Finite Group Databasebeta

← #11780189

Subgroups — #11780189

Order 1152.

351 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 80 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
1Q8×D36C211yesmaximal, characteristic
2#10588707C211yesmaximal, characteristic, complemented
3#10588530C211yesmaximal, characteristic
4#10588362C211yesmaximal, characteristic, complemented
5C9×Q8⋊C8C211yesmaximal, characteristic, complemented
6#10588161C211yesmaximal, characteristic, complemented
7#10588136C211yesmaximal, characteristic
8#68273—31nomaximal
9C2×Q8×D9—21no—
10C4×Q8×C9C2211yescharacteristic
11C2×#3674—21no—
12#63459—21no—
13#63459—22no—
14C4×D36—21no—
15C4×D36C2211yescharacteristic
16#63431—21no—
17#63431C2211yescharacteristic
18#63431C2211yescharacteristic
19C9×C4⋊C8C2211yescharacteristic
20C4×C8×C9C2211yescharacteristic
21#63380—21no—
22#63378—21no—
23#63378—22no—
24#63368—21no—
25C4⋊(C9⋊C8)C2211yescharacteristic
26Q8×D12—31no—
27#5031—31no—
28#4846—31no—
29#4682—31no—
30C3×Q8⋊C8D311yescharacteristic
31#4486—31no—
32#4459—31no—
33C2×Q8×C9D411yescharacteristic
34Q8×D9—42no—
35Q8×D9—42no—
36C2×D36D411yescharacteristic
37C2×C4×D9—22no—
38C2×C4×D9—21no—
39C2×Dic18—22no—
40C2×Dic18—21no—
41C2×Dic18—21no—
42C2×C8×C9—21no—
43C2×C8×C9—21no—
44C9×C4⋊C4—21no—
45C9×C4⋊C4D411yescharacteristic
46C42×C9—21no—
47C42×C9C2311yescharacteristic
48#3674—42no—
49(C2×D9)⋊C4—22no—
50(C2×D9)⋊C4—21no—
51C4⋊Dic9—22no—
52C4⋊Dic9—21no—
53C4⋊Dic9—21no—
54C4⋊Dic9—21no—
55C4⋊Dic9D411yescharacteristic
56C4⋊Dic9D412yes—
57C2×C9⋊C8—21no—
58#1649—91noSylow
59C2×D3×Q8—61no—
60C3×C4×Q8C2×D311yescharacteristic
61C2×#216—61no—
62#897—62no—
63#897—61no—
64C4×D12—31no—
65C4×D12—61no—
66#869—31no—
67#869—31no—
68#869—61no—
69C3×C4⋊C8C2×D311yescharacteristic
70C3×C4×C8C2×D311yescharacteristic
71#821—61no—
72#818—62no—
73#818—61no—
74#810—61no—
75C4⋊(C3⋊C8)—31no—
76C22×D9—21no—
77Q8×C9—21no—
78Q8×C9D812yes—
79C2×C4×C9—21no—
80C2×C4×C9—21no—
81C2×C4×C9C2×D411yescharacteristic
82C2×C4×C9C2×D411yescharacteristic
83C2×C4×C9C2×D411yescharacteristic
84C2×Dic9—21no—
85C2×Dic9—22no—
86C2×Dic9—41no—
87D36—21no—
88D36Q1612yescomplemented
89C4×D9—44no—
90C4×D9—41no—
91C4×D9—41no—
92Dic18—44no—
93Dic18—41no—
94Dic18—41no—
95Dic18—41no—
96Dic18—41no—
97C8×C9—22no—
98C8×C9—41no—
99C9⋊C8—41no—
100D4×Q8—91no—
101#500—91no—
102#474—91no—
103#462—91no—
104#332—91no—
105Q8⋊C8D911yescharacteristic, complemented
106D4⋊C8—91no—
107C2×C3×Q8D1211yescharacteristic
108D3×Q8—122no—
109D3×Q8—122no—
110C2×D12—31no—
111C2×C4×D3—62no—
112C2×C4×D3—61no—
113C2×Dic6—62no—
114C2×Dic6—61no—
115C2×Dic6—61no—
116C2×C3×C8—21no—
117C2×C3×C8—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4D1211yescharacteristic
120C3×C42—21no—
121C3×C42C22×D311yescharacteristic
122#216—122no—
123(C2×D3)⋊C4—62no—
124(C2×D3)⋊C4—61no—
125C4⋊Dic3—32no—
126C4⋊Dic3—31no—
127C4⋊Dic3—62no—
128C4⋊Dic3—61no—
129C4⋊Dic3—61no—
130C4⋊Dic3—61no—
131C2×C3⋊C8—61no—
132C22×C9#12011yescharacteristic
133C2×D9—22no—
134C2×D9—41no—
135C2×D9—41no—
136C4×C9—22no—
137C4×C9—21no—
138C4×C9—41no—
139C4×C9C2×D811yescharacteristic
140C4×C9C2×Q1611yescharacteristic
141C4×C9AGL(1,ℤ/8)11yescharacteristic
142C4×C9#13711yescharacteristic
143Dic9—41no—
144Dic9—42no—
145Dic9—81no—
146C22×Q8—181no—
147C2×Q16—181no—
148C4⋊Q8—91no—
149C4⋊Q8—91no—
150C4⋊Q8—181no—
151C22⋊Q8—182no—
152C22⋊Q8—181no—
153C4×Q8C2×D911yescharacteristic
154C4×D4—91no—
155C4×D4—181no—
156#107—181no—
157#107—182no—
158C4⋊C8—91no—
159C4⋊C8C2×D911yescharacteristic
160#103—181no—
161#102—181no—
162C4×C8C2×D911yescharacteristic
163C22×D3—61no—
164C3×Q8—21no—
165C3×Q8D2412yes—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4C2×D1211yescharacteristic
169C2×C3×C4D3×D411yescharacteristic
170C2×C3×C4D3×D411yescharacteristic
171C2×Dic3—62no—
172C2×Dic3—61no—
173C2×Dic3—121no—
174D12—32no—
175D12—61no—
176C4×D3—124no—
177C4×D3—121no—
178C4×D3—121no—
179Dic6—124no—
180Dic6—121no—
181Dic6—121no—
182Dic6—121no—
183Dic6—121no—
184C3×C8—22no—
185C3×C8—41no—
186C3⋊C8—121no—
187C2×C9#44711yescharacteristic
188C2×C9#45111yescharacteristic
189C2×C9#45611yescharacteristic
190D9—42no—
191C2×Q8—182no—
192C2×Q8—181no—
193C2×Q8—181no—
194C2×Q8—362no—
195C2×Q8—362no—
196C2×Q8D3611yescharacteristic
197C2×D4—91no—
198C22×C4—182no—
199C22×C4—181no—
200Q16—362no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—181no—
204C4⋊C4—21no—
205C4⋊C4—91no—
206C4⋊C4—92no—
207C4⋊C4—182no—
208C4⋊C4—181no—
209C4⋊C4—181no—
210C4⋊C4—181no—
211C4⋊C4D3611yescharacteristic
212C22⋊C4—182no—
213C22⋊C4—181no—
214C42—21no—
215C42C22×D911yescharacteristic
216C22×C3#88211yescharacteristic
217C2×D3—62no—
218C2×D3—121no—
219C2×D3—121no—
220C3×C4—22no—
221C3×C4—21no—
222C3×C4—41no—
223C3×C4C2×D2411yescharacteristic
224C3×C4#90811yescharacteristic
225C3×C4#91511yescharacteristic
226C3×C4D3×Q1611yescharacteristic
227Dic3—121no—
228Dic3—122no—
229Dic3—241no—
230C9#164911yescharacteristic, complemented, Sylow
231C23—181no—
232Q8—21no—
233Q8—364no—
234Q8—361no—
235Q8—361no—
236Q8—361no—
237Q8—361no—
238Q8D7212yes—
239D4—92no—
240D4—181no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—182no—
244C2×C4—181no—
245C2×C4—364no—
246C2×C4—361no—
247C2×C4—361no—
248C2×C4—361no—
249C2×C4C2×D3611yescharacteristic
250C2×C4D4×D911yescharacteristic
251C2×C4D4×D911yescharacteristic
252C8—22no—
253C8—41no—
254C8—361no—
255C2×C3#473211yescharacteristic
256C2×C3#480911yescharacteristic
257C2×C3#482611yescharacteristic
258D3—122no—
259C22—182no—
260C22—361no—
261C22—361no—
262C22#6344411yescharacteristic
263C4—21no—
264C4—22no—
265C4—41no—
266C4—362no—
267C4—361no—
268C4—721no—
269C4C2×D7211yescharacteristic
270C4#6347011yescharacteristic
271C4#6347711yescharacteristic
272C4Q16×D911yescharacteristic
273C3#6827311yescharacteristic
274C2—362no—
275C2#1058841411yescharacteristic
276C2#1058848911yescharacteristic
277C2#1058850611yescharacteristic
subgroupG/H
centerC22#63444
commutator, cosocleC42×C9C23
FrattiniC3×C42C22×D3
FittingC9×Q8⋊C8C2
socleC22×C3#882
prime radicalC2×D36D4
hypercenterQ8⋊C8D9
hypocenterC9#1649
LCSG ▷ C42×C9 ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #10588707C2semidirect
4. #10588362C2semidirect
5. C9×Q8⋊C8C2semidirect
6. #10588161C2semidirect
88. D36Q16semidirect
105. Q8⋊C8D9semidirect
230. C9#1649semidirect

Maximal subgroups

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

subgroupindex
Q8×D362
#105887072
#105885302
#105883622
C9×Q8⋊C82
#105881612
#105881362
#682733