Finite Group Databasebeta

← #11782005

Subgroups — #11782005

Order 1152.

362 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 86 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#10590338C211yesmaximal, characteristic
3#10590311C211yesmaximal, characteristic, complemented
4#10590300C211yesmaximal, characteristic
5C3×#4481C211yesmaximal, characteristic, complemented
6C3×#4480C211yesmaximal, characteristic, complemented
7#10589096C211yesmaximal, characteristic, complemented
8#68543—31nomaximal
9#68501—31nomaximal
10C4×C4⋊(C3⋊D3)C2211yescharacteristic
11#64078C2211yescharacteristic
12C3×#869C2211yescharacteristic
13C2×#3712—21no—
14C32×C4⋊C8C2211yescharacteristic
15#63648—21no—
16#63644—21no—
17C3×C4⋊(C3⋊C8)C2211yescharacteristic
18C3×C4⋊(C3⋊C8)C2211yescharacteristic
19C3×C4×C3⋊C8C2211yescharacteristic
20#63575—21no—
21#63575—21no—
22#63573—21no—
23#63564—21no—
24#63557—21no—
25#5084—31no—
26#5078—31no—
27#4842—31no—
28#4842—31no—
29#4842—31no—
30#4842—61no—
31#4838—31no—
32#4688—31no—
33#4682—31no—
34C3×#334—31no—
35C3×#333—31no—
36#4481—31no—
37#4481—31no—
38#4481D311yescharacteristic, complemented
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#3712—42no—
57#1695—91noSylow
58C3×C4⋊Q8—31no—
59C2×ΓL(1,𝔽25)—61no—
60C2×ΓL(1,𝔽25)—61no—
61C4×D12—31no—
62C4×D12—31no—
63C4×D12—61no—
64#869—31no—
65#869—31no—
66#869—61no—
67#869C2×D311yescharacteristic
68C3×C4⋊C8—21no—
69C3×C4⋊C8—31no—
70C3×C4⋊C8—31no—
71C3×C4⋊C8C2×D311yescharacteristic
72C3×C4⋊C8C2×D311yescharacteristic
73C3×C4×C8—31no—
74#821—61no—
75#821—61no—
76#821—61no—
77#821—62no—
78(C2×D3)⋊C8—61no—
79(C2×D3)⋊C8—61no—
80#818—61no—
81#818—61no—
82#817—61no—
83#817—61no—
84#817—61no—
85#817—62no—
86#809—61no—
87#808—61no—
88#807—61no—
89#807—61no—
90C4⋊(C3⋊C8)C2×D311yescharacteristic
91C4⋊(C3⋊C8)C2×D311yescharacteristic
92C4×C3⋊C8C2×D311yescharacteristic
93C22×C3⋊D3—21no—
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4C2×D411yescharacteristic
96C2×C32×C4#4111yescharacteristic
97C2×C3⋊Dic3—21no—
98C2×C3⋊Dic3—41no—
99C4⋊(C3⋊D3)—21no—
100C4⋊(C3⋊D3)—21no—
101C4×C3⋊D3—41no—
102C4×C3⋊D3—41no—
103#634—81no—
104C2×C3×Dic3—41no—
105C3×Dic6—81no—
106C32×C8—41no—
107C3×C3⋊C8—21no—
108C3×C3⋊C8—41no—
109C3×C3⋊C8—41no—
110C3×C3⋊C8SD1612yes—
111#500—91no—
112#476—91no—
113#476—91no—
114#436—91no—
115#334—31no—
116#333—31no—
117#333—91no—
118C2×C3×Q8—61no—
119C2×D12—31no—
120C2×D12—31no—
121C2×D12—61no—
122C2×C4×D3—61no—
123C2×C4×D3—61no—
124C2×C4×D3—121no—
125C2×Dic6—21no—
126C2×Dic6—61no—
127C2×Dic6—61no—
128C2×Dic6—121no—
129C2×C3×C8—22no—
130C2×C3×C8—21no—
131C2×C3×C8—21no—
132C2×C3×C8—31no—
133C2×C3×C8—31no—
134C2×C3×C8—61no—
135C2×C3×C8—61no—
136C3×C4⋊C4—61no—
137C3×C4⋊C4—61no—
138C3×C42—21no—
139C3×C42C22×D311yescharacteristic
140C3×C42C22×D311yescharacteristic
141(C2×D3)⋊C4—61no—
142(C2×D3)⋊C4—61no—
143(C2×D3)⋊C4—121no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—31no—
147C4⋊Dic3—31no—
148C4⋊Dic3—61no—
149C4⋊Dic3—62no—
150C4⋊Dic3—61no—
151C4⋊Dic3—61no—
152C4⋊Dic3—61no—
153C4⋊Dic3—61no—
154C4⋊Dic3—121no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—21no—
157C2×C3⋊C8D1211yescharacteristic
158C2×C3⋊C8D1211yescharacteristic
159ΓL(1,𝔽25)—122no—
160ΓL(1,𝔽25)—122no—
161C22×C32#12111yescharacteristic
162C2×C3⋊D3—41no—
163C2×C3⋊D3—41no—
164C2×C3⋊D3—41no—
165C32×C4—21no—
166C32×C4C2×SD1611yescharacteristic
167C32×C4#13711yescharacteristic
168C32×C4#13711yescharacteristic
169C32×C4#13711yescharacteristic
170C3⋊Dic3—41no—
171C3⋊Dic3—81no—
172C3×Dic3—81no—
173C2×M16—181no—
174C4⋊Q8—31no—
175C4⋊Q8—91no—
176C4×D4—91no—
177#107—181no—
178#107—181no—
179#106—181no—
180#106—181no—
181C4⋊C8—31no—
182C4⋊C8—31no—
183C4⋊C8D3211yescharacteristic
184#102—181no—
185#102—181no—
186C22⋊C8—181no—
187C4×C8—31no—
188C22×D3—61no—
189C22×D3—61no—
190C22×D3—121no—
191C3×Q8—241no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—121no—
196C2×C3×C4C2×D1211yescharacteristic
197C2×C3×C4#23511yescharacteristic
198C2×C3×C4D3×D411yescharacteristic
199C2×C3×C4D3×D411yescharacteristic
200C2×C3×C4D3×D411yescharacteristic
201C2×C3×C4#23911yescharacteristic
202C2×Dic3—41no—
203C2×Dic3—61no—
204C2×Dic3—61no—
205C2×Dic3—121no—
206C2×Dic3—122no—
207C2×Dic3—121no—
208C2×Dic3—121no—
209D12—62no—
210D12—61no—
211D12—61no—
212D12—61no—
213D12—61no—
214D12—121no—
215C4×D3—121no—
216C4×D3—121no—
217C4×D3—121no—
218C4×D3—121no—
219C4×D3—241no—
220C4×D3—241no—
221Dic6—81no—
222Dic6—242no—
223Dic6—241no—
224Dic6—241no—
225C3×C8—32no—
226C3×C8—42no—
227C3×C8—41no—
228C3×C8—41no—
229C3×C8—61no—
230C3×C8—121no—
231C3×C8—121no—
232C3⋊C8—21no—
233C3⋊C8—41no—
234C3⋊C8—41no—
235C3⋊C8#20412yes—
236C2×C32#46111yescharacteristic
237C2×C32#47011yescharacteristic
238C2×C32#47211yescharacteristic
239C3⋊D3—81no—
240C2×Q8—61no—
241C2×Q8—181no—
242C2×D4—91no—
243C22×C4—181no—
244M16—362no—
245C2×C8—21no—
246C2×C8—31no—
247C2×C8—31no—
248C2×C8—61no—
249C2×C8—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—91no—
253C4⋊C4—181no—
254C4⋊C4—181no—
255C22⋊C4—181no—
256C42C2×D3211yescharacteristic
257C22×C3—21no—
258C22×C3#88411yescharacteristic
259C22×C3#89511yescharacteristic
260C2×D3—122no—
261C2×D3—121no—
262C2×D3—121no—
263C2×D3—121no—
264C2×D3—121no—
265C2×D3—121no—
266C2×D3—121no—
267C2×D3—241no—
268C2×D3—241no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—41no—
276C3×C4—241no—
277C3×C4C2×#20411yescharacteristic
278C3×C4#90911yescharacteristic
279C3×C4D3×SD1611yescharacteristic
280C3×C4#91511yescharacteristic
281C3×C4#91511yescharacteristic
282C3×C4#91511yescharacteristic
283C3×C4#91811yescharacteristic
284C3×C4#91811yescharacteristic
285Dic3—81no—
286Dic3—121no—
287Dic3—121no—
288Dic3—241no—
289Dic3—242no—
290Dic3—241no—
291Dic3—241no—
292C32#169511yescharacteristic, complemented, Sylow
293C23—181no—
294Q8—241no—
295Q8—721no—
296D4—181no—
297D4—181no—
298C2×C4—121no—
299C2×C4—181no—
300C2×C4—361no—
301C2×C4—361no—
302C2×C4—361no—
303C2×C4#379911yescharacteristic
304C2×C4D3×D1211yescharacteristic
305C2×C4#381111yescharacteristic
306C8—32no—
307C8—41no—
308C8—61no—
309C8—121no—
310C8—121no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3#474911yescharacteristic
315C2×C3#481611yescharacteristic
316C2×C3#481811yescharacteristic
317C2×C3#484111yescharacteristic
318C2×C3#486711yescharacteristic
319C2×C3#489711yescharacteristic
320D3—242no—
321D3—241no—
322D3—241no—
323C22—361no—
324C22—361no—
325C22—361no—
326C22#6391011yescharacteristic
327C4—21no—
328C4—241no—
329C4—361no—
330C4—721no—
331C4D3×#20411yescharacteristic
332C4#6380111yescharacteristic
333C4#6393111yescharacteristic
334C4#6395111yescharacteristic
335C3—21no—
336C3#6850111yescharacteristic, complemented
337C3#6854311yescharacteristic, complemented
338C2—721no—
339C2#1059052411yescharacteristic
340C2#1059076011yescharacteristic
341C2#1059098711yescharacteristic
subgroupG/H
centerC22#63910
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32#121
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenterC4⋊C8D32
hypocenterC32#1695
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
3. #10590311C2semidirect
5. C3×#4481C2semidirect
6. C3×#4480C2semidirect
7. #10589096C2semidirect
38. #4481D3semidirect
39. #4480D3semidirect
292. C32#1695semidirect
336. C3#68501semidirect
337. C3#68543semidirect

Maximal subgroups

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

subgroupindex
#105925412
#105903382
#105903112
#105903002
C3×#44812
C3×#44802
#105890962
#685433
#685013