Finite Group Databasebeta

← #10707403

Subgroups — #10707403

Order 768.

570 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 63 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×#2840C211yesmaximal, characteristic, complemented
2#69699C212yesmaximal, complemented
3#69500C211yesmaximal, characteristic
4#67646C212yesmaximal, complemented
5#67622C211yesmaximal, characteristic, complemented
6#12123—31nomaximal, Sylow
7C2×#953—21no—
8#5493C2211yescharacteristic
9C3×#393—22no—
10C3×#393—21no—
11C3×#393—22no—
12C3×#393C2212yes—
13C3×#379—21no—
14C3×#379C2211yescharacteristic
15#5227—22no—
16C2×#806—21no—
17#4526C2211yescharacteristic
18#4525C2212yes—
19#4451—22no—
20#2840D311yescharacteristic, complemented
21#2021—31no—
22#1601—32no—
23#1351—32no—
24#1316—31no—
25C25×C3D411yescharacteristic
26C22×C3×Q8—21no—
27C22×C3×Q8D412yescomplemented
28C22×#67—21no—
29C22×#67—41no—
30C2×C3×C22⋊C4—21no—
31C2×C3×C22⋊C4—22no—
32C2×C3×C22⋊C4—21no—
33C2×C3×C22⋊C4—22no—
34C2×C3×C22⋊C4—22no—
35C2×C3×C22⋊C4D411yescharacteristic
36C2×C3×C22⋊C4C2311yescharacteristic
37#953—42no—
38#953—42no—
39C2×#217—21no—
40C2×#217—41no—
41C2×#215—42no—
42#930—22no—
43#886—41no—
44#882—41no—
45C3×#95—21no—
46C3×#95—22no—
47C3×#95—22no—
48C3×#95D412yescomplemented
49#835—42no—
50#830—22no—
51#806—42no—
52#534—31no—
53C2×#120—61no—
54#448—62no—
55C2×#99—61no—
56#393—22no—
57#393—21no—
58#393—22no—
59#393C2×D312yes—
60#379—21no—
61#379C2×D311yescharacteristic
62#342—31no—
63#327—62no—
64#323—32no—
65C24×C3—21no—
66C24×C3—21no—
67C24×C3—41no—
68C24×C3—41no—
69C24×C3—82no—
70C24×C3SD1612yescomplemented
71C24×C3C2×D411yescharacteristic
72C23×D3—21no—
73C2×C3×Q8—42no—
74C2×C3×Q8—82no—
75C2×C3×Q8—82no—
76C2×C3×Q8—81no—
77C22×C3×C4—22no—
78C22×C3×C4—21no—
79C22×C3×C4—22no—
80C22×C3×C4C2×D412yes—
81C2×#67—41no—
82C2×#67—41no—
83C2×#67—41no—
84C2×#67—44no—
85C2×#67—41no—
86C2×#67—41no—
87C2×#67—41no—
88C2×#67—84no—
89C22×Dic3—21no—
90C22×Dic3—41no—
91C2×D12—22no—
92C3×C22⋊C4—41no—
93C3×C22⋊C4—41no—
94C3×C22⋊C4—82no—
95C3×C22⋊C4—82no—
96C3×C22⋊C4—82no—
97C3×C22⋊C4—81no—
98C3×C22⋊C4—81no—
99#217—22no—
100#217—44no—
101#217—41no—
102#215—84no—
103(C2×D3)⋊C4—42no—
104C4⋊Dic3—22no—
105C2×C3⋊C8—42no—
106C25#6711yescharacteristic
107C22×Q8—21no—
108C22×Q8#6712yescomplemented
109C22×D4—61no—
110C22×D4—121no—
111C2×SD16—122no—
112#123—121no—
113#121—62no—
114#120—121no—
115#120—122no—
116#120—122no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—61no—
123C2×C22⋊C4—121no—
124C2×C22⋊C4#6711yescharacteristic
125C2×C22⋊C4C22×D311yescharacteristic
126#103—122no—
127#99—122no—
128C22⋊C8—62no—
129#95—21no—
130#95—22no—
131#95—22no—
132#95#6712yescomplemented
133C23×C3—21no—
134C23×C3—21no—
135C23×C3—22no—
136C23×C3—42no—
137C23×C3—42no—
138C23×C3—42no—
139C23×C3—42no—
140C23×C3—44no—
141C23×C3—41no—
142C23×C3—41no—
143C23×C3—81no—
144C23×C3—82no—
145C23×C3—82no—
146C23×C3—82no—
147C23×C3—84no—
148C23×C3#12011yescharacteristic
149C23×C3C2×SD1611yescharacteristic
150C23×C3#13711yescharacteristic
151C22×D3—41no—
152C22×D3—41no—
153C22×D3—41no—
154C22×D3—82no—
155C3×Q8—44no—
156C3×Q8—82no—
157C3×Q8—84no—
158C2×C3×C4—22no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—41no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168#67—81no—
169#67—81no—
170#67—88no—
171#67—81no—
172#67—81no—
173C2×Dic3—42no—
174C2×Dic3—41no—
175C2×Dic3—41no—
176C2×Dic3—41no—
177C2×Dic3—82no—
178D12—82no—
179C3⋊C8—82no—
180C24—21no—
181C24—21no—
182C24—41no—
183C24—41no—
184C24—61no—
185C24—82no—
186C24#21512yescomplemented
187C24C2×#6711yescharacteristic
188C2×Q8—42no—
189C2×Q8—82no—
190C2×Q8—82no—
191C2×Q8—81no—
192C2×D4—62no—
193C2×D4—121no—
194C2×D4—121no—
195C2×D4—121no—
196C2×D4—124no—
197C2×D4—121no—
198C2×D4—121no—
199C2×D4—121no—
200C2×D4—244no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—61no—
205C22×C4—121no—
206C22×C4C2×#6712yes—
207SD16—244no—
208C2×C8—122no—
209C4⋊C4—62no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—62no—
213C22⋊C4—82no—
214C22⋊C4—82no—
215C22⋊C4—82no—
216C22⋊C4—81no—
217C22⋊C4—81no—
218C22⋊C4—122no—
219C22⋊C4—124no—
220C22⋊C4—121no—
221C22×C3—22no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—41no—
225C22×C3—41no—
226C22×C3—42no—
227C22×C3—42no—
228C22×C3—42no—
229C22×C3—44no—
230C22×C3—42no—
231C22×C3—81no—
232C22×C3—82no—
233C22×C3—82no—
234C22×C3—84no—
235C22×C3—82no—
236C22×C3#44811yescharacteristic
237C22×C3#45311yescharacteristic
238C22×C3B(4,𝔽2)11yescharacteristic
239C2×D3—82no—
240C2×D3—81no—
241C2×D3—82no—
242C2×D3—161no—
243C3×C4—42no—
244C3×C4—81no—
245C3×C4—82no—
246C3×C4—82no—
247Dic3—82no—
248Dic3—81no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—44no—
257C23—41no—
258C23—41no—
259C23—81no—
260C23—82no—
261C23—82no—
262C23—82no—
263C23—84no—
264C23—121no—
265C23—121no—
266C23—121no—
267C23—242no—
268C23C2×#21511yescharacteristic
269C23#94211yescharacteristic
270C23#95311yescharacteristic
271Q8—44no—
272Q8—82no—
273Q8—84no—
274D4—242no—
275D4—241no—
276D4—241no—
277D4—248no—
278D4—241no—
279D4—241no—
280C2×C4—22no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—122no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—242no—
295C8—242no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—41no—
299C2×C3—41no—
300C2×C3—82no—
301C2×C3#159611yescharacteristic
302C2×C3#219912yes—
303D3—161no—
304C22—22no—
305C22—21no—
306C22—21no—
307C22—41no—
308C22—41no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—44no—
313C22—42no—
314C22—81no—
315C22—82no—
316C22—82no—
317C22—84no—
318C22—82no—
319C22—242no—
320C22—242no—
321C22—241no—
322C22—481no—
323C22#522711yescharacteristic
324C22#524111yescharacteristic
325C22#524411yescharacteristic
326C4—42no—
327C4—81no—
328C4—82no—
329C4—82no—
330C4—242no—
331C4—241no—
332C3#1212311yescharacteristic, complemented, Sylow
333C2—21no—
334C2—21no—
335C2—41no—
336C2—41no—
337C2—82no—
338C2—481no—
339C2#7189211yescharacteristic
340C2#7195012yes—
subgroupG/H
centerC22#5244
commutator, cosocleC2×C3×C22⋊C4C23
FrattiniC2×C22⋊C4C22×D3
FittingC3×#2840C2
socleC22×C3B(4,𝔽2)
prime radical#69500C2
hypercenter#2840D3
hypocenterC3#12123
LCSG ▷ C2×C3×C22⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2840

Complements

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

NH ≅ G/Ntype
1. C3×#2840C2semidirect
2. #69699C2semidirect
4. #67646C2semidirect
5. #67622C2semidirect
20. #2840D3semidirect
27. C22×C3×Q8D4semidirect
48. C3×#95D4semidirect
70. C24×C3SD16semidirect
108. C22×Q8#67semidirect
132. #95#67semidirect
186. C24#215semidirect
332. C3#12123semidirect

Maximal subgroups

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

subgroupindex
C3×#28402
#696992
#695002
#676462
#676222
#121233