Finite Group Databasebeta

← #11920406

Subgroups — #11920406

Order 1152.

602 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C23×Q8⋊C9C211yesmaximal, characteristic, complemented
2C2×#63689C211yesmaximal, characteristic, complemented
3#10589523C212yesmaximal, complemented
4#10589522C211yesmaximal, characteristic, complemented
5#10589521C212yesmaximal, complemented
6#84059—31nomaximal
7C22×Q8⋊C9—22no—
8C22×Q8⋊C9C2212yes—
9C22×Q8⋊C9C2211yescharacteristic
10#63689—22no—
11C2×#3689C2211yescharacteristic, complemented
12C2×#3688C2211yescharacteristic, complemented
13#63515—41nomaximal
14#63421C2212yescomplemented
15C23×C3×Q8D311yescharacteristic
16C2×#942—31no—
17C2×#942—31no—
18#5791—31no—
19#5228—32no—
20#5228—32no—
21#5227—32no—
22#5227—32no—
23#5225—31no—
24#5225—31no—
25#5207—31no—
26C24×C9—41no—
27C2×#611—41no—
28C2×#611—42no—
29C2×Q8⋊C9—22no—
30C2×Q8⋊C9D412yescomplemented
31C2×Q8⋊C9C2311yescharacteristic
32#3689—21no—
33#3688—21no—
34#3676—42no—
35#3676—41no—
36#3004—91noSylow
37C22×C3×Q8—22no—
38C22×C3×Q8—31no—
39C22×C3×Q8—32no—
40C22×C3×Q8—62no—
41C22×C3×Q8C2×D312yes—
42C22×C3×Q8C2×D311yescharacteristic
43C23×C3×C4—31no—
44C2×#235—31no—
45#953—61no—
46C2×#216—31no—
47C2×#216—32no—
48C2×#216—31no—
49#942—62no—
50#942—64no—
51#942—62no—
52C2×#215—31no—
53C2×#215—32no—
54C2×#215—31no—
55#930—32no—
56#929—61no—
57C4×#67—62no—
58#926—31no—
59#924—32no—
60C2×#208—31no—
61C2×#208—31no—
62#835—32no—
63#835—32no—
64#835—32no—
65#835—32no—
66#830—32no—
67#830—32no—
68C23×C9—42no—
69C23×C9—41no—
70C23×C9—84no—
71C23×C9—82no—
72C22×D9—41no—
73#611—84no—
74#611—82no—
75C2×Dic9—42no—
76C2×Dic9—41no—
77Q8⋊C9C2×D411yescharacteristic
78C23×Q8D911yescharacteristic, complemented
79C2×#137—91no—
80C2×#137—91no—
81#525—91no—
82#451—92no—
83#451—92no—
84#448—92no—
85#448—92no—
86#419—91no—
87#419—91no—
88#407—91no—
89C24×C3S411yescharacteristic
90C2×C3×Q8—22no—
91C2×C3×Q8—31no—
92C2×C3×Q8—32no—
93C2×C3×Q8—32no—
94C2×C3×Q8—34no—
95C2×C3×Q8—61no—
96C2×C3×Q8—62no—
97C2×C3×Q8—64no—
98C2×C3×Q8—62no—
99C2×C3×Q8—64no—
100C2×C3×Q8#6712yes—
101C2×C3×Q8C22×D311yescharacteristic
102C22×C3×C4—31no—
103C22×C3×C4—32no—
104C22×C3×C4—32no—
105C22×C3×C4—31no—
106C22×C3×C4—62no—
107C22×C3×C4—62no—
108C2×#67—62no—
109C2×#67—61no—
110#235—62no—
111#235—62no—
112C2×D12—31no—
113C2×C4×D3—61no—
114C2×Dic6—31no—
115#217—62no—
116#217—61no—
117#216—62no—
118#216—61no—
119#216—64no—
120#216—61no—
121#215—62no—
122#215—61no—
123#215—64no—
124#215—61no—
125(C2×D3)⋊C4—62no—
126C4⋊Dic3—32no—
127Dic3⋊C4—62no—
128C4×Dic3—61no—
129#208—62no—
130#208—62no—
131C2×C3⋊C8—31no—
132C2×C3⋊C8—31no—
133C2×C3⋊C8—32no—
134C22×C9—44no—
135C22×C9—42no—
136C22×C9—41no—
137C22×C9—82no—
138C22×C9—82no—
139C22×C9—82no—
140C22×C9—88no—
141C2×D9—81no—
142C2×D9—82no—
143Dic9—82no—
144Dic9—81no—
145C2×#41—91no—
146C22×Q8—22no—
147C22×Q8—31no—
148C22×Q8—32no—
149C22×Q8—62no—
150C22×Q8C2×D912yes—
151C22×Q8C2×D911yescharacteristic
152C23×C4—31no—
153#137—182no—
154#137—184no—
155#137—182no—
156C2×Q16—92no—
157C2×Q16—91no—
158C2×Q16—91no—
159C2×SD16—92no—
160C2×SD16—91no—
161C2×SD16—91no—
162C2×M16—91no—
163C2×M16—91no—
164#123—181no—
165C22⋊Q8—92no—
166#121—92no—
167#120—181no—
168C4×D4—182no—
169#117—91no—
170#103—92no—
171#103—92no—
172#103—92no—
173#103—92no—
174C22⋊C8—92no—
175C22⋊C8—92no—
176C23×C3—24no—
177C23×C3—22no—
178C23×C3C2×S412yes—
179C23×C3C2×S411yescharacteristic
180C22×D3—61no—
181C3×Q8—32no—
182C3×Q8—31no—
183C3×Q8—34no—
184C3×Q8—61no—
185C3×Q8—62no—
186C3×Q8—64no—
187C3×Q8C2×#6711yescharacteristic
188C2×C3×C4—32no—
189C2×C3×C4—32no—
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—32no—
193C2×C3×C4—62no—
194C2×C3×C4—62no—
195C2×C3×C4—64no—
196C2×C3×C4—62no—
197#67—124no—
198#67—122no—
199C2×Dic3—62no—
200C2×Dic3—61no—
201D12—61no—
202D12—61no—
203C4×D3—122no—
204Dic6—61no—
205Dic6—61no—
206C3⋊C8—62no—
207C3⋊C8—61no—
208C3⋊C8—61no—
209C2×C9—42no—
210C2×C9—41no—
211C2×C9—84no—
212C2×C9—82no—
213D9—161no—
214C24(C22⋊C9)⋊C211yescharacteristic
215#41—182no—
216#41—182no—
217C2×Q8—22no—
218C2×Q8—31no—
219C2×Q8—32no—
220C2×Q8—32no—
221C2×Q8—34no—
222C2×Q8—61no—
223C2×Q8—62no—
224C2×Q8—64no—
225C2×Q8—62no—
226C2×Q8—64no—
227C2×Q8—91no—
228C2×Q8#61112yescomplemented
229C2×Q8C22×D911yescharacteristic
230C2×D4—91no—
231C2×D4—182no—
232C2×D4—181no—
233C22×C4—31no—
234C22×C4—32no—
235C22×C4—32no—
236C22×C4—31no—
237C22×C4—62no—
238C22×C4—62no—
239C22×C4—181no—
240Q16—182no—
241Q16—181no—
242Q16—184no—
243Q16—181no—
244SD16—182no—
245SD16—181no—
246SD16—184no—
247SD16—181no—
248M16—182no—
249M16—182no—
250C2×C8—91no—
251C2×C8—91no—
252C2×C8—92no—
253C4⋊C4—92no—
254C4⋊C4—182no—
255C22⋊C4—182no—
256C22⋊C4—181no—
257C22⋊C4—182no—
258C42—181no—
259C22×C3—28no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—22no—
263C22×C3#98314yes—
264C22×C3GL(2,ℤ/4)12yes—
265C22×C3C22×S411yescharacteristic
266C2×D3—122no—
267C2×D3—121no—
268C3×C4—31no—
269C3×C4—31no—
270C3×C4—32no—
271C3×C4—62no—
272Dic3—122no—
273Dic3—121no—
274C9—41noSylow
275C23—24no—
276C23—22no—
277C23—181no—
278C23C2×(C22⋊C9)⋊C212yes—
279C23C2×(C22⋊C9)⋊C211yescharacteristic
280Q8—32no—
281Q8—31no—
282Q8—34no—
283Q8—61no—
284Q8—62no—
285Q8—64no—
286Q8—181no—
287Q8—181no—
288Q8C2×#61111yescharacteristic
289D4—181no—
290D4—181no—
291D4—364no—
292D4—362no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—32no—
296C2×C4—32no—
297C2×C4—32no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—64no—
301C2×C4—62no—
302C2×C4—182no—
303C2×C4—181no—
304C2×C4—362no—
305C8—182no—
306C8—181no—
307C8—181no—
308C2×C3—24no—
309C2×C3—22no—
310C2×C3C2×#98312yes—
311C2×C3C2×GL(2,ℤ/4)11yescharacteristic
312D3—241no—
313C22—28no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—362no—
318C22—361no—
319C22#6368914yescomplemented
320C22#6369412yes—
321C22C22×(C22⋊C9)⋊C211yescharacteristic
322C4—31no—
323C4—31no—
324C4—32no—
325C4—62no—
326C4—362no—
327C4—361no—
328C3#8550711yescharacteristic
329C2—24no—
330C2—22no—
331C2—721no—
332C2C2×#6368912yes—
333C2C2×#6369411yescharacteristic
subgroupG/H
centerC22C22×(C22⋊C9)⋊C2
commutator, cosocleC2×Q8⋊C9C23
Frattini, socleC22×C3C22×S4
FittingC23×C3×Q8D3
hypercenterC24(C22⋊C9)⋊C2
hypocenterQ8⋊C9C2×D4
LCSG ▷ C2×Q8⋊C9 ▷ Q8⋊C9
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×Q8⋊C9C2semidirect
2. C2×#63689C2semidirect
3. #10589523C2semidirect
4. #10589522C2semidirect
5. #10589521C2semidirect
11. C2×#3689C22semidirect
12. C2×#3688C22semidirect
14. #63421C22semidirect
30. C2×Q8⋊C9D4semidirect
78. C23×Q8D9semidirect
228. C2×Q8#611semidirect
319. C22#63689semidirect

Maximal subgroups

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

subgroupindex
C23×Q8⋊C92
C2×#636892
#105895232
#105895222
#105895212
#840593
#635154