Finite Group Databasebeta

← #11878711

Subgroups — #11878711

Order 1152.

1011 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1011 conjugacy classes of subgroups — too many to draw. Only its 172 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
1#10596196C211yesmaximal, characteristic
2#10593909C211yesmaximal, characteristic, complemented
3C3×#5093C212yesmaximal, complemented
4#10590606C214yesmaximal, complemented
5#10590565C214yesmaximal, complemented
6#10590373C212yesmaximal, complemented
7#10590297C211yesmaximal, characteristic, complemented
8#81602—32nomaximal
9C2×#3832—21no—
10C2×#3797C2212yes—
11C32×C4⋊Q8C2211yescharacteristic
12#64156—21no—
13#64104—21no—
14#64104C2212yes—
15#64102—21no—
16#64102C2212yes—
17#64083C2211yescharacteristic
18C42×C3⋊D3C2211yescharacteristic
19C3×#869C2212yes—
20#63919—21no—
21#63919—21no—
22#63900—21no—
23#63900—21no—
24#63873C2212yes—
25C2×#3717C2218yes—
26C3×#810C2218yes—
27C3×C4×C3⋊C8C2212yes—
28#63572C2212yes—
29#63557C2212yes—
30#5725—32no—
31#5725—61no—
32D3×C4⋊Q8—32no—
33C3×#487—32no—
34#5093D312yescomplemented
35#5085—32no—
36#4818—38no—
37#4806—38no—
38#4693—32no—
39#4685—32no—
40C2×C32×Q8—21no—
41#3832—44no—
42C2×C4⋊(C3⋊D3)—22no—
43C2×C4⋊(C3⋊D3)—21no—
44C2×C4⋊(C3⋊D3)—21no—
45C2×C4⋊(C3⋊D3)C2312yes—
46C2×C4×C3⋊D3—22no—
47C2×C4×C3⋊D3D411yescharacteristic
48C2×C4×C3⋊D3D411yescharacteristic
49C2×C4×C3⋊D3C2311yescharacteristic
50C2×C3×Dic6C2314yes—
51C2×#627—22no—
52C2×#624—22no—
53#3797—44no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4C2312yes—
56C32×C42C2311yescharacteristic
57#3752—22no—
58#3752—22no—
59C4×C3⋊Dic3D411yescharacteristic
60C4×C3⋊Dic3D411yescharacteristic
61C4×C3⋊Dic3C2311yescharacteristic
62C3×C4⋊Dic3—22no—
63C3×C4⋊Dic3—22no—
64C2×C3×C3⋊C8C2314yes—
65#3724—22no—
66#3724—22no—
67#3717—48no—
68#3206—91noSylow
69C2×#239—62no—
70C2×#239—62no—
71C2×D3×Q8—34no—
72C3×C4⋊Q8—21no—
73C3×C4⋊Q8—32no—
74C3×C4⋊Q8C2×D312yes—
75#947—62no—
76#947—62no—
77#944—34no—
78C2×#215—38no—
79C2×#204—38no—
80#895—34no—
81#895—62no—
82#895—62no—
83#895—62no—
84#893—34no—
85#893—62no—
86#893—62no—
87#893—62no—
88D3×C4⋊C4—62no—
89D3×C4⋊C4—62no—
90#888—62no—
91#888—62no—
92#874—32no—
93#874—61no—
94C42×D3—32no—
95C42×D3—61no—
96#869C2×D312yes—
97C3×#103—38no—
98C3×C4×C8—32no—
99(C2×D3)⋊C8—34no—
100Dic3⋊C8—34no—
101#810C2×D318yes—
102C4×C3⋊C8C2×D312yes—
103C22×C3⋊D3—22no—
104C22×C3⋊D3C2×D411yescharacteristic
105C32×Q8—42no—
106C2×C32×C4—22no—
107C2×C32×C4C2×D411yescharacteristic
108C2×C32×C4C2×D411yescharacteristic
109C2×C32×C4C2411yescharacteristic
110C2×C3⋊Dic3C2×D411yescharacteristic
111C2×C3⋊Dic3C2×D411yescharacteristic
112C2×C3⋊Dic3C2×D411yescharacteristic
113C4⋊(C3⋊D3)—24no—
114C4⋊(C3⋊D3)—44no—
115C4⋊(C3⋊D3)—42no—
116C4⋊(C3⋊D3)—42no—
117C4×C3⋊D3—22no—
118C4×C3⋊D3—22no—
119C4×C3⋊D3—44no—
120C4×C3⋊D3SD1614yescomplemented
121C2×C3×Dic3—24no—
122C3×Dic6—28no—
123#627—44no—
124#624—44no—
125C3×C3⋊C8—24no—
126#532—91no—
127C2×C4⋊Q8—91no—
128#492—92no—
129#487—32no—
130#461—94no—
131#448—94no—
132#431—91no—
133C2×C3×Q8—22no—
134C2×C3×Q8—22no—
135C2×C3×Q8—34no—
136#239—128no—
137#239—128no—
138D3×Q8—128no—
139C2×D12—34no—
140C2×D12—62no—
141C2×D12—64no—
142C2×D12—62no—
143C2×D12—62no—
144C2×D12—64no—
145C2×D12—62no—
146C2×D12—62no—
147C2×C4×D3—32no—
148C2×C4×D3—32no—
149C2×C4×D3—32no—
150C2×C4×D3—61no—
151C2×C4×D3—61no—
152C2×C4×D3—61no—
153C2×C4×D3—64no—
154C2×C4×D3—64no—
155C2×C4×D3—64no—
156C2×Dic6—64no—
157C2×Dic6C22×D314yes—
158C2×C3×C8—34no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—62no—
163C3×C4⋊C4—62no—
164C3×C4⋊C4C22×D314yes—
165C3×C42—21no—
166C3×C42C22×D312yes—
167#215—128no—
168(C2×D3)⋊C4—64no—
169(C2×D3)⋊C4—64no—
170(C2×D3)⋊C4—64no—
171(C2×D3)⋊C4—64no—
172C4⋊Dic3—22no—
173C4⋊Dic3—22no—
174Dic3⋊C4—64no—
175Dic3⋊C4—64no—
176C4×Dic3—32no—
177C4×Dic3—32no—
178C4×Dic3—32no—
179C4×Dic3—61no—
180C4×Dic3—61no—
181C4×Dic3—61no—
182C2×C3⋊C8C22×D314yes—
183#204—128no—
184C22×C32C22×D411yescharacteristic
185C2×C3⋊D3—21no—
186C2×C3⋊D3—21no—
187C2×C3⋊D3—42no—
188C2×C3⋊D3—42no—
189C2×C3⋊D3—42no—
190C2×C3⋊D3C2×SD1612yes—
191C32×C4—42no—
192C32×C4C2×SD1612yes—
193C32×C4#13712yes—
194C32×C4H(5,𝔽2)12yes—
195C3⋊Dic3—21no—
196C3⋊Dic3—21no—
197C3⋊Dic3C2×SD1612yes—
198C3×Dic3—44no—
199C2×#41—181no—
200C22×Q8—92no—
201C2×SD16—98no—
202C4⋊Q8—32no—
203C4⋊Q8—92no—
204C4⋊Q8—181no—
205C4⋊Q8—181no—
206C4⋊Q8D3211yescharacteristic
207#127—91no—
208#124—181no—
209#121—92no—
210#121—181no—
211C4×D4—92no—
212C4×D4—181no—
213C2×C4⋊C4—181no—
214C2×C4⋊C4—181no—
215C2×C42—91no—
216C4⋊C8—92no—
217#103—38no—
218C22⋊C8—92no—
219C4×C8—32no—
220C22×D3—32no—
221C22×D3—61no—
222C22×D3—64no—
223C22×D3—64no—
224C3×Q8—44no—
225C3×Q8—44no—
226C3×Q8—68no—
227C2×C3×C4—24no—
228C2×C3×C4—24no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—64no—
233C2×C3×C4D3×D412yes—
234C2×C3×C4D3×D412yes—
235C2×C3×C4C23×D312yes—
236C2×Dic3—24no—
237C2×Dic3—32no—
238C2×Dic3—32no—
239C2×Dic3—32no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243D12—68no—
244D12—124no—
245D12—124no—
246D12—124no—
247D12—128no—
248D12—128no—
249D12—124no—
250D12—124no—
251C4×D3—38no—
252C4×D3—64no—
253C4×D3—64no—
254C4×D3—64no—
255C4×D3—122no—
256C4×D3—122no—
257C4×D3—128no—
258C4×D3—128no—
259C4×D3—128no—
260Dic6—28no—
261Dic6—128no—
262C3×C8—64no—
263C3⋊C8—24no—
264C2×C32#53511yescharacteristic
265C2×C32C22×SD1611yescharacteristic
266C2×C32C2×#13711yescharacteristic
267C3⋊D3—22no—
268C3⋊D3—82no—
269#41—364no—
270C2×Q8—21no—
271C2×Q8—34no—
272C2×Q8—182no—
273C2×Q8—364no—
274C2×D4—92no—
275C2×D4—182no—
276C2×D4—181no—
277C2×D4—181no—
278C22×C4—91no—
279C22×C4—91no—
280C22×C4—91no—
281C22×C4—182no—
282C22×C4—182no—
283SD16—368no—
284C2×C8—34no—
285C4⋊C4—21no—
286C4⋊C4—62no—
287C4⋊C4—62no—
288C4⋊C4—182no—
289C4⋊C4—182no—
290C4⋊C4C2×D3212yes—
291C22⋊C4—182no—
292C22⋊C4—182no—
293C42—91no—
294C42—91no—
295C42—91no—
296C42C2×D3211yescharacteristic
297C22×C3—21no—
298C22×C3C2×D3×D412yes—
299C2×D3—34no—
300C2×D3—62no—
301C2×D3—62no—
302C2×D3—62no—
303C2×D3—121no—
304C2×D3—121no—
305C2×D3—124no—
306C2×D3—124no—
307C2×D3—124no—
308C2×D3—124no—
309C2×D3—124no—
310C2×D3—124no—
311C3×C4—22no—
312C3×C4—22no—
313C3×C4—22no—
314C3×C4—44no—
315C3×C4—44no—
316C3×C4—124no—
317C3×C4D3×SD1614yes—
318C3×C4#91814yes—
319C3×C4#100914yes—
320Dic3—34no—
321Dic3—44no—
322Dic3—62no—
323Dic3—62no—
324Dic3—62no—
325Dic3—121no—
326Dic3—121no—
327C32#320611yescharacteristic, complemented, Sylow
328C23—91no—
329C23—182no—
330Q8—42no—
331Q8—68no—
332Q8—364no—
333D4—184no—
334D4—364no—
335D4—362no—
336D4—362no—
337C2×C4—22no—
338C2×C4—64no—
339C2×C4—91no—
340C2×C4—94no—
341C2×C4—91no—
342C2×C4—91no—
343C2×C4—182no—
344C2×C4—182no—
345C2×C4—364no—
346C2×C4—364no—
347C2×C4#381111yescharacteristic
348C2×C4#381111yescharacteristic
349C2×C4C22×D3211yescharacteristic
350C8—64no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3#567412yes—
355C2×C3C2×D3×SD1612yes—
356C2×C3C2×#91812yes—
357D3—64no—
358D3—122no—
359D3—244no—
360D3—244no—
361C22—92no—
362C22—181no—
363C22—181no—
364C22—362no—
365C22—362no—
366C22—362no—
367C22C2×#381111yescharacteristic
368C4—42no—
369C4—92no—
370C4—124no—
371C4—181no—
372C4—181no—
373C4#6393012yes—
374C4#6394512yes—
375C4#6432012yes—
376C3—21no—
377C3#8160212yescomplemented
378C2—182no—
379C2—722no—
380C2C2×#6393011yescharacteristic
381C2C2×#6394511yescharacteristic
382C2#1059543311yescharacteristic
subgroupG/H
centerC22C2×#3811
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊Q8C22
socleC22×C32C22×D4
prime radical#10596196C2
hypercenterC4⋊Q8D32
hypocenterC32#3206
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #10593909C2semidirect
3. C3×#5093C2semidirect
4. #10590606C2semidirect
5. #10590565C2semidirect
6. #10590373C2semidirect
7. #10590297C2semidirect
34. #5093D3semidirect
120. C4×C3⋊D3SD16semidirect
327. C32#3206semidirect
377. C3#81602semidirect

Maximal subgroups

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

subgroupindex
#105961962
#105939092
C3×#50932
#105906062
#105905652
#105903732
#105902972
#816023