Finite Group Databasebeta

← #10691260

Subgroups — #10691260

Order 768.

690 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2312C211yesmaximal, characteristic, complemented
2#82130C211yesmaximal, characteristic
3C2×#4525C211yesmaximal, characteristic, complemented
4#69509C212yesmaximal, complemented
5#67625C212yesmaximal, complemented
6#9205—31nomaximal, Sylow
7C22×C3×C22⋊C4C2211yescharacteristic
8C2×#953—21no—
9C2×#930C2211yescharacteristic
10C3×#388—24no—
11C3×#380C2212yes—
12C3×#379—21no—
13C3×C4×C22⋊C4—21no—
14#5213—21no—
15#5212C2211yescharacteristic
16C2×#830C2212yes—
17#4961—22no—
18#4960C412yescomplemented
19#4953C412yescomplemented
20#4525—22no—
21#4451—24no—
22#2617—31no—
23#2312D311yescharacteristic, complemented
24#1519—32no—
25C2×#323—31no—
26#1289—32no—
27C25×C3D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×#67—21no—
31C22×#67—22no—
32C22×D12C2×C411yescharacteristic
33C2×C3×C4⋊C4—22no—
34C2×C3×C22⋊C4—22no—
35C2×C3×C22⋊C4—22no—
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4C2×C412yes—
40C2×C3×C22⋊C4D412yes—
41C2×C3×C42—22no—
42#953—44no—
43C2×#217—22no—
44C2×#217—21no—
45#930—22no—
46#930—22no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—22no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3C2×C411yescharacteristic
51C22×C3⋊C8—22no—
52C3×#95—22no—
53C3×#95D414yescomplemented
54C3⋊#95—22no—
55C3⋊#95C2×C412yes—
56#830—22no—
57#830—22no—
58C2×#121—31no—
59C2×#120—61no—
60C22×C22⋊C4C2×D311yescharacteristic
61C2×C22⋊C8—32no—
62#397—32no—
63#394—62no—
64#392—32no—
65#388—24no—
66#386—61no—
67#380—31no—
68#380C2×D312yes—
69#379—21no—
70C4×C22⋊C4—21no—
71#327—64no—
72#323—62no—
73C24×C3—21no—
74C24×C3—21no—
75C24×C3—22no—
76C24×C3—22no—
77C24×C3—81no—
78C24×C3—81no—
79C24×C3C22⋊C412yes—
80C24×C3C2×D411yescharacteristic
81C23×D3—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—24no—
85C22×C3×C4—81no—
86C22×C3×C4—81no—
87C22×C3×C4C22⋊C412yes—
88C22×C3×C4C22×C411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D412yes—
91C22×C3×C4#4112yes—
92C2×#67—44no—
93C2×#67—41no—
94C2×#67—41no—
95C2×#67—42no—
96C2×#67—42no—
97C2×#67—82no—
98C2×#67—82no—
99C22×Dic3—21no—
100C22×Dic3—22no—
101C2×D12—22no—
102C2×D12—81no—
103C3×C4⋊C4—82no—
104C3×C22⋊C4—24no—
105C3×C22⋊C4—24no—
106C3×C22⋊C4—42no—
107C3×C22⋊C4—42no—
108C3×C22⋊C4—82no—
109C3×C22⋊C4—82no—
110C3×C42—82no—
111#217—44no—
112#217—42no—
113(C2×D3)⋊C4—42no—
114(C2×D3)⋊C4—82no—
115C4⋊Dic3—22no—
116C4⋊Dic3—81no—
117C2×C3⋊C8—42no—
118C2×C3⋊C8—82no—
119C25#6711yescharacteristic
120C22×D4—31no—
121C22×D4—61no—
122C22×D4—62no—
123C23×C4#6711yescharacteristic
124C23×C4C22×D311yescharacteristic
125C22×C8—62no—
126#121—62no—
127#121—62no—
128#120—124no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—61no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—62no—
138C2×C22⋊C4—61no—
139C2×C22⋊C4—61no—
140C2×C22⋊C4—62no—
141C2×C22⋊C4C4×D312yes—
142C2×C22⋊C4D1212yes—
143C2×C42—22no—
144C22⋊C8—62no—
145C22⋊C8—62no—
146#95—22no—
147#95—32no—
148#95—62no—
149#95#6714yescomplemented
150C23×C3—22no—
151C23×C3—22no—
152C23×C3—22no—
153C23×C3—41no—
154C23×C3—41no—
155C23×C3—42no—
156C23×C3—42no—
157C23×C3—42no—
158C23×C3—42no—
159C23×C3—42no—
160C23×C3—42no—
161C23×C3—81no—
162C23×C3—81no—
163C23×C3—84no—
164C23×C3—84no—
165C23×C3C2×C22⋊C411yescharacteristic
166C23×C3#12011yescharacteristic
167C23×C3#12311yescharacteristic
168C23×C3#13512yes—
169C23×C3AGL(1,ℤ/8)12yes—
170C22×D3—42no—
171C22×D3—42no—
172C22×D3—41no—
173C22×D3—41no—
174C22×D3—41no—
175C2×C3×C4—22no—
176C2×C3×C4—42no—
177C2×C3×C4—41no—
178C2×C3×C4—42no—
179C2×C3×C4—41no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—44no—
183C2×C3×C4—44no—
184C2×C3×C4—44no—
185C2×C3×C4—81no—
186C2×C3×C4C4×D412yes—
187C2×C3×C4#12112yes—
188#67—88no—
189#67—82no—
190#67—82no—
191C2×Dic3—42no—
192C2×Dic3—42no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—42no—
196C2×Dic3—41no—
197D12—82no—
198C3⋊C8—82no—
199C24—21no—
200C24—21no—
201C24—22no—
202C24—22no—
203C24—61no—
204C24—81no—
205C24—81no—
206C24(C2×D3)⋊C412yes—
207C24C2×#6711yescharacteristic
208C2×D4—62no—
209C2×D4—124no—
210C2×D4—121no—
211C2×D4—122no—
212C2×D4—122no—
213C2×D4—121no—
214C2×D4—241no—
215C2×D4—242no—
216C2×D4—242no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—62no—
221C22×C4—61no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4(C2×D3)⋊C412yes—
225C22×C4C2×C4×D311yescharacteristic
226C22×C4C2×D1211yescharacteristic
227C22×C4#23512yes—
228C22×C4C2×#6712yes—
229C2×C8—122no—
230C2×C8—242no—
231C4⋊C4—62no—
232C4⋊C4—82no—
233C4⋊C4—241no—
234C22⋊C4—24no—
235C22⋊C4—24no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—82no—
239C22⋊C4—82no—
240C22⋊C4—124no—
241C22⋊C4—122no—
242C22⋊C4—122no—
243C22⋊C4—242no—
244C42—82no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—42no—
252C22×C3—42no—
253C22×C3—44no—
254C22×C3—42no—
255C22×C3—42no—
256C22×C3—42no—
257C22×C3—82no—
258C22×C3—84no—
259C22×C3—84no—
260C22×C3#38611yescharacteristic
261C22×C3#41611yescharacteristic
262C22×C3#41811yescharacteristic
263C22×C3#44912yes—
264C22×C3#48212yes—
265C2×D3—82no—
266C2×D3—82no—
267C2×D3—81no—
268C2×D3—81no—
269C2×D3—81no—
270C3×C4—22no—
271C3×C4—22no—
272C3×C4—42no—
273C3×C4—84no—
274Dic3—82no—
275Dic3—81no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—41no—
280C23—41no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—81no—
288C23—81no—
289C23—84no—
290C23—84no—
291C23—122no—
292C23—122no—
293C23—121no—
294C23—121no—
295C23—121no—
296C23C2×(C2×D3)⋊C411yescharacteristic
297C23#92911yescharacteristic
298C23#94912yes—
299C23#95012yes—
300C23#95311yescharacteristic
301D4—242no—
302D4—248no—
303D4—242no—
304D4—242no—
305C2×C4—22no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—81no—
316C2×C4—122no—
317C2×C4—122no—
318C2×C4—121no—
319C2×C4—122no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4C4×#6712yes—
323C2×C4#93012yes—
324C8—242no—
325C2×C3—21no—
326C2×C3—22no—
327C2×C3—21no—
328C2×C3—22no—
329C2×C3—82no—
330C2×C3#187511yescharacteristic
331C2×C3#191111yescharacteristic
332C2×C3#191311yescharacteristic
333D3—161no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—42no—
341C22—42no—
342C22—44no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—82no—
347C22—84no—
348C22—84no—
349C22—242no—
350C22—242no—
351C22—241no—
352C22—241no—
353C22—241no—
354C22#500111yescharacteristic
355C22#500211yescharacteristic
356C22#500312yes—
357C22#521311yescharacteristic
358C22#523812yes—
359C4—22no—
360C4—22no—
361C4—42no—
362C4—84no—
363C4—242no—
364C4—241no—
365C3#920511yescharacteristic, complemented, Sylow
366C2—22no—
367C2—21no—
368C2—21no—
369C2—22no—
370C2—82no—
371C2—481no—
372C2#7158611yescharacteristic
373C2#7165211yescharacteristic
374C2#7165611yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2312C2
socleC22×C3#386
prime radical#82130C2
cosocleC23×C3×C4C23
hypercenter#2312D3
hypocenterC3#9205
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
1. C3×#2312C2semidirect
3. C2×#4525C2semidirect
4. #69509C2semidirect
5. #67625C2semidirect
18. #4960C4semidirect
19. #4953C4semidirect
23. #2312D3semidirect
53. C3×#95D4semidirect
149. #95#67semidirect
365. C3#9205semidirect

Maximal subgroups

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

subgroupindex
C3×#23122
#821302
C2×#45252
#695092
#676252
#92053