristic
71C8×D4—24no—
72#431—21no—
73C2×#99—61no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8C2×D312yes—
76#400—31no—
77#399—31no—
78#395—31no—
79#393—31no—
80#387—61no—
81#386—61no—
82#385—61no—
83#380—32no—
84#380—61no—
85C2×#95—31no—
86#342—61no—
87#328—64no—
88C24×C3C22⋊C411yescharacteristic
89C24×C3C2×D411yescharacteristic
90C2×C3×D4—22no—
91C2×C3×D4—22no—
92C2×C3×D4SD1612yes—
93C2×C3×D4Q1612yes—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—81no—
97C22×C3×C4—81no—
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C2×D412yes—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4112yes—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—42no—
110C22×Dic3—42no—
111C22×Dic3—42no—
112C22×Dic3—81no—
113C2×Dic6—22no—
114C2×Dic6—81no—
115C2×C3×C8—24no—
116C2×C3×C8—82no—
117C2×C3×C8—82no—
118C3×C4⋊C4—22no—
119C3×C22⋊C4—22no—
120C3×C22⋊C4—22no—
121C3×C42—21no—
122C3×C42—21no—
123#217—42no—
124#217—42no—
125#217—42no—
126#217—42no—
127#217—81no—
128C4⋊Dic3—22no—
129C4⋊Dic3—82no—
130Dic3⋊C4—42no—
131Dic3⋊C4—41no—
132Dic3⋊C4—41no—
133Dic3⋊C4—81no—
134C22×Q8—31no—
135C22×D4D1211yescharacteristic
136C23×C4—61no—
137C23×C4#6711yescharacteristic
138C23×C4C22×D311yescharacteristic
139C22×C8—22no—
140C22×C8—22no—
141C22⋊Q8—62no—
142C22⋊Q8—62no—
143C4×D4—22no—
144C4×D4—22no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—62no—
150C2×C4⋊C4C4×D311yescharacteristic
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C22⋊C4—62no—
154C2×C22⋊C4—61no—
155C2×C22⋊C4C4×D311yescharacteristic
156C2×C22⋊C4D1211yescharacteristic
157C2×C42#6711yescharacteristic
158C4⋊C8—22no—
159#99—122no—
160C22⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8#6714yes—
163C4×C8—22no—
164#95—31no—
165#95—31no—
166#95—61no—
167#95—61no—
168#95—61no—
169#95—61no—
170#95—61no—
171#95—62no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3—41no—
178C23×C3#10312yes—
179C23×C3C2×C22⋊C411yescharacteristic
180C23×C3#12112yes—
181C23×C3C2×SD1611yescharacteristic
182C23×C3C2×Q1611yescharacteristic
183C23×C3#13512yes—
184C3×D4—24no—
185C3×D4—24no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—42no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—81no—
198C2×C3×C4#10312yes—
199C2×C3×C4C4×D412yes—
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12311yescharacteristic
202C2×Dic3—22no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—84no—
215C2×Dic3—82no—
216C2×Dic3—82no—
217C2×Dic3—82no—
218Dic6—82no—
219C3×C8—44no—
220C24(C2×D3)⋊C411yescharacteristic
221C24C2×D1211yescharacteristic
222C2×Q8—62no—
223C2×Q8—241no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4#20412yes—
227C2×D4Dic1212yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—81no—
235C22×C4—81no—
236C22×C4—122no—
237C22×C4—122no—
238C22×C4—122no—
239C22×C4—241no—
240C22×C4(C2×D3)⋊C411yescharacteristic
241C22×C4(C2×D3)⋊C411yescharacteristic
242C22×C4(C2×D3)⋊C411yescharacteristic
243C22×C4C2×C4×D311yescharacteristic
244C22×C4#23512yes—
245C22×C4C2×#6712yes—
246C22×C4C2×#6711yescharacteristic
247C2×C8—24no—
248C2×C8—82no—
249C2×C8—82no—
250C4⋊C4—22no—
251C4⋊C4—62no—
252C4⋊C4—122no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C4⋊C4—241no—
256C4⋊C4—242no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—122no—
260C22⋊C4—122no—
261C22⋊C4—122no—
262C22⋊C4—122no—
263C22⋊C4—241no—
264C42—21no—
265C42—21no—
266C22×C3—21no—
267C22×C3—22no—
268C22×C3—21no—
269C22×C3—22no—
270C22×C3—22no—
271C22×C3—42no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—42no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—81no—
278C22×C3#38611yescharacteristic
279C22×C3C2×#10311yescharacteristic
280C22×C3#41611yescharacteristic
281C22×C3#46512yes—
282C22×C3#46712yes—
283C3×C4—22no—
284C3×C4—21no—
285C3×C4—22no—
286C3×C4—21no—
287C3×C4—41no—
288C3×C4—41no—
289Dic3—42no—
290Dic3—81no—
291Dic3—81no—
292Dic3—81no—
293Dic3—162no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23#81612yes—
301C23C2×#20411yescharacteristic
302C23#90412yes—
303C23C2×Dic1211yescharacteristic
304C23C2×(C2×D3)⋊C411yescharacteristic
305C23#93012yes—
306Q8—242no—
307D4—24no—
308D4—24no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—62no—
321C2×C4—81no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—244no—
334C2×C4—242no—
335C2×C4—242no—
336C2×C4—242no—
337C2×C4#81612yes—
338C2×C4C4×#6712yes—
339C2×C4#92911yescharacteristic
340C2×C4#95311yescharacteristic
341C8—44no—
342C2×C3—21no—
343C2×C3—22no—
344C2×C3—21no—
345C2×C3—22no—
346C2×C3—42no—
347C2×C3#188411yescharacteristic
348C2×C3#189211yescharacteristic
349C2×C3#191311yescharacteristic
350C22—21no—
351C22—22no—
352C22—21no—
353C22—22no—
354C22—22no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—42no—
359C22—41no—
360C22—41no—
361C22—81no—
362C22C2×#81611yescharacteristic
363C22#511311yescharacteristic
364C22#511412yes—
365C22#511612yes—
366C22#521311yescharacteristic
367C4—22no—
368C4—21no—
369C4—21no—
370C4—22no—
371C4—41no—
372C4—41no—
373C4—122no—
374C4—241no—
375C4—241no—
376C4—241no—
377C4—482no—
378C3#938311yescharacteristic, complemented, Sylow
379C2—21no—
380C2—22no—
381C2—21no—
382C2—22no—
383C2—42no—
384C2#7165611yescharacteristic
385C2#7197211yescharacteristic
386C2#7202111yescharacteristic
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#9383
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. #69946C2semidirect
5. #67675C2semidirect
18. #4986C4semidirect
19. #4979C4semidirect
20. #4949C4semidirect
21. #4947C4semidirect
378. C3#9383semidirect

Maximal subgroups

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

subgroupindex
C3×#29722
#823852
#716192
#699462
#676752
#93833