Finite Group Databasebeta

← #11781997

Subgroups — #11781997

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#10592783C211yesmaximal, characteristic
2C3×#5078C211yesmaximal, characteristic, complemented
3#10590338C211yesmaximal, characteristic
4#10590300C211yesmaximal, characteristic
5C3×#4546C211yesmaximal, characteristic, complemented
6#10589097C211yesmaximal, characteristic, complemented
7#10589095C211yesmaximal, characteristic, complemented
8#71374—31nomaximal
9#68501—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C4×C4⋊(C3⋊D3)C2211yescharacteristic
12C3×#869C2211yescharacteristic
13C2×#3712—21no—
14#63636—21no—
15#63635—21no—
16C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#807—21no—
18C3×#807—21no—
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4×C3⋊C8C2211yescharacteristic
22#63573—21no—
23#63564—21no—
24#63557—21no—
25C3×#500—31no—
26#5084—31no—
27#5084—31no—
28#5084—31no—
29#5084—61no—
30#5078D311yescharacteristic, complemented
31#4842—31no—
32#4838—31no—
33#4688—31no—
34C3×#333—31no—
35#4546—31no—
36#4546D311yescharacteristic, complemented
37#4481—31no—
38#4480—31no—
39#4457—31no—
40C2×C32×Q8—21no—
41C2×C4⋊(C3⋊D3)D411yescharacteristic
42C2×C4×C3⋊D3—21no—
43C2×C3×Dic6—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C42C2311yescharacteristic
47#3752—21no—
48C4⋊(C3⋊Dic3)D411yescharacteristic
49C2×C3⋊(C3⋊C8)—21no—
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—21no—
59C3×C4⋊Q8—31no—
60C3×C4⋊Q8C2×D311yescharacteristic
61C3×C4⋊Q8C2×D311yescharacteristic
62C2×ΓL(1,𝔽25)—61no—
63C2×ΓL(1,𝔽25)—61no—
64C4×D12—31no—
65C4×D12—31no—
66C4×D12—61no—
67#869C2×D311yescharacteristic
68C3×#107—61no—
69C3×#107—61no—
70C3×C4⋊C8—31no—
71C3×C4⋊C8—31no—
72C3×C4×C8—31no—
73#821—61no—
74(C2×D3)⋊C8—61no—
75(C2×D3)⋊C8—61no—
76#817—61no—
77#809—61no—
78#809—61no—
79#809—61no—
80#809—62no—
81#808—61no—
82#808—61no—
83#808—61no—
84#808—62no—
85#807—21no—
86#807—21no—
87C4⋊(C3⋊C8)—31no—
88C4⋊(C3⋊C8)—31no—
89C4⋊(C3⋊C8)—61no—
90C4⋊(C3⋊C8)C2×D311yescharacteristic
91C4⋊(C3⋊C8)C2×D311yescharacteristic
92C4×C3⋊C8C2×D311yescharacteristic
93C22×C3⋊D3—21no—
94C32×Q8—81no—
95C2×C32×C4—41no—
96C2×C32×C4C2×D411yescharacteristic
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4#4111yescharacteristic
99C2×C3⋊Dic3—21no—
100C4⋊(C3⋊D3)—21no—
101C4⋊(C3⋊D3)—21no—
102C4×C3⋊D3—41no—
103C4×C3⋊D3—41no—
104C2×C3×Dic3—41no—
105C3×Dic6—81no—
106C3⋊(C3⋊C8)—41no—
107C3×C3⋊C8—21no—
108C3×C3⋊C8—41no—
109C3×C3⋊C8—41no—
110C3×C3⋊C8SD1612yes—
111#500—31no—
112#476—91no—
113#476—91no—
114#436—91no—
115#334—91no—
116#333—31no—
117#333—91no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—41no—
121C2×C3×Q8—61no—
122C2×D12—31no—
123C2×D12—31no—
124C2×D12—61no—
125C2×C4×D3—61no—
126C2×C4×D3—61no—
127C2×C4×D3—121no—
128C2×Dic6—21no—
129C2×C3×C8—31no—
130C2×C3×C8—31no—
131C2×C3×C8—61no—
132C2×C3×C8—61no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—61no—
140C3×C4⋊C4—61no—
141C3×C42—21no—
142C3×C42C22×D311yescharacteristic
143C3×C42C22×D311yescharacteristic
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—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—62no—
155C2×C3⋊C8—61no—
156C2×C3⋊C8—61no—
157C2×C3⋊C8#6711yescharacteristic
158C2×C3⋊C8#6711yescharacteristic
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×C4—81no—
167C32×C4C2×SD1611yescharacteristic
168C32×C4#13711yescharacteristic
169C32×C4#13711yescharacteristic
170C32×C4#13711yescharacteristic
171C3⋊Dic3—41no—
172C3×Dic3—81no—
173C2×M16—181no—
174C4⋊Q8—31no—
175C4⋊Q8D3211yescharacteristic
176C4×D4—91no—
177#107—61no—
178#107—61no—
179#106—181no—
180#106—181no—
181C4⋊C8—31no—
182C4⋊C8—31no—
183C4⋊C8—91no—
184#102—181no—
185#102—181no—
186C22⋊C8—181no—
187C4×C8—31no—
188C22×D3—61no—
189C22×D3—61no—
190C22×D3—121no—
191C3×Q8—81no—
192C3×Q8—81no—
193C3×Q8—82no—
194C3×Q8—241no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—42no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—121no—
202C2×C3×C4#23511yescharacteristic
203C2×C3×C4D3×D411yescharacteristic
204C2×C3×C4D3×D411yescharacteristic
205C2×C3×C4D3×D411yescharacteristic
206C2×C3×C4#23711yescharacteristic
207C2×C3×C4C2×#6711yescharacteristic
208C2×Dic3—41no—
209C2×Dic3—61no—
210C2×Dic3—61no—
211C2×Dic3—121no—
212D12—62no—
213D12—61no—
214D12—61no—
215D12—61no—
216D12—61no—
217D12—121no—
218C4×D3—121no—
219C4×D3—121no—
220C4×D3—121no—
221C4×D3—121no—
222C4×D3—241no—
223C4×D3—241no—
224Dic6—81no—
225C3×C8—32no—
226C3×C8—61no—
227C3×C8—121no—
228C3×C8—121no—
229C3⋊C8—21no—
230C3⋊C8—41no—
231C3⋊C8—41no—
232C3⋊C8—122no—
233C3⋊C8—121no—
234C3⋊C8—121no—
235C3⋊C8#21512yes—
236C2×C32#46111yescharacteristic
237C2×C32#47011yescharacteristic
238C2×C32#47211yescharacteristic
239C3⋊D3—81no—
240C2×Q8—21no—
241C2×Q8—61no—
242C2×D4—91no—
243C22×C4—181no—
244M16—362no—
245C2×C8—31no—
246C2×C8—31no—
247C2×C8—61no—
248C2×C8—61no—
249C2×C8—181no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—91no—
255C22⋊C4—181no—
256C42C2×D3211yescharacteristic
257C22×C3—21no—
258C22×C3#88411yescharacteristic
259C22×C3#93811yescharacteristic
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—82no—
277C3×C4—81no—
278C3×C4—81no—
279C3×C4—241no—
280C3×C4D3×SD1611yescharacteristic
281C3×C4#91511yescharacteristic
282C3×C4#91811yescharacteristic
283C3×C4#91811yescharacteristic
284C3×C4#91811yescharacteristic
285C3×C4#91811yescharacteristic
286C3×C4C2×#21511yescharacteristic
287C3×C4#94211yescharacteristic
288Dic3—81no—
289Dic3—121no—
290Dic3—121no—
291Dic3—241no—
292C32#169511yescharacteristic, complemented, Sylow
293C23—181no—
294Q8—81no—
295Q8—241no—
296D4—181no—
297D4—181no—
298C2×C4—41no—
299C2×C4—121no—
300C2×C4—181no—
301C2×C4—361no—
302C2×C4—361no—
303C2×C4#380411yescharacteristic
304C2×C4D3×#6711yescharacteristic
305C2×C4#381111yescharacteristic
306C8—32no—
307C8—61no—
308C8—121no—
309C8—121no—
310C8—361no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3#474911yescharacteristic
315C2×C3#481611yescharacteristic
316C2×C3#481811yescharacteristic
317C2×C3#508311yescharacteristic
318C2×C3#516011yescharacteristic
319C2×C3#518911yescharacteristic
320D3—242no—
321D3—241no—
322D3—241no—
323C22—361no—
324C22—361no—
325C22—361no—
326C22#6395811yescharacteristic
327C4—21no—
328C4—81no—
329C4—241no—
330C4—361no—
331C4D3×#21511yescharacteristic
332C4#6394511yescharacteristic
333C4#6394811yescharacteristic
334C4#6395111yescharacteristic
335C3—21no—
336C3#6850111yescharacteristic, complemented
337C3#7137411yescharacteristic, complemented
338C2—721no—
339C2#1059140611yescharacteristic
340C2#1059156511yescharacteristic
341C2#1059160011yescharacteristic
subgroupG/H
centerC22#63958
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenterC4⋊Q8D32
hypocenterC32#1695
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. C3×#5078C2semidirect
5. C3×#4546C2semidirect
6. #10589097C2semidirect
7. #10589095C2semidirect
30. #5078D3semidirect
36. #4546D3semidirect
292. C32#1695semidirect
336. C3#68501semidirect
337. C3#71374semidirect

Maximal subgroups

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

subgroupindex
#105927832
C3×#50782
#105903382
#105903002
C3×#45462
#105890972
#105890952
#713743
#685013