Finite Group Databasebeta

← #11779960

Subgroups — #11779960

Order 1152.

470 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 90 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#10593956C211yesmaximal, characteristic
2#10592789C211yesmaximal, characteristic, complemented
3C3×#5093C211yesmaximal, characteristic, complemented
4C3×#5087C211yesmaximal, characteristic
5#10589095C211yesmaximal, characteristic, complemented
6#10589085C211yesmaximal, characteristic, complemented
7#10589081C211yesmaximal, characteristic
8#71238—31nomaximal
9#71113—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C2×#3756—21no—
12C4×#634C2211yescharacteristic
13C3×C4×D12C2211yescharacteristic
14C3×#869C2211yescharacteristic
15#63898—21no—
16D3×C4⋊Dic3—21no—
17#63887—22no—
18#63872—21no—
19#63857—21no—
20#63662—21no—
21C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22C3×#810—22no—
23C3×#809—21no—
24C3×#807—21no—
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4×C3⋊C8C2211yescharacteristic
27#5730—31no—
28#5543—31no—
29C3×#487—31no—
30C3×#474—31no—
31#5093D311yescharacteristic, complemented
32#5090—31no—
33#5090—31no—
34#5090—61no—
35#5087D311yescharacteristic
36#4546—31no—
37D4⋊(C3⋊C8)—31no—
38#4484—31no—
39#4481—31no—
40#4459—31no—
41#4456—31no—
42C2×C32×Q8—21no—
43C2×#634D411yescharacteristic
44C2×C3×D12D411yescharacteristic
45C2×C3×C4×D3—21no—
46C2×C3×Dic6D412yes—
47C2×D3×Dic3—22no—
48C32×C4⋊C4—21no—
49C32×C4⋊C4—21no—
50C32×C42C2311yescharacteristic
51#3756—42no—
52C4⋊(C3⋊Dic3)D411yescharacteristic
53#3750—21no—
54C4×C3⋊Dic3—21no—
55C2×C3⋊(C3⋊C8)—21no—
56C3×(C2×D3)⋊C4—21no—
57C3×C4⋊Dic3—21no—
58C3×C4⋊Dic3—21no—
59C3×C4⋊Dic3D411yescharacteristic
60C2×C3×C3⋊C8—21no—
61C2×C3×C3⋊C8—21no—
62Dic3⋊Dic3—22no—
63Dic3⋊Dic3—22no—
64(C2×D3)⋊Dic3—22no—
65#1644—91noSylow
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—31no—
68C3×C4⋊Q8C2×D311yescharacteristic
69C3×C4⋊Q8C2×D311yescharacteristic
70C3×C4×D4—31no—
71#946—62no—
72C2×#216—61no—
73C2×#216—61no—
74C2×#216—62no—
75D4×Dic3—61no—
76C2×C4⋊Dic3—61no—
77#924—61no—
78#897—61no—
79#893—61no—
80D3×C4⋊C4—61no—
81#890—61no—
82#890—61no—
83#878—62no—
84C4×D12C2×D311yescharacteristic
85#869C2×D311yescharacteristic
86C4×Dic6—31no—
87C4×Dic6—31no—
88C4×Dic6—61no—
89C3×#107—61no—
90C3×C4⋊C8—31no—
91C3×#103—62no—
92C3×#102—61no—
93C3×C4×C8—31no—
94#835—61no—
95#835—61no—
96#835—62no—
97#810—22no—
98#809—21no—
99#807—21no—
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—41no—
107C32×Q8—41no—
108C2×C32×C4—41no—
109C2×C32×C4C2×D411yescharacteristic
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4C2×D411yescharacteristic
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—21no—
114#634—21no—
115#634—21no—
116C2×C3×Dic3—22no—
117C2×C3×Dic3—21no—
118C3×D12—21no—
119C3×D12Q1612yescomplemented
120C3×C4×D3—41no—
121C3×C4×D3—41no—
122C3×Dic6—42no—
123D3×Dic3—82no—
124C3⋊(C3⋊C8)—41no—
125C3×C3⋊C8—41no—
126C3×C3⋊C8—41no—
127#554—91no—
128#487—31no—
129#474—31no—
130#462—91no—
131#333—91no—
132Q8⋊C8—91no—
133D4⋊C8—91no—
134C2×C3×Q8—22no—
135C2×C3×Q8—21no—
136C2×C3×Q8—21no—
137C2×C3×Q8—32no—
138C2×C3×D4—31no—
139C22×C3×C4—61no—
140C22×Dic3—62no—
141C2×D12#6711yescharacteristic
142C2×C4×D3—21no—
143C2×C4×D3—62no—
144C2×Dic6—31no—
145C2×Dic6—31no—
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—21no—
154C3×C4⋊C4—31no—
155C3×C4⋊C4—41no—
156C3×C4⋊C4—41no—
157C3×C4⋊C4—61no—
158C3×C4⋊C4—61no—
159C3×C22⋊C4—61no—
160C3×C42—21no—
161C3×C42C22×D311yescharacteristic
162C3×C42C22×D311yescharacteristic
163#217—62no—
164#216—124no—
165#216—122no—
166#216—122no—
167(C2×D3)⋊C4—21no—
168(C2×D3)⋊C4—62no—
169C4⋊Dic3—21no—
170C4⋊Dic3—21no—
171C4⋊Dic3—31no—
172C4⋊Dic3—31no—
173C4⋊Dic3—61no—
174C4⋊Dic3—62no—
175C4⋊Dic3—62no—
176C4⋊Dic3#6711yescharacteristic
177Dic3⋊C4—62no—
178Dic3⋊C4—62no—
179Dic3⋊C4—61no—
180Dic3⋊C4—61no—
181Dic3⋊C4—121no—
182C4×Dic3—61no—
183C4×Dic3—61no—
184C4×Dic3—121no—
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—81no—
196C32×C4C2×Q1611yescharacteristic
197C32×C4#13511yescharacteristic
198C32×C4AGL(1,ℤ/8)11yescharacteristic
199C32×C4#13711yescharacteristic
200C3⋊Dic3—41no—
201C3⋊Dic3—41no—
202C3×Dic3—41no—
203C3×Dic3—42no—
204C2×Q16—181no—
205C4⋊Q8—31no—
206C4⋊Q8D3211yescharacteristic
207#125—181no—
208C22⋊Q8—182no—
209C22⋊Q8—181no—
210C4×Q8—91no—
211C4×D4—31no—
212C4×D4—181no—
213C2×C4⋊C4—181no—
214#107—61no—
215C4⋊C8—31no—
216C4⋊C8—91no—
217#103—62no—
218#103—181no—
219#102—61no—
220C4×C8—31no—
221C23×C3—61no—
222C22×D3—21no—
223C3×Q8—42no—
224C3×Q8—42no—
225C3×Q8—41no—
226C3×Q8—41no—
227C3×Q8—41no—
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—42no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—61no—
239C2×C3×C4—62no—
240C2×C3×C4—121no—
241C2×C3×C4—121no—
242C2×C3×C4D3×D411yescharacteristic
243C2×C3×C4D3×D411yescharacteristic
244C2×C3×C4D3×D411yescharacteristic
245C2×C3×C4D3×D411yescharacteristic
246C2×C3×C4C2×#6711yescharacteristic
247C2×C3×C4C2×#6711yescharacteristic
248C2×Dic3—21no—
249C2×Dic3—22no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—121no—
255C2×Dic3—121no—
256C2×Dic3—242no—
257D12—21no—
258D12#21612yescomplemented
259C4×D3—41no—
260C4×D3—41no—
261C4×D3—242no—
262Dic6—42no—
263Dic6—62no—
264Dic6—61no—
265Dic6—61no—
266Dic6—61no—
267Dic6—61no—
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—21no—
281C2×Q8—32no—
282C2×Q8—91no—
283C2×D4—31no—
284C22×C4—61no—
285C22×C4—182no—
286Q16—362no—
287C2×C8—61no—
288C2×C8—61no—
289C2×C8—181no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—31no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—91no—
296C4⋊C4—182no—
297C4⋊C4—182no—
298C4⋊C4—181no—
299C22⋊C4—61no—
300C22⋊C4—182no—
301C42—181no—
302C42C2×D3211yescharacteristic
303C22×C3—21no—
304C22×C3—62no—
305C22×C3—121no—
306C22×C3—121no—
307C22×C3#93711yescharacteristic
308C22×C3#93711yescharacteristic
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—82no—
320C3×C4—81no—
321C3×C4—81no—
322C3×C4—121no—
323C3×C4—122no—
324C3×C4#91111yescharacteristic
325C3×C4#91211yescharacteristic
326C3×C4#91411yescharacteristic
327C3×C4#91611yescharacteristic
328C3×C4D3×Q1611yescharacteristic
329C3×C4#91811yescharacteristic
330C3×C4C2×#21611yescharacteristic
331C3×C4#95111yescharacteristic
332Dic3—41no—
333Dic3—42no—
334Dic3—122no—
335Dic3—121no—
336Dic3—121no—
337Dic3—121no—
338Dic3—121no—
339Dic3—241no—
340C32#164411yescharacteristic, complemented, Sylow
341C23—61no—
342Q8—41no—
343Q8—41no—
344Q8—122no—
345Q8—181no—
346Q8—181no—
347D4—32no—
348D4—61no—
349C2×C4—41no—
350C2×C4—61no—
351C2×C4—62no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—181no—
355C2×C4—181no—
356C2×C4—722no—
357C2×C4D3×#6711yescharacteristic
358C2×C4D3×#6711yescharacteristic
359C2×C4#381111yescharacteristic
360C8—121no—
361C8—121no—
362C8—361no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—122no—
367C2×C3#505011yescharacteristic
368C2×C3#506111yescharacteristic
369C2×C3#515811yescharacteristic
370C2×C3#517211yescharacteristic
371C2×C3#518611yescharacteristic
372C2×C3#520211yescharacteristic
373D3—42no—
374C22—62no—
375C22—121no—
376C22—121no—
377C22#6398311yescharacteristic
378C4—21no—
379C4—81no—
380C4—121no—
381C4—122no—
382C4—361no—
383C4—361no—
384C4#6393611yescharacteristic
385C4#6393711yescharacteristic
386C4D3×#21611yescharacteristic
387C4#6394311yescharacteristic
388C3—21no—
389C3#7111311yescharacteristic, complemented
390C3#7123811yescharacteristic, complemented
391C2—122no—
392C2#1059152011yescharacteristic
393C2#1059165311yescharacteristic
394C2#1059165911yescharacteristic
subgroupG/H
centerC22#63983
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. #10592789C2semidirect
3. C3×#5093C2semidirect
5. #10589095C2semidirect
6. #10589085C2semidirect
31. #5093D3semidirect
119. C3×D12Q16semidirect
258. D12#216semidirect
340. C32#1644semidirect
389. C3#71113semidirect
390. C3#71238semidirect

Maximal subgroups

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

subgroupindex
#105939562
#105927892
C3×#50932
C3×#50872
#105890952
#105890852
#105890812
#712383
#711133