Finite Group Databasebeta

← #11781859

Subgroups — #11781859

Order 1152.

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

normal subgroups only

This group has 352 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
1C3×#4838C211yesmaximal, characteristic
2#10590365C211yesmaximal, characteristic
3#10590303C211yesmaximal, characteristic, complemented
4#10590266C211yesmaximal, characteristic
5#10589352C211yesmaximal, characteristic, complemented
6C3×#4481C211yesmaximal, characteristic, complemented
7#10589099C211yesmaximal, characteristic, complemented
8#70872—31nomaximal
9#68500—31nomaximal
10#64078C2211yescharacteristic
11C2×C3×ΓL(1,𝔽25)—21no—
12C3×C4×D12C2211yescharacteristic
13C3×#869C2211yescharacteristic
14C2×#3716—21no—
15C32×C4⋊C8C2211yescharacteristic
16C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×(C2×D3)⋊C8—21no—
18C3×C4⋊(C3⋊C8)C2211yescharacteristic
19C3×C4×C3⋊C8C2211yescharacteristic
20#63574—21no—
21#63573—21no—
22#63573—21no—
23#63567—21no—
24#63562—21no—
25C3×#436—31no—
26#5084—31no—
27#5080—31no—
28#5019—31no—
29#5008—31no—
30#4841—31no—
31#4838D311yescharacteristic
32#4680—31no—
33C3×#333—31no—
34#4486—31no—
35#4486—31no—
36#4486—61no—
37#4481D311yescharacteristic, complemented
38#4479—31no—
39#4458—31no—
40C2×#634—21no—
41C2×C3×D12D411yescharacteristic
42C2×C3×C4×D3—21no—
43C2×C3×Dic6—21no—
44C2×C32×C8—21no—
45C32×C42C2311yescharacteristic
46C4⋊(C3⋊Dic3)—21no—
47C4⋊(C3⋊Dic3)—21no—
48C2×C3⋊(C3⋊C8)—21no—
49C3×(C2×D3)⋊C4—21no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3D411yescharacteristic
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56C3×ΓL(1,𝔽25)—42no—
57#3716—42no—
58#1691—91noSylow
59C2×C3×M16—61no—
60C3×C4⋊Q8—31no—
61C3×C4×D4—31no—
62C2×#214—61no—
63C2×#204—61no—
64C2×ΓL(1,𝔽25)—21no—
65C4×D12C2×D311yescharacteristic
66#869—31no—
67#869—31no—
68#869—61no—
69#869C2×D311yescharacteristic
70C3×C4⋊C8—21no—
71C3×C4⋊C8—31no—
72C3×C4⋊C8C2×D311yescharacteristic
73C3×C4⋊C8C2×D311yescharacteristic
74C3×C22⋊C8—61no—
75C3×C4×C8—31no—
76#832—61no—
77(C2×D3)⋊C8—21no—
78#817—61no—
79#817—61no—
80#816—61no—
81#810—61no—
82#809—61no—
83#808—61no—
84#808—61no—
85#808—61no—
86#808—61no—
87C4⋊(C3⋊C8)—31no—
88C4⋊(C3⋊C8)—31no—
89C4⋊(C3⋊C8)—61no—
90C4⋊(C3⋊C8)C2×D311yescharacteristic
91C4×C3⋊C8C2×D311yescharacteristic
92C22×C3×D3—21no—
93C2×C32×C4C2×D411yescharacteristic
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4#4111yescharacteristic
96C2×C3⋊Dic3—41no—
97#634—41no—
98#634—41no—
99C2×C3×Dic3—21no—
100C2×C3×Dic3—41no—
101C3×D12—21no—
102C3×D12—21no—
103C3×C4×D3—41no—
104C3×C4×D3—41no—
105C3×Dic6—81no—
106C32×C8—41no—
107C3⋊(C3⋊C8)—41no—
108C3×C3⋊C8—21no—
109C3×C3⋊C8—41no—
110C3×C3⋊C8SD1612yescomplemented
111#498—91no—
112#476—91no—
113#461—91no—
114#436—31no—
115#335—91no—
116#333—31no—
117#332—91no—
118C2×C3×Q8—61no—
119C2×C3×D4—31no—
120C22×C3×C4—61no—
121C2×D12D1211yescharacteristic
122C2×C4×D3—21no—
123C2×Dic6—21no—
124C2×Dic6—61no—
125C2×Dic6—61no—
126C2×Dic6—121no—
127C3×M16—122no—
128C2×C3×C8—21no—
129C2×C3×C8—21no—
130C2×C3×C8—31no—
131C2×C3×C8—31no—
132C2×C3×C8—41no—
133C2×C3×C8—61no—
134C3×C4⋊C4—31no—
135C3×C4⋊C4—61no—
136C3×C4⋊C4—61no—
137C3×C22⋊C4—61no—
138C3×C42—21no—
139C3×C42C22×D311yescharacteristic
140C3×C42C22×D311yescharacteristic
141#214—122no—
142(C2×D3)⋊C4—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—62no—
146C4⋊Dic3—61no—
147C4⋊Dic3—61no—
148C4⋊Dic3—61no—
149C4⋊Dic3—61no—
150C4⋊Dic3—121no—
151C4⋊Dic3D1211yescharacteristic
152C2×C3⋊C8—21no—
153C2×C3⋊C8—61no—
154C2×C3⋊C8—61no—
155C2×C3⋊C8—121no—
156C2×C3⋊C8#6711yescharacteristic
157C2×C3⋊C8#6711yescharacteristic
158#204—122no—
159ΓL(1,𝔽25)—42no—
160C22×C32#12111yescharacteristic
161C2×C3×D3—41no—
162C2×C3×D3—41no—
163C2×C3×D3—41no—
164C32×C4—21no—
165C32×C4C2×SD1611yescharacteristic
166C32×C4AGL(1,ℤ/8)11yescharacteristic
167C32×C4#13711yescharacteristic
168C32×C4#13711yescharacteristic
169C3⋊Dic3—81no—
170C3×Dic3—41no—
171C3×Dic3—81no—
172C2×SD16—181no—
173C2×M16—61no—
174C4⋊Q8—31no—
175C4⋊Q8—91no—
176C4×D4—31no—
177#106—181no—
178#106—181no—
179#106—181no—
180C4⋊C8—31no—
181C4⋊C8—91no—
182C4⋊C8D3211yescharacteristic
183#103—181no—
184#102—181no—
185C22⋊C8—61no—
186C4×C8—31no—
187C23×C3—61no—
188C22×D3—21no—
189C3×Q8—241no—
190C3×D4—61no—
191C3×D4—61no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—61no—
196C2×C3×C4—121no—
197C2×C3×C4—121no—
198C2×C3×C4—121no—
199C2×C3×C4C2×D1211yescharacteristic
200C2×C3×C4#23511yescharacteristic
201C2×C3×C4#23511yescharacteristic
202C2×C3×C4D3×D411yescharacteristic
203C2×C3×C4D3×D411yescharacteristic
204C2×C3×C4C2×#6711yescharacteristic
205C2×Dic3—21no—
206C2×Dic3—41no—
207C2×Dic3—122no—
208C2×Dic3—121no—
209C2×Dic3—121no—
210D12—21no—
211D12—21no—
212C4×D3—41no—
213C4×D3—41no—
214Dic6—81no—
215Dic6—121no—
216Dic6—121no—
217Dic6—121no—
218Dic6—121no—
219Dic6—241no—
220Dic6—241no—
221C3×C8—32no—
222C3×C8—41no—
223C3×C8—41no—
224C3×C8—61no—
225C3×C8—81no—
226C3×C8—121no—
227C3⋊C8—21no—
228C3⋊C8—41no—
229C3⋊C8—121no—
230C3⋊C8—121no—
231C3⋊C8—241no—
232C3⋊C8#21412yescomplemented
233C2×C32#46111yescharacteristic
234C2×C32#46811yescharacteristic
235C2×C32#47111yescharacteristic
236C3×D3—81no—
237C2×Q8—61no—
238C2×Q8—181no—
239C2×D4—31no—
240C22×C4—61no—
241SD16—362no—
242M16—122no—
243C2×C8—21no—
244C2×C8—31no—
245C2×C8—31no—
246C2×C8—61no—
247C2×C8—181no—
248C4⋊C4—31no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—181no—
252C4⋊C4—181no—
253C22⋊C4—61no—
254C42C2×D3211yescharacteristic
255C22×C3—21no—
256C22×C3—121no—
257C22×C3—121no—
258C22×C3—121no—
259C22×C3#88411yescharacteristic
260C22×C3#93011yescharacteristic
261C2×D3—41no—
262C2×D3—41no—
263C2×D3—41no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—41no—
271C3×C4—121no—
272C3×C4—241no—
273C3×C4#90811yescharacteristic
274C3×C4#90911yescharacteristic
275C3×C4D3×SD1611yescharacteristic
276C3×C4#91411yescharacteristic
277C3×C4#91811yescharacteristic
278C3×C4#91811yescharacteristic
279C3×C4C2×#21411yescharacteristic
280C3×C4#95111yescharacteristic
281Dic3—41no—
282Dic3—81no—
283Dic3—242no—
284Dic3—241no—
285Dic3—241no—
286C32#169111yescharacteristic, complemented, Sylow
287C23—61no—
288Q8—241no—
289Q8—361no—
290Q8—361no—
291D4—61no—
292D4—61no—
293C2×C4—61no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—361no—
298C2×C4#379811yescharacteristic
299C2×C4D3×D1211yescharacteristic
300C2×C4D3×#6711yescharacteristic
301C8—32no—
302C8—41no—
303C8—61no—
304C8—121no—
305C8—361no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—241no—
310C2×C3#474811yescharacteristic
311C2×C3#481211yescharacteristic
312C2×C3#481811yescharacteristic
313C2×C3#501911yescharacteristic
314C2×C3#513411yescharacteristic
315C2×C3#514111yescharacteristic
316D3—81no—
317C22—121no—
318C22—121no—
319C22—121no—
320C22#6390611yescharacteristic
321C4—21no—
322C4—121no—
323C4—241no—
324C4—721no—
325C4#6379411yescharacteristic
326C4#6380111yescharacteristic
327C4D3×#21411yescharacteristic
328C4#6394311yescharacteristic
329C3—21no—
330C3#6850011yescharacteristic, complemented
331C3#7087211yescharacteristic, complemented
332C2—241no—
333C2#1059055911yescharacteristic
334C2#1059085011yescharacteristic
335C2#1059099611yescharacteristic
subgroupG/H
centerC22#63906
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32#121
prime radicalC2×C3×D12D4
hypercenterC4⋊C8D32
hypocenterC32#1691
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. #10590303C2semidirect
5. #10589352C2semidirect
6. C3×#4481C2semidirect
7. #10589099C2semidirect
37. #4481D3semidirect
110. C3×C3⋊C8SD16semidirect
232. C3⋊C8#214semidirect
286. C32#1691semidirect
330. C3#68500semidirect
331. C3#70872semidirect

Maximal subgroups

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

subgroupindex
C3×#48382
#105903652
#105903032
#105902662
#105893522
C3×#44812
#105890992
#708723
#685003