Finite Group Databasebeta

← #11781043

Subgroups — #11781043

Order 1152.

386 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 88 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#10592541C211yesmaximal, characteristic
2#10590379C211yesmaximal, characteristic
3#10590327C211yesmaximal, characteristic
4#10590303C211yesmaximal, characteristic, complemented
5C3×#4482C211yesmaximal, characteristic, complemented
6C3×#4480C211yesmaximal, characteristic, complemented
7#10589096C211yesmaximal, characteristic, complemented
8#68537—31nomaximal
9#68442—31nomaximal
10C4×C4⋊(C3⋊D3)C2211yescharacteristic
11#64078C2211yescharacteristic
12C3×#869C2211yescharacteristic
13C2×#3717—21no—
14C2×#3710—21no—
15C32×C4⋊C8C2211yescharacteristic
16#63648—21no—
17#63644—21no—
18C3×C4⋊(C3⋊C8)C2211yescharacteristic
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4×C3⋊C8C2211yescharacteristic
21#63573—21no—
22#63573—21no—
23#63566—21no—
24#63557—21no—
25#5083—31no—
26#5080—31no—
27#4842—31no—
28#4842—31no—
29#4842—61no—
30#4841—31no—
31#4834—31no—
32C8×D12—31no—
33#4680—31no—
34C3×#334—31no—
35C3×#332—31no—
36#4482D311yescharacteristic, complemented
37#4481—31no—
38#4481—31no—
39#4480D311yescharacteristic, complemented
40C2×C4⋊(C3⋊D3)D411yescharacteristic
41C2×C4×C3⋊D3—21no—
42C2×#634—21no—
43C2×C3×Dic6—21no—
44C2×C32×C8—21no—
45C32×C42C2311yescharacteristic
46#3752—21no—
47C4⋊(C3⋊Dic3)—21no—
48C4⋊(C3⋊Dic3)—21no—
49C4⋊(C3⋊Dic3)D411yescharacteristic
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—21no—
52C2×C3×C3⋊C8—21no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56#3717—42no—
57#3710—42no—
58#1671—91noSylow
59C3×C4⋊Q8—31no—
60C2×#215—61no—
61C2×#204—61no—
62C2×D3×C8—61no—
63C2×D3×C8—61no—
64C4×D12—31no—
65C4×D12—31no—
66C4×D12—61no—
67#869—31no—
68#869—31no—
69#869—61no—
70#869C2×D311yescharacteristic
71C3×C4⋊C8—21no—
72C3×C4⋊C8—31no—
73C3×C4⋊C8—31no—
74C3×C4⋊C8C2×D311yescharacteristic
75C3×C4⋊C8C2×D311yescharacteristic
76C3×C4×C8—31no—
77#835—61no—
78#821—61no—
79#821—61no—
80#821—62no—
81(C2×D3)⋊C8—61no—
82(C2×D3)⋊C8—61no—
83#817—61no—
84#817—61no—
85#817—61no—
86#817—61no—
87#817—62no—
88#816—61no—
89#808—61no—
90#808—61no—
91C4⋊(C3⋊C8)C2×D311yescharacteristic
92C4⋊(C3⋊C8)C2×D311yescharacteristic
93C4×C3⋊C8C2×D311yescharacteristic
94C22×C3⋊D3—21no—
95C2×C32×C4C2×D411yescharacteristic
96C2×C32×C4C2×D411yescharacteristic
97C2×C32×C4#4111yescharacteristic
98C2×C3⋊Dic3—21no—
99C2×C3⋊Dic3—41no—
100C4⋊(C3⋊D3)—21no—
101C4⋊(C3⋊D3)SD1612yes—
102C4×C3⋊D3—41no—
103C4×C3⋊D3—41no—
104#634—81no—
105C2×C3×Dic3—41no—
106C3×Dic6—41no—
107C3×Dic6—41no—
108C32×C8—41no—
109C3×C3⋊C8—21no—
110C3×C3⋊C8—41no—
111C3×C3⋊C8—41no—
112C3×C3⋊C8SD1612yescomplemented
113#498—91no—
114#476—91no—
115#461—91no—
116C8×D4—91no—
117#334—31no—
118#333—91no—
119#332—31no—
120C2×C3×Q8—61no—
121C2×D12—31no—
122C2×D12—31no—
123C2×D12—61no—
124C2×C4×D3—61no—
125C2×C4×D3—61no—
126C2×C4×D3—121no—
127C2×Dic6—21no—
128C2×Dic6—61no—
129C2×Dic6—61no—
130C2×Dic6—121no—
131C2×C3×C8—22no—
132C2×C3×C8—21no—
133C2×C3×C8—21no—
134C2×C3×C8—31no—
135C2×C3×C8—31no—
136C2×C3×C8—61no—
137C2×C3×C8—61no—
138C3×C4⋊C4—61no—
139C3×C4⋊C4—61no—
140C3×C42—21no—
141C3×C42C22×D311yescharacteristic
142C3×C42C22×D311yescharacteristic
143#215—122no—
144(C2×D3)⋊C4—61no—
145(C2×D3)⋊C4—61no—
146(C2×D3)⋊C4—121no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—31no—
150C4⋊Dic3—31no—
151C4⋊Dic3—61no—
152C4⋊Dic3—62no—
153C4⋊Dic3—61no—
154C4⋊Dic3—61no—
155C4⋊Dic3—61no—
156C4⋊Dic3—61no—
157C4⋊Dic3—121no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8D1211yescharacteristic
161C2×C3⋊C8D1211yescharacteristic
162#204—122no—
163D3×C8—122no—
164D3×C8—122no—
165C22×C32#12111yescharacteristic
166C2×C3⋊D3—22no—
167C2×C3⋊D3—41no—
168C2×C3⋊D3—41no—
169C32×C4—21no—
170C32×C4C2×SD1611yescharacteristic
171C32×C4C2×SD1611yescharacteristic
172C32×C4#13511yescharacteristic
173C32×C4#13711yescharacteristic
174C3⋊Dic3—41no—
175C3⋊Dic3—81no—
176C3×Dic3—81no—
177C2×SD16—181no—
178C22×C8—181no—
179C4⋊Q8—31no—
180C4⋊Q8—91no—
181C4×D4—91no—
182#106—181no—
183#106—181no—
184#106—181no—
185C4⋊C8—31no—
186C4⋊C8—31no—
187C4⋊C8D3211yescharacteristic
188#103—181no—
189#102—181no—
190C22⋊C8—181no—
191C4×C8—31no—
192C22×D3—61no—
193C22×D3—61no—
194C22×D3—121no—
195C3×Q8—121no—
196C3×Q8—121no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—121no—
201C2×C3×C4C2×D1211yescharacteristic
202C2×C3×C4#23511yescharacteristic
203C2×C3×C4D3×D411yescharacteristic
204C2×C3×C4D3×D411yescharacteristic
205C2×C3×C4D3×D411yescharacteristic
206C2×C3×C4#23911yescharacteristic
207C2×Dic3—41no—
208C2×Dic3—61no—
209C2×Dic3—61no—
210C2×Dic3—121no—
211C2×Dic3—122no—
212C2×Dic3—121no—
213C2×Dic3—121no—
214D12—32no—
215D12—32no—
216D12—62no—
217D12—61no—
218D12—61no—
219D12—121no—
220C4×D3—121no—
221C4×D3—121no—
222C4×D3—121no—
223C4×D3—121no—
224C4×D3—241no—
225C4×D3—241no—
226Dic6—41no—
227Dic6—41no—
228Dic6—242no—
229Dic6—241no—
230Dic6—241no—
231C3×C8—32no—
232C3×C8—42no—
233C3×C8—41no—
234C3×C8—41no—
235C3×C8—61no—
236C3×C8—121no—
237C3×C8—121no—
238C3⋊C8—21no—
239C3⋊C8—41no—
240C3⋊C8—41no—
241C3⋊C8#20412yescomplemented
242C2×C32#46111yescharacteristic
243C2×C32#46511yescharacteristic
244C2×C32#47111yescharacteristic
245C3⋊D3—42no—
246C2×Q8—61no—
247C2×Q8—181no—
248C2×D4—91no—
249C22×C4—181no—
250SD16—362no—
251C2×C8—21no—
252C2×C8—31no—
253C2×C8—31no—
254C2×C8—61no—
255C2×C8—61no—
256C2×C8—362no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—91no—
260C4⋊C4—181no—
261C4⋊C4—181no—
262C22⋊C4—181no—
263C42C2×D3211yescharacteristic
264C22×C3—21no—
265C22×C3#88411yescharacteristic
266C22×C3#89511yescharacteristic
267C2×D3—62no—
268C2×D3—62no—
269C2×D3—122no—
270C2×D3—121no—
271C2×D3—121no—
272C2×D3—121no—
273C2×D3—121no—
274C2×D3—241no—
275C2×D3—241no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—41no—
283C3×C4—241no—
284C3×C4C2×#20411yescharacteristic
285C3×C4#90911yescharacteristic
286C3×C4#91211yescharacteristic
287C3×C4D3×SD1611yescharacteristic
288C3×C4D3×SD1611yescharacteristic
289C3×C4D3×SD1611yescharacteristic
290C3×C4#91611yescharacteristic
291C3×C4#91811yescharacteristic
292Dic3—81no—
293Dic3—121no—
294Dic3—121no—
295Dic3—241no—
296Dic3—242no—
297Dic3—241no—
298Dic3—241no—
299C32#167111yescharacteristic, complemented, Sylow
300C23—181no—
301Q8—121no—
302Q8—121no—
303Q8—721no—
304D4—92no—
305D4—181no—
306C2×C4—121no—
307C2×C4—181no—
308C2×C4—361no—
309C2×C4—361no—
310C2×C4—361no—
311C2×C4#379911yescharacteristic
312C2×C4D3×D1211yescharacteristic
313C2×C4#381111yescharacteristic
314C8—32no—
315C8—41no—
316C8—61no—
317C8—121no—
318C8—121no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3#475411yescharacteristic
323C2×C3#477811yescharacteristic
324C2×C3#481811yescharacteristic
325C2×C3#484111yescharacteristic
326C2×C3#486411yescharacteristic
327C2×C3#489811yescharacteristic
328D3—122no—
329D3—122no—
330D3—242no—
331C22—182no—
332C22—361no—
333C22—361no—
334C22#6391011yescharacteristic
335C4—21no—
336C4—241no—
337C4—361no—
338C4—721no—
339C4D3×#20411yescharacteristic
340C4#6380111yescharacteristic
341C4#6393611yescharacteristic
342C4#6394011yescharacteristic
343C3—21no—
344C3#6844211yescharacteristic, complemented
345C3#6853711yescharacteristic, complemented
346C2—362no—
347C2#1059050711yescharacteristic
348C2#1059076011yescharacteristic
349C2#1059099011yescharacteristic
subgroupG/H
centerC22#63910
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32#121
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenterC4⋊C8D32
hypocenterC32#1671
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
4. #10590303C2semidirect
5. C3×#4482C2semidirect
6. C3×#4480C2semidirect
7. #10589096C2semidirect
36. #4482D3semidirect
39. #4480D3semidirect
112. C3×C3⋊C8SD16semidirect
241. C3⋊C8#204semidirect
299. C32#1671semidirect
344. C3#68442semidirect
345. C3#68537semidirect

Maximal subgroups

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

subgroupindex
#105925412
#105903792
#105903272
#105903032
C3×#44822
C3×#44802
#105890962
#685373
#684423