Finite Group Databasebeta

← #11791080

Subgroups — #11791080

Order 1152.

681 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 681 conjugacy classes of subgroups — too many to draw. Only its 164 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
1C22×#3718C211yesmaximal, characteristic
2C2×#63630C211yesmaximal, characteristic
3C3×#4929C212yesmaximal
4C2×#63574C211yesmaximal, characteristic
5C2×#63565C212yesmaximal
6#69678—32nomaximal
7C22×C3×Dic6C2212yes—
8C2×C32×C42C2211yescharacteristic
9C22×C3⋊(C3⋊C8)C2211yescharacteristic
10C22×C3⋊(C3⋊C8)C2211yescharacteristic
11C2×C3×C4⋊Dic3C2212yes—
12C2×#3718—22no—
13C2×#3718—22no—
14C2×#3718C414yescomplemented
15#63630—22no—
16C3×C3⋊#95—24no—
17#63574—22no—
18#63565—24no—
19C22×#216—32no—
20C3×#391—32no—
21C2×#835—32no—
22C2×#810—32no—
23C2×#808—32no—
24#4929D312yes—
25C2×#803—32no—
26C2×#803—61no—
27C22×C32×C4D412yes—
28C22×C32×C4C2311yescharacteristic
29C22×C3×Dic3—22no—
30C22×C3×Dic3—22no—
31C2×C3×Dic6—22no—
32C2×C3×Dic6—22no—
33C2×C3×Dic6—22no—
34C2×C3×Dic6—22no—
35C2×C3×Dic6—22no—
36C2×C3×Dic6C2×C414yes—
37C32×C42—21no—
38C32×C42—21no—
39C2×C3⋊(C3⋊C8)—21no—
40C2×C3⋊(C3⋊C8)—21no—
41C2×C3⋊(C3⋊C8)C2×C412yes—
42C2×C3⋊(C3⋊C8)D412yes—
43C2×C3⋊(C3⋊C8)D412yes—
44C2×C3⋊(C3⋊C8)D412yes—
45C3×C4⋊Dic3—22no—
46C3×C4⋊Dic3—22no—
47#3718—28no—
48#1971—91noSylow
49C22×C3×Q8—32no—
50C22×Dic6C2×D312yes—
51C2×C3×C4⋊C4—32no—
52C2×C3×C42—21no—
53C2×C3×C42C2×D312yes—
54C2×#216—38no—
55C2×#216—64no—
56C2×#216—64no—
57C2×C4⋊Dic3C2×D312yes—
58C22×C3⋊C8—32no—
59C22×C3⋊C8—32no—
60C22×C3⋊C8—61no—
61C22×C3⋊C8—61no—
62C3×#95—64no—
63#835—64no—
64C3⋊#95—24no—
65#810—64no—
66#808—64no—
67#803—64no—
68#803—64no—
69C23×C32C2×D411yescharacteristic
70C2×C32×C4—22no—
71C2×C32×C4—24no—
72C2×C32×C4C22×C411yescharacteristic
73C2×C32×C4C2×D411yescharacteristic
74C2×C32×C4C2×D411yescharacteristic
75C2×C32×C4C2×D411yescharacteristic
76C2×C32×C4#4111yescharacteristic
77C2×C32×C4#4111yescharacteristic
78C2×C3×Dic3—24no—
79C2×C3×Dic3—42no—
80C2×C3×Dic3—42no—
81C2×C3×Dic3—42no—
82C2×C3×Dic3—44no—
83C3×Dic6—24no—
84C3×Dic6—24no—
85C3×Dic6—42no—
86C3×Dic6—42no—
87C3⋊(C3⋊C8)—22no—
88C3⋊(C3⋊C8)C22⋊C414yes—
89C22×Q16—91no—
90C2×#106—91no—
91C2×#103—92no—
92C2×#97—91no—
93#391—32no—
94C2×C3×Q8—34no—
95C2×C3×Q8—62no—
96C2×C3×Q8—62no—
97C2×C3×Q8—62no—
98C2×C3×Q8—62no—
99C2×C3×Q8—62no—
100C22×C3×C4—22no—
101C22×C3×C4—21no—
102C22×C3×C4—62no—
103C22×C3×C4—62no—
104C22×C3×C4#6714yes—
105C22×C3×C4C22×D312yes—
106C22×Dic3—22no—
107C22×Dic3—22no—
108C2×Dic6—22no—
109C2×Dic6—22no—
110C2×Dic6—22no—
111C2×Dic6—22no—
112C2×Dic6—22no—
113C2×Dic6C4×D314yes—
114C3×C4⋊C4—62no—
115C3×C4⋊C4—62no—
116C3×C42—22no—
117C3×C42—22no—
118C3×C42—22no—
119C3×C42—22no—
120#216—616no—
121C4⋊Dic3—22no—
122C4⋊Dic3—22no—
123C2×C3⋊C8—34no—
124C2×C3⋊C8—34no—
125C2×C3⋊C8—34no—
126C2×C3⋊C8—34no—
127C2×C3⋊C8—62no—
128C2×C3⋊C8—62no—
129C2×C3⋊C8—62no—
130C2×C3⋊C8—62no—
131C2×C3⋊C8—62no—
132C2×C3⋊C8—62no—
133C2×C3⋊C8—62no—
134C2×C3⋊C8—62no—
135C22×C32C4×D411yescharacteristic
136C22×C32#12111yescharacteristic
137C22×C32#12711yescharacteristic
138C22×C32#13714yes—
139C32×C4—42no—
140C32×C4C2×C22⋊C411yescharacteristic
141C32×C4C4×D411yescharacteristic
142C32×C4#12111yescharacteristic
143C32×C4#12411yescharacteristic
144C3×Dic3—44no—
145C3×Dic3—82no—
146C22×Q8—32no—
147C2×Q16—94no—
148C2×Q16—182no—
149C2×Q16—182no—
150C22×C8—91no—
151C22×C8—91no—
152C2×C4⋊C4—32no—
153C2×C42D3211yescharacteristic
154#106—182no—
155#103—184no—
156#97—182no—
157#95—64no—
158C23×C3—21no—
159C23×C3C2×#6712yes—
160C3×Q8—64no—
161C3×Q8—64no—
162C3×Q8—122no—
163C3×Q8—122no—
164C2×C3×C4—24no—
165C2×C3×C4—24no—
166C2×C3×C4—24no—
167C2×C3×C4—24no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—44no—
175C2×C3×C4—64no—
176C2×C3×C4—122no—
177C2×C3×C4—122no—
178C2×C3×C4—122no—
179C2×C3×C4—124no—
180C2×C3×C4C2×C4×D312yes—
181C2×C3×C4#23512yes—
182C2×C3×C4D3×D412yes—
183C2×C3×C4D3×D412yes—
184C2×C3×C4D3×D412yes—
185C2×C3×C4#23712yes—
186C2×Dic3—24no—
187C2×Dic3—42no—
188C2×Dic3—42no—
189C2×Dic3—42no—
190C2×Dic3—44no—
191Dic6—24no—
192Dic6—24no—
193Dic6—42no—
194Dic6—42no—
195C3⋊C8—38no—
196C3⋊C8—64no—
197C3⋊C8—64no—
198C3⋊C8—64no—
199C2×C32#39011yescharacteristic
200C2×C32#44112yes—
201C2×C32#47012yes—
202C2×C32#49712yes—
203C2×Q8—34no—
204C2×Q8—62no—
205C2×Q8—62no—
206C2×Q8—62no—
207C2×Q8—62no—
208C2×Q8—62no—
209C22×C4—62no—
210C22×C4—62no—
211C22×C4#62512yes—
212C22×C4C2×D3211yescharacteristic
213Q16—188no—
214C2×C8—92no—
215C2×C8—92no—
216C2×C8—92no—
217C2×C8—92no—
218C2×C8—181no—
219C2×C8—181no—
220C4⋊C4—62no—
221C4⋊C4—62no—
222C42—21no—
223C42—21no—
224C22×C3—24no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3C4×#6712yes—
229C22×C3#93912yes—
230C22×C3#94012yes—
231C22×C3#94214yes—
232C22×C3#95114yes—
233C3×C4—21no—
234C3×C4—21no—
235C3×C4—21no—
236C3×C4—21no—
237C3×C4—44no—
238C3×C4—44no—
239C3×C4—124no—
240C3×C4—242no—
241C3×C4D3×C22⋊C412yes—
242C3×C4#67⋊C412yes—
243C3×C4#88412yes—
244C3×C4#88512yes—
245Dic3—44no—
246Dic3—82no—
247C32#197111yescharacteristic, complemented, Sylow
248C23C2×#62511yescharacteristic
249Q8—64no—
250Q8—64no—
251Q8—122no—
252Q8—122no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—64no—
256C2×C4—122no—
257C2×C4—122no—
258C2×C4—124no—
259C2×C4—122no—
260C2×C4#379811yescharacteristic
261C2×C4C4×D3211yescharacteristic
262C2×C4#380211yescharacteristic
263C2×C4#380211yescharacteristic
264C2×C4#380511yescharacteristic
265C2×C4#381111yescharacteristic
266C8—94no—
267C8—182no—
268C2×C3—21no—
269C2×C3—22no—
270C2×C3—22no—
271C2×C3—22no—
272C2×C3#495712yes—
273C2×C3#503014yes—
274C2×C3#505214yes—
275C2×C3#506012yes—
276C2×C3#508612yes—
277C22#6382714yes—
278C22C4×#62511yescharacteristic
279C22#6391411yescharacteristic
280C22#6398011yescharacteristic
281C4—42no—
282C4—124no—
283C4—242no—
284C4#6390211yescharacteristic
285C4#6391211yescharacteristic
286C4#6391511yescharacteristic
287C4#6392111yescharacteristic
288C3—21no—
289C3#6967812yescomplemented
290C2#1059037612yes—
291C2#1059038112yes—
292C2#1059118912yes—
293C2#1059138511yescharacteristic
subgroupG/H
centerC23C2×#625
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×C42C22
socle, prime radicalC23×C32C2×D4
cosocleC22×C32×C4C23
hypercenterC2×C42D32
hypocenterC32#1971
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
14. C2×#3718C4semidirect
247. C32#1971semidirect
289. C3#69678semidirect

Maximal subgroups

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

subgroupindex
C22×#37182
C2×#636302
C3×#49292
C2×#635742
C2×#635652
#696783