Finite Group Databasebeta

← #11781931

Subgroups — #11781931

Order 1152.

381 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 84 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#10592777C211yesmaximal, characteristic, complemented
2C3×#5083C211yesmaximal, characteristic
3#10590365C211yesmaximal, characteristic
4#10590339C211yesmaximal, characteristic
5C3×#4547C211yesmaximal, characteristic, complemented
6#10589098C211yesmaximal, characteristic, complemented
7#10589094C211yesmaximal, characteristic, complemented
8#71041—31nomaximal
9#68451—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11#64078C2211yescharacteristic
12C2×C3×#215—21no—
13C3×C4×D12C2211yescharacteristic
14C2×#3716—21no—
15C2×#3711—21no—
16#63634—21no—
17#63634—21no—
18C4×C3⋊(C3⋊C8)C2211yescharacteristic
19C3×#835—21no—
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×C4⋊(C3⋊C8)C2211yescharacteristic
23#63567—21no—
24(C2×D3)⋊(C3⋊C8)—21no—
25C3×#461—31no—
26#5083D311yescharacteristic
27#5078—31no—
28#5078—31no—
29#5078—61no—
30#5019—31no—
31#5012—31no—
32#4841—31no—
33#4838—31no—
34C3×#332—31no—
35#4547D311yescharacteristic, complemented
36#4486—31no—
37#4482—31no—
38#4480—31no—
39#4480—31no—
40C2×C32×Q8—21no—
41C2×#634—21no—
42C2×C3×D12D411yescharacteristic
43C2×C3×C4×D3—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C42C2311yescharacteristic
47C4⋊(C3⋊Dic3)—21no—
48C4⋊(C3⋊Dic3)—21no—
49C2×C3⋊(C3⋊C8)D411yescharacteristic
50C2×C3⋊(C3⋊C8)D411yescharacteristic
51C3×#215—42no—
52C3×(C2×D3)⋊C4—21no—
53C3×C4⋊Dic3D411yescharacteristic
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8—21no—
56C2×C3×C3⋊C8—21no—
57#3716—42no—
58#3711—42no—
59#1693—91noSylow
60C2×C3×SD16—61no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8C2×D311yescharacteristic
63C3×C4⋊Q8C2×D311yescharacteristic
64C3×C4×D4—31no—
65C2×#215—21no—
66C2×#214—61no—
67C2×#208—61no—
68C2×#204—61no—
69C2×ΓL(1,𝔽25)—61no—
70C4×D12C2×D311yescharacteristic
71#869—31no—
72#869—31no—
73#869—61no—
74C3×C4⋊C8—31no—
75C3×C4⋊C8—31no—
76C3×C4⋊C8—31no—
77C3×#103—61no—
78#835—21no—
79#830—61no—
80(C2×D3)⋊C8—61no—
81#816—61no—
82#810—61no—
83#807—61no—
84#807—61no—
85#807—61no—
86#807—61no—
87#807—62no—
88#807—62no—
89C4⋊(C3⋊C8)C2×D311yescharacteristic
90C4⋊(C3⋊C8)C2×D311yescharacteristic
91C4⋊(C3⋊C8)C2×D311yescharacteristic
92C4×C3⋊C8—31no—
93C4×C3⋊C8—31no—
94C4×C3⋊C8—61no—
95C22×C3×D3—21no—
96C32×Q8—41no—
97C32×Q8—41no—
98C2×C32×C4—41no—
99C2×C32×C4C2×D411yescharacteristic
100C2×C32×C4C2×D411yescharacteristic
101C2×C32×C4#4111yescharacteristic
102C2×C3⋊Dic3—41no—
103#634—41no—
104#634—41no—
105C2×C3×Dic3—21no—
106C3×D12—21no—
107C3×D12—21no—
108C3×C4×D3—41no—
109C3×C4×D3—41no—
110C3⋊(C3⋊C8)—21no—
111C3⋊(C3⋊C8)SD1612yescomplemented
112C3×C3⋊C8—41no—
113C3×C3⋊C8—41no—
114C3×C3⋊C8—41no—
115#500—91no—
116#461—31no—
117#461—91no—
118#436—91no—
119#334—91no—
120#332—31no—
121#332—91no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—41no—
125C2×C3×D4—31no—
126C22×C3×C4—61no—
127C2×D12#6711yescharacteristic
128C2×C4×D3—21no—
129C2×Dic6—61no—
130C2×Dic6—61no—
131C2×Dic6—121no—
132C3×SD16—122no—
133C2×C3×C8—61no—
134C2×C3×C8—61no—
135C2×C3×C8—61no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—31no—
142C3×C4⋊C4—41no—
143C3×C22⋊C4—61no—
144C3×C42—21no—
145C3×C42C22×D311yescharacteristic
146C3×C42C22×D311yescharacteristic
147#215—42no—
148#214—122no—
149(C2×D3)⋊C4—21no—
150C4⋊Dic3—62no—
151C4⋊Dic3—61no—
152C4⋊Dic3—61no—
153C4⋊Dic3—61no—
154C4⋊Dic3—61no—
155C4⋊Dic3—121no—
156C4⋊Dic3#6711yescharacteristic
157#208—122no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8—21no—
161C2×C3⋊C8—31no—
162C2×C3⋊C8—31no—
163C2×C3⋊C8—31no—
164C2×C3⋊C8—31no—
165C2×C3⋊C8—61no—
166C2×C3⋊C8—61no—
167#204—122no—
168ΓL(1,𝔽25)—122no—
169C22×C32#12111yescharacteristic
170C2×C3×D3—41no—
171C2×C3×D3—41no—
172C2×C3×D3—41no—
173C32×C4—21no—
174C32×C4—81no—
175C32×C4C2×SD1611yescharacteristic
176C32×C4AGL(1,ℤ/8)11yescharacteristic
177C32×C4AGL(1,ℤ/8)11yescharacteristic
178C32×C4#13711yescharacteristic
179C3⋊Dic3—81no—
180C3×Dic3—41no—
181C2×SD16—61no—
182C2×SD16—181no—
183C2×M16—181no—
184C4⋊Q8—91no—
185C4⋊Q8D3211yescharacteristic
186C4×D4—31no—
187#107—181no—
188#107—181no—
189C4⋊C8—31no—
190C4⋊C8—31no—
191C4⋊C8—31no—
192#103—61no—
193#103—181no—
194C22⋊C8—181no—
195C4×C8—91no—
196C23×C3—61no—
197C22×D3—21no—
198C3×Q8—41no—
199C3×Q8—41no—
200C3×Q8—41no—
201C3×Q8—41no—
202C3×Q8—81no—
203C3×Q8—81no—
204C3×D4—61no—
205C3×D4—61no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—42no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—61no—
213C2×C3×C4—121no—
214C2×C3×C4—121no—
215C2×C3×C4#23511yescharacteristic
216C2×C3×C4D3×D411yescharacteristic
217C2×C3×C4D3×D411yescharacteristic
218C2×C3×C4D3×D411yescharacteristic
219C2×C3×C4#23711yescharacteristic
220C2×C3×C4C2×#6711yescharacteristic
221C2×Dic3—21no—
222C2×Dic3—122no—
223C2×Dic3—121no—
224C2×Dic3—121no—
225D12—21no—
226D12—21no—
227C4×D3—41no—
228C4×D3—41no—
229Dic6—121no—
230Dic6—121no—
231Dic6—121no—
232Dic6—121no—
233Dic6—241no—
234Dic6—241no—
235C3×C8—121no—
236C3×C8—121no—
237C3×C8—121no—
238C3⋊C8—32no—
239C3⋊C8—32no—
240C3⋊C8—41no—
241C3⋊C8—41no—
242C3⋊C8—41no—
243C3⋊C8—62no—
244C3⋊C8—62no—
245C3⋊C8—61no—
246C3⋊C8—61no—
247C2×C32#46111yescharacteristic
248C2×C32#46911yescharacteristic
249C2×C32#47311yescharacteristic
250C3×D3—81no—
251C2×Q8—21no—
252C2×Q8—181no—
253C2×D4—31no—
254C22×C4—61no—
255SD16—122no—
256SD16—362no—
257M16—362no—
258C2×C8—61no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—91no—
262C2×C8—91no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—31no—
266C4⋊C4—181no—
267C4⋊C4—181no—
268C22⋊C4—61no—
269C42C2×D3211yescharacteristic
270C22×C3—21no—
271C22×C3—121no—
272C22×C3—121no—
273C22×C3—121no—
274C22×C3#88411yescharacteristic
275C22×C3#93911yescharacteristic
276C2×D3—41no—
277C2×D3—41no—
278C2×D3—41no—
279C3×C4—22no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—82no—
287C3×C4—81no—
288C3×C4—81no—
289C3×C4—121no—
290C3×C4#91111yescharacteristic
291C3×C4D3×SD1611yescharacteristic
292C3×C4D3×SD1611yescharacteristic
293C3×C4#91411yescharacteristic
294C3×C4#91511yescharacteristic
295C3×C4#91811yescharacteristic
296C3×C4#93211yescharacteristic
297C3×C4#94911yescharacteristic
298Dic3—41no—
299Dic3—242no—
300Dic3—241no—
301Dic3—241no—
302C32#169311yescharacteristic, complemented, Sylow
303C23—61no—
304Q8—41no—
305Q8—41no—
306Q8—361no—
307Q8—361no—
308D4—61no—
309D4—61no—
310C2×C4—41no—
311C2×C4—61no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—361no—
315C2×C4#380411yescharacteristic
316C2×C4D3×#6711yescharacteristic
317C2×C4#381111yescharacteristic
318C8—92no—
319C8—121no—
320C8—121no—
321C8—121no—
322C8—181no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—241no—
327C2×C3#475511yescharacteristic
328C2×C3#478211yescharacteristic
329C2×C3#481811yescharacteristic
330C2×C3#503911yescharacteristic
331C2×C3#516211yescharacteristic
332C2×C3#520411yescharacteristic
333D3—81no—
334C22—121no—
335C22—121no—
336C22—121no—
337C22#6397111yescharacteristic
338C4—21no—
339C4—81no—
340C4—121no—
341C4—721no—
342C4#6392511yescharacteristic
343C4#6393411yescharacteristic
344C4#6394011yescharacteristic
345C4#6395111yescharacteristic
346C3—21no—
347C3#6845111yescharacteristic, complemented
348C3#7104111yescharacteristic, complemented
349C2—241no—
350C2#1059145311yescharacteristic
351C2#1059156911yescharacteristic
352C2#1059164411yescharacteristic
subgroupG/H
centerC22#63971
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radicalC2×C3×D12D4
hypercenterC4⋊Q8D32
hypocenterC32#1693
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
1. #10592777C2semidirect
5. C3×#4547C2semidirect
6. #10589098C2semidirect
7. #10589094C2semidirect
35. #4547D3semidirect
111. C3⋊(C3⋊C8)SD16semidirect
302. C32#1693semidirect
347. C3#68451semidirect
348. C3#71041semidirect

Maximal subgroups

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

subgroupindex
#105927772
C3×#50832
#105903652
#105903392
C3×#45472
#105890982
#105890942
#710413
#684513