Finite Group Databasebeta

← #11779439

Subgroups — #11779439

Order 1152.

463 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 96 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×#5579C211yesmaximal, characteristic
2C3×#5021C211yesmaximal, characteristic, complemented
3#10590387C211yesmaximal, characteristic, complemented
4#10590266C211yesmaximal, characteristic
5#10589401C211yesmaximal, characteristic, complemented
6#10589095C211yesmaximal, characteristic, complemented
7#10589080C211yesmaximal, characteristic
8#71924—31nomaximal
9#71156—31nomaximal
10C32×C4×Q8C2211yescharacteristic
11C3×#946—21no—
12C3×#896—22no—
13C3×D3×C4⋊C4—21no—
14C3×C4×D12—21no—
15C3×C4×D12C2211yescharacteristic
16C3×#870—21no—
17C3×#869C2211yescharacteristic
18C3×#869C2211yescharacteristic
19C2×#3718—22no—
20C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C4×C3⋊(C3⋊C8)C2211yescharacteristic
22C3×#835—21no—
23C3×#807—21no—
24C3×C4⋊(C3⋊C8)C2211yescharacteristic
25#63574—21no—
26#63562—21no—
27C3×#554—31no—
28#5579D311yescharacteristic
29C3×#477—31no—
30#5092—31no—
31#5092—31no—
32#5084—31no—
33#5021D311yescharacteristic, complemented
34#5008—31no—
35#4546—31no—
36#4535—31no—
37#4535—31no—
38#4535—61no—
39D4⋊(C3⋊C8)—31no—
40#4483—31no—
41#4481—31no—
42C2×C32×Q8D411yescharacteristic
43C2×C3×D12D411yescharacteristic
44C2×C3×C4×D3—21no—
45C2×C3×C4×D3—22no—
46C2×C3×Dic6—21no—
47C2×C3×Dic6D412yes—
48C32×C4⋊C4—21no—
49C32×C4⋊C4D411yescharacteristic
50C32×C42—21no—
51C32×C42C2311yescharacteristic
52C2×C3⋊(C3⋊C8)—21no—
53C2×C3⋊(C3⋊C8)—21no—
54C3×(C2×D3)⋊C4—22no—
55C3×(C2×D3)⋊C4—21no—
56C3×C4⋊Dic3—21no—
57C3×C4⋊Dic3—21no—
58C3×C4⋊Dic3—21no—
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3D411yescharacteristic
61C3×Dic3⋊C4—22no—
62C3×Dic3⋊C4—22no—
63C2×C3×C3⋊C8—21no—
64#3718—44no—
65#1632—91noSylow
66C3×C4⋊Q8—31no—
67C3×C4⋊Q8—31no—
68C3×#125—61no—
69C3×C22⋊Q8—62no—
70C3×C22⋊Q8—61no—
71C3×C4×Q8—21no—
72C3×C4×Q8C2×D311yescharacteristic
73C3×C4×Q8C2×D311yescharacteristic
74C3×C4×D4—31no—
75C3×C4×D4—61no—
76C2×C3×C4⋊C4—61no—
77#946—21no—
78C2×#216—62no—
79C2×#216—62no—
80#896—22no—
81D3×C4⋊C4—21no—
82C4×D12—21no—
83C4×D12C2×D311yescharacteristic
84#870—21no—
85#869C2×D311yescharacteristic
86#869C2×D311yescharacteristic
87C3×#107—61no—
88C3×C4⋊C8—31no—
89C3×#103—61no—
90#835—21no—
91#832—61no—
92#809—61no—
93#808—61no—
94#808—61no—
95#807—21no—
96C4⋊(C3⋊C8)—31no—
97C4⋊(C3⋊C8)—31no—
98C4⋊(C3⋊C8)—61no—
99C4⋊(C3⋊C8)C2×D311yescharacteristic
100C4×C3⋊C8—31no—
101C4×C3⋊C8—31no—
102C4×C3⋊C8—61no—
103C22×C3×D3—21no—
104C32×Q8—21no—
105C32×Q8—21no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4C2×D411yescharacteristic
109C2×C32×C4C2×D411yescharacteristic
110C2×C32×C4C2×D411yescharacteristic
111C2×C3×Dic3—21no—
112C2×C3×Dic3—22no—
113C2×C3×Dic3—41no—
114C3×D12—21no—
115C3×D12Q1612yescomplemented
116C3×C4×D3—41no—
117C3×C4×D3—41no—
118C3×C4×D3—82no—
119C3×Dic6—42no—
120C3×Dic6—42no—
121C3⋊(C3⋊C8)—22no—
122C3⋊(C3⋊C8)—41no—
123C3×C3⋊C8—41no—
124#554—31no—
125#494—91no—
126#477—31no—
127#476—91no—
128#333—91no—
129Q8⋊C8—91no—
130D4⋊C8—91no—
131C2×C3×Q8—21no—
132C2×C3×Q8—32no—
133C2×C3×Q8—61no—
134C2×C3×Q8#6711yescharacteristic
135C2×C3×Q8#6711yescharacteristic
136C2×C3×D4—31no—
137C22×C3×C4—62no—
138C22×C3×C4—61no—
139C2×D12#6711yescharacteristic
140C2×C4×D3—21no—
141C2×C4×D3—22no—
142C2×Dic6—21no—
143C2×Dic6#6712yes—
144C2×C3×C8—61no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—31no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—62no—
151C3×C4⋊C4—62no—
152C3×C4⋊C4—61no—
153C3×C4⋊C4—61no—
154C3×C4⋊C4—61no—
155C3×C4⋊C4—61no—
156C3×C4⋊C4#6711yescharacteristic
157C3×C4⋊C4#6711yescharacteristic
158C3×C22⋊C4—62no—
159C3×C22⋊C4—61no—
160C3×C42—21no—
161C3×C42—21no—
162C3×C42—21no—
163C3×C42—41no—
164C3×C42C22×D311yescharacteristic
165C3×C42C22×D311yescharacteristic
166#216—124no—
167#216—124no—
168(C2×D3)⋊C4—22no—
169(C2×D3)⋊C4—21no—
170C4⋊Dic3—21no—
171C4⋊Dic3—21no—
172C4⋊Dic3—21no—
173C4⋊Dic3—21no—
174C4⋊Dic3#6711yescharacteristic
175Dic3⋊C4—22no—
176Dic3⋊C4—22no—
177C2×C3⋊C8—21no—
178C2×C3⋊C8—61no—
179C2×C3⋊C8—61no—
180C2×C3⋊C8—61no—
181C2×C3⋊C8—61no—
182C2×C3⋊C8—121no—
183C2×C3⋊C8—121no—
184C22×C32#12011yescharacteristic
185C2×C3×D3—22no—
186C2×C3×D3—41no—
187C2×C3×D3—41no—
188C32×C4—21no—
189C32×C4—41no—
190C32×C4—41no—
191C32×C4C2×Q1611yescharacteristic
192C32×C4#13511yescharacteristic
193C32×C4#13711yescharacteristic
194C32×C4#13711yescharacteristic
195C3×Dic3—41no—
196C3×Dic3—42no—
197C3×Dic3—81no—
198C2×Q16—182no—
199C4⋊Q8—31no—
200C4⋊Q8—31no—
201#125—61no—
202C22⋊Q8—62no—
203C22⋊Q8—61no—
204C4×Q8D3211yescharacteristic
205C4×D4—31no—
206C4×D4—61no—
207C2×C4⋊C4—61no—
208#107—61no—
209#106—181no—
210C4⋊C8—31no—
211C4⋊C8—91no—
212#103—61no—
213#102—181no—
214C4×C8—91no—
215C23×C3—61no—
216C22×D3—21no—
217C3×Q8—22no—
218C3×Q8—21no—
219C3×Q8—21no—
220C3×Q8—21no—
221C3×Q8—21no—
222C3×Q8—41no—
223C3×Q8—122no—
224C3×Q8—122no—
225C3×D4—32no—
226C3×D4—61no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—61no—
237C2×C3×C4—62no—
238C2×C3×C4—121no—
239C2×C3×C4—121no—
240C2×C3×C4—121no—
241C2×C3×C4—242no—
242C2×C3×C4D3×D411yescharacteristic
243C2×C3×C4D3×D411yescharacteristic
244C2×C3×C4C2×#6711yescharacteristic
245C2×C3×C4C2×#6711yescharacteristic
246C2×C3×C4C2×#6711yescharacteristic
247C2×C3×C4C2×#6711yescharacteristic
248C2×Dic3—21no—
249C2×Dic3—22no—
250C2×Dic3—41no—
251D12—21no—
252D12#21612yescomplemented
253C4×D3—41no—
254C4×D3—41no—
255C4×D3—82no—
256Dic6—42no—
257Dic6—42no—
258C3×C8—121no—
259C3⋊C8—41no—
260C3⋊C8—62no—
261C3⋊C8—62no—
262C3⋊C8—122no—
263C3⋊C8—121no—
264C3⋊C8—121no—
265C3⋊C8—241no—
266C2×C32#45111yescharacteristic
267C2×C32#45211yescharacteristic
268C2×C32#45311yescharacteristic
269C3×D3—42no—
270C2×Q8—32no—
271C2×Q8—61no—
272C2×Q8#62511yescharacteristic
273C2×D4—31no—
274C22×C4—62no—
275C22×C4—61no—
276Q16—364no—
277C2×C8—61no—
278C2×C8—181no—
279C2×C8—181no—
280C4⋊C4—21no—
281C4⋊C4—31no—
282C4⋊C4—62no—
283C4⋊C4—62no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4#62511yescharacteristic
289C22⋊C4—62no—
290C22⋊C4—61no—
291C42—21no—
292C42C2×D3211yescharacteristic
293C22×C3—21no—
294C22×C3—62no—
295C22×C3—121no—
296C22×C3—121no—
297C22×C3#93711yescharacteristic
298C22×C3#95311yescharacteristic
299C2×D3—22no—
300C2×D3—41no—
301C2×D3—41no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—41no—
309C3×C4—42no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—41no—
314C3×C4—81no—
315C3×C4—122no—
316C3×C4—121no—
317C3×C4—241no—
318C3×C4D3×Q1611yescharacteristic
319C3×C4#91811yescharacteristic
320C3×C4#94211yescharacteristic
321C3×C4#94211yescharacteristic
322C3×C4C2×#21611yescharacteristic
323C3×C4#95011yescharacteristic
324C3×C4#95011yescharacteristic
325C3×C4#95111yescharacteristic
326Dic3—41no—
327Dic3—42no—
328Dic3—81no—
329C32#163211yescharacteristic, complemented, Sylow
330C23—61no—
331Q8—21no—
332Q8—21no—
333Q8—122no—
334Q8—122no—
335D4—32no—
336D4—61no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—61no—
340C2×C4—62no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—242no—
345C2×C4C2×#62511yescharacteristic
346C2×C4D3×#6711yescharacteristic
347C2×C4D3×#6711yescharacteristic
348C8—121no—
349C8—182no—
350C8—361no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—122no—
355C2×C3#504711yescharacteristic
356C2×C3#518611yescharacteristic
357C2×C3#519811yescharacteristic
358C2×C3#522811yescharacteristic
359C2×C3#523911yescharacteristic
360C2×C3#524111yescharacteristic
361D3—42no—
362C22—62no—
363C22—121no—
364C22—121no—
365C22#6397611yescharacteristic
366C4—21no—
367C4—41no—
368C4—41no—
369C4—122no—
370C4—121no—
371C4—241no—
372C4#6382211yescharacteristic
373C4#6382711yescharacteristic
374C4D3×#21611yescharacteristic
375C4#6394811yescharacteristic
376C3—21no—
377C3#7115611yescharacteristic, complemented
378C3#7192411yescharacteristic, complemented
379C2—122no—
380C2#1059117011yescharacteristic
381C2#1059159411yescharacteristic
382C2#1059166511yescharacteristic
subgroupG/H
centerC22#63976
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×Q8C22
socleC22×C32#120
prime radicalC2×C3×D12D4
hypercenterC4×Q8D32
hypocenterC32#1632
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#5021C2semidirect
3. #10590387C2semidirect
5. #10589401C2semidirect
6. #10589095C2semidirect
33. #5021D3semidirect
115. C3×D12Q16semidirect
252. D12#216semidirect
329. C32#1632semidirect
377. C3#71156semidirect
378. C3#71924semidirect

Maximal subgroups

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

subgroupindex
C3×#55792
C3×#50212
#105903872
#105902662
#105894012
#105890952
#105890802
#719243
#711563