Finite Group Databasebeta

← #11779965

Subgroups — #11779965

Order 1152.

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

normal subgroups only

This group has 470 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
2#10592730C211yesmaximal, characteristic, complemented
3#10590287C211yesmaximal, characteristic, complemented
4#10590272C211yesmaximal, characteristic
5C3×#4535C211yesmaximal, characteristic, complemented
6#10589095C211yesmaximal, characteristic, complemented
7#10589081C211yesmaximal, characteristic
8#71926—31nomaximal
9#68270—31nomaximal
10C32×C4×Q8C2211yescharacteristic
11C2×#3756—21no—
12#64078C2211yescharacteristic
13C3×#946—21no—
14C3×#896—22no—
15C3×D3×C4⋊C4—21no—
16C3×C4×D12—21no—
17C3×C4×D12C2211yescharacteristic
18C3×#870—21no—
19C3×#869C2211yescharacteristic
20#63637—21no—
21C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22C3×C4⋊(C3⋊C8)C2211yescharacteristic
23C3×C4×C3⋊C8C2211yescharacteristic
24#63575—21no—
25#63566—22no—
26#63563—21no—
27C3×#554—31no—
28#5579D311yescharacteristic
29#5079—31no—
30#5031—31no—
31#5031—31no—
32#5031—61no—
33#5007—31no—
34#4846—31no—
35#4681—31no—
36C3×Q8⋊C8—31no—
37#4546—31no—
38#4535D311yescharacteristic, complemented
39D4⋊(C3⋊C8)—31no—
40#4481—31no—
41#4459—31no—
42C2×C32×Q8D411yescharacteristic
43C2×#634—21no—
44C2×C3×D12D411yescharacteristic
45C2×C3×C4×D3—21no—
46C2×C3×C4×D3—22no—
47C2×C3×Dic6D412yes—
48C32×C4⋊C4—21no—
49C32×C4⋊C4D411yescharacteristic
50C32×C42—21no—
51C32×C42C2311yescharacteristic
52#3756—42no—
53C4⋊(C3⋊Dic3)—21no—
54C4⋊(C3⋊Dic3)—21no—
55C2×C3⋊(C3⋊C8)—21no—
56C3×(C2×D3)⋊C4—22no—
57C3×(C2×D3)⋊C4—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—
64C2×C3×C3⋊C8—21no—
65#1644—91noSylow
66C3×C4⋊Q8—31no—
67C3×#125—61no—
68C3×C22⋊Q8—62no—
69C3×C22⋊Q8—61no—
70C3×C4×Q8—21no—
71C3×C4×Q8C2×D311yescharacteristic
72C3×C4×Q8C2×D311yescharacteristic
73C3×C4×D4—31no—
74C3×C4×D4—61no—
75C2×C3×C4⋊C4—61no—
76#946—21no—
77C2×#216—62no—
78C2×#216—61no—
79C2×#216—61no—
80#896—22no—
81D3×C4⋊C4—21no—
82C4×D12—21no—
83C4×D12C2×D311yescharacteristic
84#870—21no—
85#869—31no—
86#869—31no—
87#869—61no—
88#869C2×D311yescharacteristic
89C3×C4⋊C8—31no—
90C3×C4×C8—31no—
91#835—62no—
92#832—61no—
93#821—61no—
94#818—61no—
95#816—62no—
96#810—62no—
97#810—61no—
98#810—61no—
99#807—61no—
100C4⋊(C3⋊C8)—31no—
101C4⋊(C3⋊C8)—31no—
102C4⋊(C3⋊C8)—61no—
103C4⋊(C3⋊C8)C2×D311yescharacteristic
104C4×C3⋊C8C2×D311yescharacteristic
105C22×C3×D3—21no—
106C32×Q8—21no—
107C32×Q8—21no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4C2×D411yescharacteristic
112C2×C32×C4C2×D411yescharacteristic
113C2×C3⋊Dic3—41no—
114#634—41no—
115#634—41no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—22no—
118C3×D12—21no—
119C3×D12Q1612yescomplemented
120C3×C4×D3—41no—
121C3×C4×D3—41no—
122C3×C4×D3—82no—
123C3×Dic6—42no—
124C3⋊(C3⋊C8)—41no—
125C3×C3⋊C8—41no—
126C3×C3⋊C8—41no—
127#554—31no—
128#487—91no—
129#474—91no—
130#462—91no—
131#333—91no—
132Q8⋊C8—31no—
133D4⋊C8—91no—
134C2×C3×Q8—21no—
135C2×C3×Q8—32no—
136C2×C3×Q8D1211yescharacteristic
137C2×C3×Q8#6711yescharacteristic
138C2×C3×D4—31no—
139C22×C3×C4—62no—
140C22×C3×C4—61no—
141C2×D12#6711yescharacteristic
142C2×C4×D3—21no—
143C2×C4×D3—22no—
144C2×Dic6—62no—
145C2×Dic6—61no—
146C2×Dic6—61no—
147C2×Dic6#6712yes—
148C2×C3×C8—61no—
149C2×C3×C8—61no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—31no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—62no—
156C3×C4⋊C4—62no—
157C3×C4⋊C4—61no—
158C3×C4⋊C4—61no—
159C3×C4⋊C4D1211yescharacteristic
160C3×C4⋊C4#6711yescharacteristic
161C3×C22⋊C4—62no—
162C3×C22⋊C4—61no—
163C3×C42—21no—
164C3×C42—21no—
165C3×C42—21no—
166C3×C42—41no—
167C3×C42C22×D311yescharacteristic
168C3×C42C22×D311yescharacteristic
169#216—124no—
170#216—122no—
171#216—122no—
172(C2×D3)⋊C4—22no—
173(C2×D3)⋊C4—21no—
174C4⋊Dic3—21no—
175C4⋊Dic3—21no—
176C4⋊Dic3—61no—
177C4⋊Dic3—61no—
178C4⋊Dic3—61no—
179C4⋊Dic3—61no—
180C4⋊Dic3—121no—
181C4⋊Dic3—121no—
182C4⋊Dic3#6711yescharacteristic
183Dic3⋊C4—22no—
184Dic3⋊C4—22no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8—62no—
188C2×C3⋊C8—61no—
189C2×C3⋊C8—61no—
190C22×C32#12011yescharacteristic
191C2×C3×D3—22no—
192C2×C3×D3—41no—
193C2×C3×D3—41no—
194C32×C4—21no—
195C32×C4—41no—
196C32×C4—41no—
197C32×C4C2×Q1611yescharacteristic
198C32×C4#13511yescharacteristic
199C32×C4AGL(1,ℤ/8)11yescharacteristic
200C32×C4#13711yescharacteristic
201C3⋊Dic3—81no—
202C3×Dic3—41no—
203C3×Dic3—42no—
204C2×Q16—181no—
205C4⋊Q8—31no—
206C4⋊Q8—91no—
207#125—61no—
208C22⋊Q8—62no—
209C22⋊Q8—61no—
210C4×Q8D3211yescharacteristic
211C4×D4—31no—
212C4×D4—61no—
213C2×C4⋊C4—61no—
214#107—181no—
215C4⋊C8—31no—
216C4⋊C8—91no—
217#103—182no—
218#103—181no—
219#102—181no—
220C4×C8—31no—
221C23×C3—61no—
222C22×D3—21no—
223C3×Q8—22no—
224C3×Q8—21no—
225C3×Q8—21no—
226C3×Q8—21no—
227C3×Q8—21no—
228C3×Q8—41no—
229C3×Q8—122no—
230C3×D4—32no—
231C3×D4—61no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—61no—
242C2×C3×C4—62no—
243C2×C3×C4—121no—
244C2×C3×C4—121no—
245C2×C3×C4—242no—
246C2×C3×C4C2×D1211yescharacteristic
247C2×C3×C4D3×D411yescharacteristic
248C2×C3×C4D3×D411yescharacteristic
249C2×C3×C4C2×#6711yescharacteristic
250C2×C3×C4C2×#6711yescharacteristic
251C2×C3×C4C2×#6711yescharacteristic
252C2×Dic3—21no—
253C2×Dic3—22no—
254C2×Dic3—122no—
255C2×Dic3—121no—
256C2×Dic3—121no—
257D12—21no—
258D12#21612yescomplemented
259C4×D3—41no—
260C4×D3—41no—
261C4×D3—82no—
262Dic6—42no—
263Dic6—122no—
264Dic6—122no—
265Dic6—121no—
266Dic6—121no—
267Dic6—121no—
268Dic6—121no—
269C3×C8—121no—
270C3×C8—121no—
271C3⋊C8—41no—
272C3⋊C8—41no—
273C3⋊C8—122no—
274C3⋊C8—121no—
275C3⋊C8—121no—
276C2×C32#44911yescharacteristic
277C2×C32#45111yescharacteristic
278C2×C32#45511yescharacteristic
279C3×D3—42no—
280C2×Q8—32no—
281C2×Q8—181no—
282C2×Q8#62611yescharacteristic
283C2×D4—31no—
284C22×C4—62no—
285C22×C4—61no—
286Q16—362no—
287C2×C8—61no—
288C2×C8—61no—
289C2×C8—181no—
290C4⋊C4—21no—
291C4⋊C4—31no—
292C4⋊C4—62no—
293C4⋊C4—62no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—181no—
297C4⋊C4—181no—
298C4⋊C4#62611yescharacteristic
299C22⋊C4—62no—
300C22⋊C4—61no—
301C42—21no—
302C42C2×D3211yescharacteristic
303C22×C3—21no—
304C22×C3—62no—
305C22×C3—121no—
306C22×C3—121no—
307C22×C3#88211yescharacteristic
308C22×C3#95311yescharacteristic
309C2×D3—22no—
310C2×D3—41no—
311C2×D3—41no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—41no—
319C3×C4—42no—
320C3×C4—41no—
321C3×C4—41no—
322C3×C4—41no—
323C3×C4—41no—
324C3×C4—81no—
325C3×C4—121no—
326C3×C4—122no—
327C3×C4#90411yescharacteristic
328C3×C4#90811yescharacteristic
329C3×C4#91511yescharacteristic
330C3×C4D3×Q1611yescharacteristic
331C3×C4#94211yescharacteristic
332C3×C4C2×#21611yescharacteristic
333C3×C4#94911yescharacteristic
334C3×C4#95011yescharacteristic
335Dic3—41no—
336Dic3—42no—
337Dic3—242no—
338Dic3—241no—
339Dic3—241no—
340C32#164411yescharacteristic, complemented, Sylow
341C23—61no—
342Q8—21no—
343Q8—21no—
344Q8—122no—
345Q8—361no—
346Q8—361no—
347D4—32no—
348D4—61no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—61no—
352C2×C4—62no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—242no—
356C2×C4—361no—
357C2×C4C2×#62611yescharacteristic
358C2×C4D3×#6711yescharacteristic
359C2×C4D3×#6711yescharacteristic
360C8—121no—
361C8—121no—
362C8—361no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—122no—
367C2×C3#473411yescharacteristic
368C2×C3#480911yescharacteristic
369C2×C3#482511yescharacteristic
370C2×C3#522811yescharacteristic
371C2×C3#523811yescharacteristic
372C2×C3#524211yescharacteristic
373D3—42no—
374C22—62no—
375C22—121no—
376C22—121no—
377C22#6397711yescharacteristic
378C4—21no—
379C4—41no—
380C4—41no—
381C4—121no—
382C4—122no—
383C4—721no—
384C4#6382511yescharacteristic
385C4#6382911yescharacteristic
386C4D3×#21611yescharacteristic
387C4#6394611yescharacteristic
388C3—21no—
389C3#6827011yescharacteristic, complemented
390C3#7192611yescharacteristic, complemented
391C2—122no—
392C2#1059116311yescharacteristic
393C2#1059159511yescharacteristic
394C2#1059168811yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×Q8C22
socleC22×C32#120
prime radicalC2×C3×D12D4
hypercenterC4×Q8D32
hypocenterC32#1644
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. #10592730C2semidirect
3. #10590287C2semidirect
5. C3×#4535C2semidirect
6. #10589095C2semidirect
38. #4535D3semidirect
119. C3×D12Q16semidirect
258. D12#216semidirect
340. C32#1644semidirect
389. C3#68270semidirect
390. C3#71926semidirect

Maximal subgroups

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

subgroupindex
C3×#55792
#105927302
#105902872
#105902722
C3×#45352
#105890952
#105890812
#719263
#682703