Finite Group Databasebeta

← #11792771

Subgroups — #11792771

Order 1152.

763 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 763 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C2×C3×#951C212yesmaximal
2C22×#3718C211yesmaximal, characteristic, complemented
3#10592933C211yesmaximal, characteristic
4C2×#63560C212yesmaximal, complemented
5#10589165C211yesmaximal, characteristic, complemented
6#72109—32nomaximal
7C2×C32×#41C2211yescharacteristic
8C22×C3×Dic6C2212yes—
9C2×#3748C2211yescharacteristic
10C22×C3⋊(C3⋊C8)C2211yescharacteristic
11C3×#951—24no—
12C3×#951—24no—
13C2×C3×#216—22no—
14C2×C3×#214—22no—
15C2×C3×#208C2212yes—
16C2×#3718—24no—
17C2×#3718—22no—
18#63657—22no—
19#63560—24no—
20C2×C3×#137—32no—
21C2×#951D312yes—
22C22×#216—32no—
23#5234—32no—
24#5234—61no—
25C2×#836—32no—
26C2×#824—32no—
27#4558—32no—
28C32×#41—22no—
29C32×#41—22no—
30C2×C32×Q8D411yescharacteristic
31C2×C32×D4—21no—
32C2×C32×D4D411yescharacteristic
33C22×C32×C4—21no—
34C22×C32×C4C2311yescharacteristic
35C22×C3×Dic3—22no—
36C2×C3×Dic6—22no—
37C2×C3×Dic6—22no—
38C2×C3×Dic6—22no—
39C2×C3×Dic6—22no—
40C2×C3×Dic6—22no—
41C2×C3×Dic6D414yes—
42#3748—22no—
43C2×C3⋊(C3⋊C8)—22no—
44C2×C3⋊(C3⋊C8)—21no—
45C2×C3⋊(C3⋊C8)D412yes—
46C3×#216—28no—
47C3×#214—28no—
48C3×#208—22no—
49C3×#208D414yes—
50C2×C3×C3⋊C8—22no—
51#3718—28no—
52#2020—91noSylow
53C2×C3×#41—21no—
54C2×C3×#41C2×D312yes—
55C22×C3×Q8—32no—
56C22×Dic6C2×D312yes—
57C3×#137—64no—
58C3×#137—64no—
59C2×C3×Q16—62no—
60C2×C3×SD16—62no—
61C2×C3×M16—32no—
62#951—24no—
63#951—24no—
64C2×#216—22no—
65C2×#216—64no—
66C2×#216—64no—
67C2×#216—64no—
68C2×#214—22no—
69C2×#208—32no—
70C2×#208—61no—
71C2×#208C2×D312yes—
72C22×C3⋊C8—32no—
73C22×C3⋊C8—61no—
74#836—64no—
75#830—64no—
76#830—64no—
77#824—64no—
78C23×C32—21no—
79C23×C32#4111yescharacteristic
80C32×Q8—22no—
81C32×D4—22no—
82C32×D4—42no—
83C2×C32×C4—21no—
84C2×C32×C4—41no—
85C2×C32×C4—41no—
86C2×C32×C4C2×D411yescharacteristic
87C2×C32×C4C2×D412yes—
88C2×C32×C4C2×D412yes—
89C2×C32×C4C2×D411yescharacteristic
90C2×C3×Dic3—24no—
91C2×C3×Dic3—42no—
92C2×C3×Dic3—42no—
93C3×Dic6—24no—
94C3×Dic6—24no—
95C3×Dic6—42no—
96C3×Dic6—42no—
97C3⋊(C3⋊C8)—22no—
98C3⋊(C3⋊C8)—41no—
99C3×C3⋊C8—24no—
100C2×#137—32no—
101C22×Q16—91no—
102C2×#101—92no—
103#408—91no—
104#341—91no—
105C3×#41—22no—
106C3×#41—22no—
107C3×#41—22no—
108C3×#41—22no—
109C3×#41—42no—
110C3×#41—42no—
111C2×C3×Q8—21no—
112C2×C3×Q8—34no—
113C2×C3×Q8—62no—
114C2×C3×Q8—62no—
115C2×C3×Q8—62no—
116C2×C3×Q8—62no—
117C2×C3×Q8—62no—
118C2×C3×Q8#6712yes—
119C2×C3×D4—22no—
120C2×C3×D4—22no—
121C2×C3×D4—21no—
122C2×C3×D4#6712yes—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—21no—
126C22×C3×C4—62no—
127C22×C3×C4C22×D312yes—
128C22×Dic3—22no—
129C2×Dic6—22no—
130C2×Dic6—22no—
131C2×Dic6—22no—
132C2×Dic6—22no—
133C2×Dic6—22no—
134C2×Dic6#6714yes—
135C3×Q16—68no—
136C3×SD16—68no—
137C3×M16—34no—
138C3×M16—62no—
139C2×C3×C8—62no—
140#216—28no—
141#216—616no—
142#214—28no—
143#208—22no—
144#208—62no—
145#208—62no—
146#208—122no—
147#208#6714yes—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—34no—
150C2×C3⋊C8—62no—
151C2×C3⋊C8—62no—
152C2×C3⋊C8—62no—
153C2×C3⋊C8—62no—
154C2×C3⋊C8—62no—
155C2×C3⋊C8—122no—
156C22×C32—22no—
157C22×C32—41no—
158C22×C32—42no—
159C22×C32#12112yes—
160C22×C32#12111yescharacteristic
161C32×C4—41no—
162C32×C4#12011yescharacteristic
163C32×C4#12012yes—
164C32×C4#12711yescharacteristic
165C3×Dic3—44no—
166C2×#41D3211yescharacteristic
167C22×Q8—32no—
168#137—64no—
169#137—64no—
170C2×Q16—62no—
171C2×Q16—182no—
172C2×Q16—184no—
173C2×SD16—62no—
174C2×M16—32no—
175C2×M16—91no—
176C22×C8—91no—
177#101—184no—
178C22⋊C8—182no—
179C23×C3—22no—
180C23×C3—22no—
181C23×C3—21no—
182C23×C3#23712yes—
183C3×Q8—22no—
184C3×Q8—22no—
185C3×Q8—42no—
186C3×Q8—64no—
187C3×Q8—64no—
188C3×Q8—122no—
189C3×Q8—122no—
190C3×D4—22no—
191C3×D4—22no—
192C3×D4—42no—
193C3×D4—44no—
194C3×D4—42no—
195C3×D4—42no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—21no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—21no—
202C2×C3×C4—42no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—64no—
207C2×C3×C4—122no—
208C2×C3×C4—122no—
209C2×C3×C4D3×D412yes—
210C2×C3×C4D3×D412yes—
211C2×C3×C4D3×D412yes—
212C2×C3×C4C2×#6712yes—
213C2×C3×C4C2×#6712yes—
214C2×C3×C4C2×#6712yes—
215C2×Dic3—24no—
216C2×Dic3—42no—
217C2×Dic3—42no—
218Dic6—24no—
219Dic6—24no—
220Dic6—42no—
221Dic6—42no—
222C3×C8—64no—
223C3⋊C8—24no—
224C3⋊C8—64no—
225C3⋊C8—122no—
226C3⋊C8—122no—
227C3⋊C8—122no—
228C2×C32—22no—
229C2×C32—81no—
230C2×C32#39211yescharacteristic
231C2×C32#47312yes—
232#41—22no—
233#41—22no—
234C2×Q8—34no—
235C2×Q8—62no—
236C2×Q8—62no—
237C2×Q8—62no—
238C2×Q8—62no—
239C2×Q8—62no—
240C2×Q8#62511yescharacteristic
241C2×D4—21no—
242C2×D4#62511yescharacteristic
243C22×C4—21no—
244C22×C4—62no—
245C22×C4C2×D3211yescharacteristic
246Q16—68no—
247Q16—188no—
248SD16—68no—
249M16—34no—
250M16—62no—
251M16—182no—
252C2×C8—62no—
253C2×C8—92no—
254C2×C8—182no—
255C2×C8—181no—
256C22×C3—24no—
257C22×C3—22no—
258C22×C3—21no—
259C22×C3—42no—
260C22×C3—42no—
261C22×C3—44no—
262C22×C3—44no—
263C22×C3—42no—
264C22×C3#93812yes—
265C22×C3#93912yes—
266C22×C3#93912yes—
267C3×C4—22no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—42no—
271C3×C4—42no—
272C3×C4—124no—
273C3×C4#93712yes—
274C3×C4#93712yes—
275C3×C4#94012yes—
276C3×C4#95312yes—
277Dic3—44no—
278C32#202011yescharacteristic, complemented, Sylow
279C23—21no—
280C23#380511yescharacteristic
281Q8—22no—
282Q8—64no—
283Q8—64no—
284Q8—122no—
285Q8—122no—
286D4—22no—
287D4—42no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—64no—
292C2×C4—122no—
293C2×C4—122no—
294C2×C4C2×#62511yescharacteristic
295C2×C4D3×#6712yes—
296C2×C4D3×#6712yes—
297C2×C4#381111yescharacteristic
298C8—64no—
299C8—182no—
300C8—361no—
301C2×C3—22no—
302C2×C3—22no—
303C2×C3—22no—
304C2×C3—21no—
305C2×C3—42no—
306C2×C3—82no—
307C2×C3—82no—
308C2×C3#520414yes—
309C2×C3#522312yes—
310C22—22no—
311C22—42no—
312C22—41no—
313C22#6397012yes—
314C22#6398011yescharacteristic
315C4—41no—
316C4—124no—
317C4#6397612yes—
318C4#6397911yescharacteristic
319C4#6398311yescharacteristic
320C3—21no—
321C3#7210912yescomplemented
322C2—22no—
323C2—81no—
324C2#1059168312yes—
325C2#1059172311yescharacteristic
subgroupG/H
centerC22#63980
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
Fitting, prime radicalC2×C32×#41C22
socleC22×C32#121
hypercenterC2×#41D32
hypocenterC32#2020
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C22×#3718C2semidirect
4. C2×#63560C2semidirect
5. #10589165C2semidirect
278. C32#2020semidirect
321. C3#72109semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9512
C22×#37182
#105929332
C2×#635602
#105891652
#721093