Finite Group Databasebeta

← #11798729

Subgroups — #11798729

Order 1152.

487 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 487 conjugacy classes of subgroups — too many to draw. Only its 68 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×#5672C211yesmaximal, characteristic, complemented
2C3×#5061C211yesmaximal, characteristic
3#10591533C212yesmaximal
4#10589407C211yesmaximal, characteristic, complemented
5#10589176C212yesmaximal, complemented
6#71958—31nomaximal
7#70992—31nomaximal
8C3×#1004C2211yescharacteristic
9C32×#124C2211yescharacteristic
10C3×#944—21no—
11C3×#942—21no—
12C3×#935C2212yes—
13C3×#879—22no—
14C3×#878—22no—
15C3×#870—21no—
16#63961—22no—
17#63661C2212yes—
18C3×#833C2211yescharacteristic
19C3×#805—21no—
20#63580—22no—
21C3×#563—31no—
22#5672D311yescharacteristic, complemented
23C3×#455—31no—
24#5243—32no—
25#5061D311yescharacteristic
26#5053—32no—
27#4541—31no—
28#4541—31no—
29#4541—61no—
30#4539—32no—
31#4476—32no—
32C2×C32×Q8D411yescharacteristic
33C2×C32×D4C2311yescharacteristic
34C2×C3×#67—22no—
35C22×C3×Dic3D412yes—
36C3×#237—41no—
37C3×D3×D4—41no—
38C3×#235—21no—
39C2×C3×Dic6—21no—
40C32×C22⋊C4—22no—
41C32×C42D411yescharacteristic
42#3757—22no—
43C3×#217—22no—
44C3×#217D412yes—
45C3×#216—41no—
46C3×#215—41no—
47C3×C4⋊Dic3—21no—
48C3×C4⋊Dic3—21no—
49C3×Dic3⋊C4—22no—
50C3×Dic3⋊C4—22no—
51C3×Dic3⋊C4—22no—
52C3×C4×Dic3—21no—
53C3×#208—21no—
54#3724—42no—
55(C2×D3)⋊Dic3—42no—
56#2201—91noSylow
57C3×H(5,𝔽2)—31no—
58#1004C2×D311yescharacteristic
59C3×#137—61no—
60C3×C4⋊Q8—61no—
61C3×#126—62no—
62C3×#125—61no—
63C3×#124—21no—
64C3×#124C2×D311yescharacteristic
65C3×#124C2×D311yescharacteristic
66C3×#123—32no—
67C3×C22⋊Q8—62no—
68#944—21no—
69#942—21no—
70#935—62no—
71#935C2×D312yes—
72#929—62no—
73#879—22no—
74#878—22no—
75#870—21no—
76C3×#104—61no—
77C3×#100—31no—
78#834—32no—
79#834—32no—
80#834—62no—
81#834—62no—
82#833C2×D311yescharacteristic
83#806—62no—
84#805—21no—
85C23×C32C2×D412yes—
86C22×C3×D3—22no—
87C32×Q8—41no—
88C32×D4—41no—
89C2×C32×C4—21no—
90C2×C32×C4—21no—
91C2×C32×C4C2×D411yescharacteristic
92C2×C3⋊Dic3—42no—
93C3×#67—42no—
94C3×#67—41no—
95C2×C3×Dic3—22no—
96C2×C3×Dic3—22no—
97C2×C3×Dic3—22no—
98C2×C3×Dic3—42no—
99C3×D12—21no—
100C3×C4×D3—41no—
101C3×Dic6—21no—
102C3×Dic6—81no—
103C3×C3⋊C8—41no—
104#563—31no—
105#458—92no—
106#455—31no—
107#354—91no—
108#352—92no—
109C3×#41—61no—
110C3×#41—121no—
111C2×C3×Q8—21no—
112C2×C3×Q8—61no—
113C2×C3×Q8#6711yescharacteristic
114C2×C3×Q8#6711yescharacteristic
115C2×C3×D4—21no—
116C2×C3×D4—62no—
117C2×C3×D4—121no—
118C2×C3×D4C22×D311yescharacteristic
119C2×C3×D4C22×D311yescharacteristic
120C22×C3×C4—32no—
121C2×#67—22no—
122C22×Dic3#6712yes—
123#237—41no—
124D3×D4—41no—
125#235—21no—
126C2×Dic6—21no—
127C3×Q16—121no—
128C3×SD16—121no—
129C3×M16—61no—
130C3×C4⋊C4—62no—
131C3×C4⋊C4—62no—
132C3×C4⋊C4—61no—
133C3×C4⋊C4—61no—
134C3×C4⋊C4—62no—
135C3×C22⋊C4—22no—
136C3×C22⋊C4—22no—
137C3×C22⋊C4—32no—
138C3×C22⋊C4—42no—
139C3×C22⋊C4—62no—
140C3×C42—21no—
141C3×C42—61no—
142C3×C42#6711yescharacteristic
143C3×C42#6711yescharacteristic
144#217—22no—
145#217—62no—
146#217—62no—
147#217—122no—
148#217—122no—
149#217#6712yes—
150#216—41no—
151#215—41no—
152(C2×D3)⋊C4—122no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155Dic3⋊C4—22no—
156Dic3⋊C4—22no—
157Dic3⋊C4—22no—
158Dic3⋊C4—122no—
159Dic3⋊C4—122no—
160C4×Dic3—21no—
161#208—21no—
162C22×C32—22no—
163C22×C32—42no—
164C22×C32#12011yescharacteristic
165C2×C3×D3—41no—
166C2×C3×D3—82no—
167C32×C4—21no—
168C32×C4—41no—
169C32×C4—41no—
170C3⋊Dic3—82no—
171C3×Dic3—42no—
172C3×Dic3—41no—
173C3×Dic3—42no—
174H(5,𝔽2)—31no—
175#137—61no—
176C4⋊Q8—61no—
177#126—62no—
178#125—61no—
179#124D3211yescharacteristic
180#123—32no—
181#123—182no—
182C22⋊Q8—62no—
183#104—61no—
184#100—31no—
185#99—92no—
186#99—182no—
187C23×C3—22no—
188C23×C3—62no—
189C23×C3D3×D412yes—
190C23×C3C2×#6712yes—
191C22×D3—22no—
192C3×Q8—41no—
193C3×Q8—41no—
194C3×Q8—61no—
195C3×Q8—81no—
196C3×Q8—241no—
197C3×D4—41no—
198C3×D4—41no—
199C3×D4—61no—
200C3×D4—81no—
201C3×D4—122no—
202C3×D4—121no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—62no—
211C2×C3×C4—62no—
212C2×C3×C4—62no—
213C2×C3×C4—121no—
214C2×C3×C4—122no—
215C2×C3×C4C2×#6711yescharacteristic
216C2×C3×C4C2×#6711yescharacteristic
217#67—42no—
218#67—41no—
219C2×Dic3—22no—
220C2×Dic3—22no—
221C2×Dic3—22no—
222C2×Dic3—42no—
223C2×Dic3—122no—
224C2×Dic3—122no—
225C2×Dic3—242no—
226D12—21no—
227C4×D3—41no—
228Dic6—21no—
229Dic6—81no—
230C3×C8—121no—
231C3⋊C8—41no—
232C2×C32—21no—
233C2×C32—42no—
234C2×C32B(4,𝔽2)11yescharacteristic
235C3×D3—81no—
236#41—61no—
237#41—121no—
238C2×Q8—61no—
239C2×Q8#62511yescharacteristic
240C2×D4—62no—
241C2×D4—121no—
242C2×D4C2×D3211yescharacteristic
243C22×C4—32no—
244Q16—121no—
245SD16—121no—
246M16—61no—
247C4⋊C4—61no—
248C4⋊C4—62no—
249C4⋊C4—62no—
250C4⋊C4—61no—
251C4⋊C4—62no—
252C4⋊C4—362no—
253C22⋊C4—22no—
254C22⋊C4—32no—
255C22⋊C4—62no—
256C22⋊C4—182no—
257C22⋊C4—362no—
258C42—61no—
259C42#62511yescharacteristic
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—21no—
263C22×C3—42no—
264C22×C3—42no—
265C22×C3—42no—
266C22×C3—82no—
267C22×C3—121no—
268C22×C3—242no—
269C22×C3#93711yescharacteristic
270C22×C3#95311yescharacteristic
271C2×D3—41no—
272C2×D3—82no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—81no—
281C3×C4—81no—
282C3×C4—122no—
283C3×C4—121no—
284C3×C4—122no—
285Dic3—42no—
286Dic3—42no—
287Dic3—41no—
288Dic3—242no—
289Dic3—242no—
290Dic3—482no—
291C32#220111yescharacteristic, complemented, Sylow
292C23—62no—
293C23D3×#6712yes—
294Q8—41no—
295Q8—61no—
296Q8—241no—
297D4—41no—
298D4—61no—
299D4—122no—
300D4—121no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—62no—
304C2×C4—62no—
305C2×C4—62no—
306C2×C4—121no—
307C2×C4—122no—
308C2×C4—362no—
309C2×C4C2×#62511yescharacteristic
310C8—121no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—41no—
315C2×C3—42no—
316C2×C3—42no—
317C2×C3—82no—
318C2×C3—241no—
319C2×C3#503211yescharacteristic
320C2×C3#524411yescharacteristic
321D3—81no—
322C22—22no—
323C22—42no—
324C22—121no—
325C22—242no—
326C22#6397611yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—122no—
331C4—121no—
332C4—122no—
333C4—722no—
334C3—21no—
335C3#7099211yescharacteristic, complemented
336C3#7195811yescharacteristic, complemented
337C2—21no—
338C2—42no—
339C2—241no—
340C2#1059153011yescharacteristic
subgroupG/H
centerC2#10591530
commutator, cosocleC2×C32×D4C23
FrattiniC2×D4C2×D32
FittingC32×#124C22
socleC2×C32B(4,𝔽2)
prime radicalC3×#1004C22
hypercenter#124D32
hypocenterC32#2201
LCSG ▷ C2×C32×D4 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C3×#5672C2semidirect
4. #10589407C2semidirect
5. #10589176C2semidirect
22. #5672D3semidirect
291. C32#2201semidirect
335. C3#70992semidirect
336. C3#71958semidirect

Maximal subgroups

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

subgroupindex
C3×#56722
C3×#50612
#105915332
#105894072
#105891762
#719583
#709923