Finite Group Databasebeta

← #11869013

Subgroups — #11869013

Order 1152.

516 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 109 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#10596642C211yesmaximal, characteristic
2#10596543C211yesmaximal, characteristic, complemented
3#10593814C211yesmaximal, characteristic
4C2×#63782C211yesmaximal, characteristic
5#10593559C212yesmaximal, complemented
6#10593558C212yesmaximal, complemented
7#10593557C212yesmaximal, complemented
8#10593557C212yesmaximal, complemented
9#10593544C211yesmaximal, characteristic, complemented
10#10593543C211yesmaximal, characteristic, complemented
11#10593540C211yesmaximal, characteristic, complemented
12C2×D4×C3⋊D3C2211yescharacteristic
13C2×C22⋊#157—22no—
14D4×#157—21no—
15D4×#157—21no—
16D4×#157C2212yes—
17D4×#157C2212yes—
18#64286C2211yescharacteristic
19#64286C2211yescharacteristic
20#64286C2211yescharacteristic
21C2×C4⋊#157C2211yescharacteristic
22#64283C2211yescharacteristic
23C2×#3787C2211yescharacteristic
24C22⋊PSU(3,𝔽2)—22no—
25C4⋊PSU(3,𝔽2)—21no—
26C4⋊PSU(3,𝔽2)C2212yescomplemented
27#64243—21no—
28#64243C2212yescomplemented
29#63782C2212yes—
30#63782C2212yes—
31#63782C2212yes—
32#63782C2212yes—
33#63746C2211yescharacteristic, complemented
34#63746C2211yescharacteristic, complemented
35#63746C2212yescomplemented
36#63745C2211yescharacteristic, complemented
37#63745C2211yescharacteristic, complemented
38#63745C2212yescomplemented
39#63744C2212yescomplemented
40#63744C2212yescomplemented
41C23×C3⋊D3—21no—
42C22×#157—21no—
43C22×#157—22no—
44C22×#157—22no—
45C2×PSU(3,𝔽2)—22no—
46C2×C32×D4Q811yescharacteristic, complemented
47C2×#638—21no—
48D4×C3⋊D3—21no—
49D4×C3⋊D3—21no—
50D4×C3⋊D3Q812yescomplemented
51D4×C3⋊D3C2312yes—
52C2×C4⋊(C3⋊D3)Q811yescharacteristic, complemented
53C2×C4×C3⋊D3C2311yescharacteristic
54C22⋊#157—21no—
55C22⋊#157—21no—
56C22⋊#157—22no—
57C22⋊#157—22no—
58C22⋊#157D412yes—
59#3792—21no—
60C2×#622—21no—
61C4⋊#157—21no—
62C4⋊#157D411yescharacteristic
63C4⋊#157D411yescharacteristic
64C4⋊#157C2311yescharacteristic
65C4⋊#157C2311yescharacteristic
66C4⋊#157C2312yes—
67C4⋊#157C2311yescharacteristic
68C4⋊#157C2311yescharacteristic
69C4×#157—21no—
70C4×#157—21no—
71C4×#157C2312yes—
72#3788—21no—
73#3787—21no—
74#3787C2311yescharacteristic
75#3787C2312yes—
76#3787C2311yescharacteristic
77#3777—22no—
78#3777—22no—
79#3777—22no—
80#3081—91nomaximal, Sylow
81C2×D3×D4—121no—
82C23×C32—21no—
83C22×C3⋊D3—21no—
84C22×C3⋊D3—22no—
85C22×C3⋊D3—22no—
86C22×C3⋊D3—81no—
87C22×C3⋊D3—81no—
88C22×C3⋊D3C2×D411yescharacteristic
89C2×#157—21no—
90C2×#157—22no—
91C2×#157—21no—
92C2×#157—21no—
93C2×#157—21no—
94C2×#157—82no—
95C2×#157—82no—
96C2×#157—81no—
97C2×#157C2×D412yes—
98C2×#157C2×D412yes—
99PSU(3,𝔽2)—82no—
100C32×D4—21no—
101C32×D4C2×Q812yes—
102C2×C32×C4C2×Q811yescharacteristic
103#638—22no—
104#638—22no—
105C2×C3⋊Dic3#4111yescharacteristic
106C4⋊(C3⋊D3)—21no—
107C4⋊(C3⋊D3)C2×Q812yes—
108C4×C3⋊D3C2×D411yescharacteristic
109C4×C3⋊D3C2×Q811yescharacteristic
110C4×C3⋊D3#4111yescharacteristic
111C4×C3⋊D3C2411yescharacteristic
112#622—21no—
113#622—21no—
114#622—22no—
115#544—91no—
116#518—91no—
117#478—92no—
118#478—92no—
119#476—92no—
120#474—92no—
121#428—91no—
122#427—91no—
123#424—91no—
124#418—91no—
125C2×#102—91no—
126C23×D3—122no—
127C2×C3×D4—41no—
128C2×#67—122no—
129D3×D4—122no—
130D3×D4—122no—
131D3×D4—122no—
132D3×D4—122no—
133C2×D12—121no—
134C2×C4×D3—121no—
135C22×C32—22no—
136C22×C32—81no—
137C22×C32C22⋊Q811yescharacteristic
138C2×C3⋊D3—21no—
139C2×C3⋊D3—21no—
140C2×C3⋊D3—22no—
141C2×C3⋊D3—42no—
142C2×C3⋊D3—42no—
143C2×C3⋊D3—82no—
144C2×C3⋊D3—81no—
145C2×C3⋊D3C22⋊Q811yescharacteristic
146C2×C3⋊D3C22×D411yescharacteristic
147#157—22no—
148#157—22no—
149#157—41no—
150#157—42no—
151#157—41no—
152#157—41no—
153#157—41no—
154C32×C4C22⋊Q811yescharacteristic
155C32×C4C22×Q811yescharacteristic
156C3⋊Dic3C22⋊Q811yescharacteristic
157C3⋊Dic3C2×#4111yescharacteristic
158C22×D4—91no—
159C2×M16—91no—
160C22×C8—91no—
161C4⋊Q8—92no—
162C4⋊Q8—181no—
163#125—92no—
164#125—181no—
165C22⋊Q8—182no—
166C4×D4—92no—
167C4×D4—92no—
168C4×D4—181no—
169C4×D4—181no—
170#117—91no—
171#117—91no—
172#117—91no—
173C2×C4⋊C4—91no—
174C2×C22⋊C4—182no—
175#107—91no—
176#107—91no—
177#107—92no—
178#106—91no—
179#106—91no—
180#106—92no—
181C4⋊C8—92no—
182C4⋊C8—92no—
183#102—92no—
184#102—92no—
185#102—92no—
186#102—92no—
187C23×C3—42no—
188C22×D3—121no—
189C22×D3—122no—
190C22×D3—124no—
191C22×D3—124no—
192C22×D3—244no—
193C22×D3—244no—
194C3×D4—42no—
195C3×D4—42no—
196C2×C3×C4—41no—
197#67—124no—
198#67—124no—
199C2×Dic3—121no—
200D12—122no—
201D12—122no—
202C4×D3—121no—
203C4×D3—121no—
204C4×D3—121no—
205C4×D3—121no—
206C2×C32—21no—
207C2×C32—42no—
208C2×C32C2×C22⋊Q811yescharacteristic
209C3⋊D3—21no—
210C3⋊D3—42no—
211C3⋊D3#57611yescharacteristic
212C3⋊D3#57711yescharacteristic
213C24—181no—
214C2×Q8—182no—
215C2×D4—92no—
216C2×D4—92no—
217C2×D4—91no—
218C2×D4—181no—
219C2×D4—181no—
220C2×D4—181no—
221C2×D4PSU(3,𝔽2)11yescharacteristic, complemented
222C22×C4—91no—
223C22×C4—182no—
224C22×C4—182no—
225C22×C4—181no—
226M16—181no—
227M16—181no—
228C2×C8—91no—
229C2×C8—92no—
230C2×C8—91no—
231C2×C8—181no—
232C2×C8—181no—
233C4⋊C4—91no—
234C4⋊C4—91no—
235C4⋊C4—92no—
236C4⋊C4—91no—
237C4⋊C4—91no—
238C4⋊C4—91no—
239C4⋊C4—91no—
240C4⋊C4—181no—
241C4⋊C4—182no—
242C4⋊C4—182no—
243C4⋊C4—182no—
244C22⋊C4—92no—
245C22⋊C4—182no—
246C22⋊C4—182no—
247C22⋊C4—181no—
248C22⋊C4—181no—
249C42—92no—
250C42—181no—
251C42—181no—
252C22×C3—44no—
253C22×C3—41no—
254C22×C3—84no—
255C2×D3—121no—
256C2×D3—121no—
257C2×D3—122no—
258C2×D3—122no—
259C2×D3—124no—
260C2×D3—244no—
261C2×D3—244no—
262C2×D3—248no—
263C2×D3—244no—
264C3×C4—41no—
265C3×C4—41no—
266Dic3—121no—
267Dic3—121no—
268C32#308111yescharacteristic, complemented, Sylow
269C23—21no—
270C23—91no—
271C23—182no—
272C23—182no—
273C23—181no—
274C23—721no—
275C23—721no—
276Q8—722no—
277D4—21no—
278D4—92no—
279D4—182no—
280D4—182no—
281D4—181no—
282D4C2×PSU(3,𝔽2)12yes—
283C2×C4—91no—
284C2×C4—91no—
285C2×C4—91no—
286C2×C4—91no—
287C2×C4—91no—
288C2×C4—92no—
289C2×C4—92no—
290C2×C4—181no—
291C2×C4—181no—
292C2×C4—181no—
293C2×C4—181no—
294C2×C4—182no—
295C2×C4—722no—
296C2×C4—722no—
297C2×C4—721no—
298C2×C4C2×PSU(3,𝔽2)11yescharacteristic
299C8—181no—
300C8—181no—
301C8—182no—
302C2×C3—42no—
303C2×C3—41no—
304C2×C3—84no—
305D3—121no—
306D3—121no—
307D3—122no—
308D3—244no—
309C22—22no—
310C22—81no—
311C22—91no—
312C22—91no—
313C22—181no—
314C22—181no—
315C22—182no—
316C22—362no—
317C22—362no—
318C22—721no—
319C22—722no—
320C22C22⋊PSU(3,𝔽2)11yescharacteristic
321C4—91no—
322C4—91no—
323C4—182no—
324C4—182no—
325C4—361no—
326C4—361no—
327C4—362no—
328C4—361no—
329C4—361no—
330C4C22⋊PSU(3,𝔽2)11yescharacteristic
331C4C22×PSU(3,𝔽2)11yescharacteristic
332C3—41no—
333C2—21no—
334C2—42no—
335C2—91no—
336C2—91no—
337C2—181no—
338C2—362no—
339C2C2×C22⋊PSU(3,𝔽2)11yescharacteristic
subgroupG/H
centerC2C2×C22⋊PSU(3,𝔽2)
commutator, cosocleC4×C3⋊D3C24
FrattiniC4C22×PSU(3,𝔽2)
FittingC2×C32×D4Q8
socleC2×C32C2×C22⋊Q8
prime radicalC2×D4×C3⋊D3C22
hypercenterC2×D4PSU(3,𝔽2)
hypocenterC32#3081
LCSG ▷ C4×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #10596543C2semidirect
5. #10593559C2semidirect
6. #10593558C2semidirect
7. #10593557C2semidirect
8. #10593557C2semidirect
9. #10593544C2semidirect
10. #10593543C2semidirect
11. #10593540C2semidirect
26. C4⋊PSU(3,𝔽2)C22semidirect
28. #64243C22semidirect
33. #63746C22semidirect
34. #63746C22semidirect
35. #63746C22semidirect
36. #63745C22semidirect
37. #63745C22semidirect
38. #63745C22semidirect
39. #63744C22semidirect
40. #63744C22semidirect
46. C2×C32×D4Q8semidirect
50. D4×C3⋊D3Q8semidirect
52. C2×C4⋊(C3⋊D3)Q8semidirect
221. C2×D4PSU(3,𝔽2)semidirect
268. C32#3081semidirect

Maximal subgroups

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

subgroupindex
#105966422
#105965432
#105938142
C2×#637822
#105935592
#105935582
#105935572
#105935442
#105935432
#105935402
#30819