Finite Group Databasebeta

← #11808734

Subgroups — #11808734

Order 1152.

646 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#64286C211yesmaximal, characteristic
2C2×C4×PSU(3,𝔽2)C211yesmaximal, characteristic, complemented
3C4×C22⋊#157C212yesmaximal
4#10593570C211yesmaximal, characteristic
5#10593567C212yesmaximal
6#10593566C211yesmaximal, characteristic
7#10593556C211yesmaximal, characteristic, complemented
8#10593552C212yesmaximal, complemented
9#10593551C211yesmaximal, characteristic, complemented
10C4×#3777C212yesmaximal, complemented
11C4×#3777C211yesmaximal, characteristic, complemented
12C22×PSU(3,𝔽2)C2211yescharacteristic
13C22×C4×C3⋊D3C2211yescharacteristic
14C2×C22⋊#157C2212yes—
15C2×C22⋊#157C2212yes—
16C2×C22⋊#157C2211yescharacteristic
17C2×C22⋊#157C2211yescharacteristic
18#64286—24no—
19C2×C4⋊#157C2211yescharacteristic
20C2×C4⋊#157C2211yescharacteristic
21C2×C4×#157C2212yes—
22C2×C4×#157C2211yescharacteristic
23C2×C4×#157C2212yes—
24C2×C4×#157C2211yescharacteristic
25C2×#3777C2212yes—
26C2×#3777C2212yes—
27C2×#3777C2212yes—
28C2×#3777C2211yescharacteristic
29C2×#3777C2212yes—
30C4×PSU(3,𝔽2)—24no—
31#63781C2212yes—
32#63781C2212yes—
33#63747C2212yes—
34#63747C2211yescharacteristic
35#63747C2212yes—
36#63747C2211yescharacteristic
37C23×C3⋊D3C2311yescharacteristic
38C22×#157C2312yes—
39C22×#157C2311yescharacteristic
40C22×#157C2311yescharacteristic
41C22×#157C2312yes—
42C22×#157C2311yescharacteristic
43C22×#157C2311yescharacteristic
44C22×#157C2312yes—
45C22×#157C2312yes—
46C2×PSU(3,𝔽2)—22no—
47C2×PSU(3,𝔽2)—24no—
48C22×C32×C4Q811yescharacteristic, complemented
49C22×C3⋊Dic3Q811yescharacteristic, complemented
50C2×C4×C3⋊D3—22no—
51C2×C4×C3⋊D3—22no—
52C2×C4×C3⋊D3Q811yescharacteristic, complemented
53C2×C4×C3⋊D3Q811yescharacteristic, complemented
54C2×C4×C3⋊D3C2311yescharacteristic
55C2×C4×C3⋊D3C2311yescharacteristic
56C22⋊#157—22no—
57C22⋊#157—22no—
58C22⋊#157—24no—
59C22⋊#157—24no—
60C4⋊#157—22no—
61C4⋊#157—22no—
62C4×#157—22no—
63C4×#157—24no—
64C4×#157—24no—
65C4×#157D414yes—
66#3777—24no—
67#3777—22no—
68#3777—24no—
69#3777—24no—
70#3777—24no—
71#2396—91nomaximal, Sylow
72C22×C4×D3—121no—
73C23×C32C2×Q811yescharacteristic
74C22×C3⋊D3—21no—
75C22×C3⋊D3—22no—
76C22×C3⋊D3—22no—
77C22×C3⋊D3—21no—
78C22×C3⋊D3C2×Q811yescharacteristic
79C22×C3⋊D3C2411yescharacteristic
80C2×#157—22no—
81C2×#157—22no—
82C2×#157—22no—
83C2×#157—22no—
84C2×#157—22no—
85C2×#157—22no—
86C2×#157—22no—
87C2×#157—21no—
88C2×#157—21no—
89C2×#157—22no—
90C2×#157—22no—
91C2×#157—22no—
92C2×#157—22no—
93C2×#157—22no—
94C2×#157—22no—
95C2×#157—22no—
96C2×#157C2×D412yes—
97C2×#157C2×D412yes—
98C2×#157#4114yes—
99C2×#157#4114yes—
100PSU(3,𝔽2)—28no—
101C2×C32×C4—22no—
102C2×C32×C4C2×Q811yescharacteristic
103C2×C32×C4C2×Q811yescharacteristic
104C2×C3⋊Dic3—22no—
105C2×C3⋊Dic3C2×Q811yescharacteristic
106C2×C3⋊Dic3C2×Q811yescharacteristic
107C4×C3⋊D3—21no—
108C4×C3⋊D3—21no—
109C4×C3⋊D3—22no—
110C4×C3⋊D3—22no—
111C4×C3⋊D3C2×D412yes—
112C4×C3⋊D3#4112yes—
113C2×C4×Q8—91no—
114C2×#117—91no—
115#400—91no—
116#399—91no—
117#398—92no—
118#396—92no—
119#395—91no—
120#393—91no—
121C4×C4⋊C4—92no—
122C4×C4⋊C4—91no—
123C4×C22⋊C4—92no—
124C23×D3—121no—
125C22×C3×C4—41no—
126C22×Dic3—121no—
127C2×C4×D3—121no—
128C2×C4×D3—121no—
129C2×C4×D3—121no—
130C2×C4×D3—121no—
131C2×C4×D3—124no—
132C2×C4×D3—124no—
133C22×C32—21no—
134C22×C32—22no—
135C22×C32C22×Q811yescharacteristic
136C2×C3⋊D3—21no—
137C2×C3⋊D3—22no—
138C2×C3⋊D3—41no—
139C2×C3⋊D3—42no—
140C2×C3⋊D3—41no—
141C2×C3⋊D3C22×D411yescharacteristic
142C2×C3⋊D3C2×#4112yes—
143C2×C3⋊D3C2×#4111yescharacteristic
144C2×C3⋊D3C2×#4112yes—
145#157—22no—
146#157—24no—
147#157—24no—
148#157—22no—
149#157—41no—
150#157—42no—
151#157—42no—
152#157—41no—
153C32×C4—21no—
154C32×C4C22⋊Q812yes—
155C3⋊Dic3—21no—
156C3⋊Dic3C22⋊Q812yes—
157C22×Q8—91no—
158C23×C4—91no—
159C4×Q8—184no—
160#117—184no—
161C2×C4⋊C4—91no—
162C2×C4⋊C4—91no—
163C2×C4⋊C4—92no—
164C2×C4⋊C4—91no—
165C2×C4⋊C4—92no—
166C2×C4⋊C4—92no—
167C2×C4⋊C4—92no—
168C2×C22⋊C4—91no—
169C2×C22⋊C4—91no—
170C2×C22⋊C4—92no—
171C2×C22⋊C4—92no—
172C2×C42—91no—
173C2×C42—91no—
174C2×C42—92no—
175C2×C42—92no—
176#95—92no—
177#95—91no—
178#95—91no—
179#95—92no—
180#95—92no—
181#95—92no—
182C23×C3—41no—
183C22×D3—121no—
184C22×D3—121no—
185C22×D3—122no—
186C22×D3—122no—
187C22×D3—124no—
188C22×D3—124no—
189C2×C3×C4—44no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×Dic3—121no—
193C2×Dic3—121no—
194C2×Dic3—124no—
195C4×D3—122no—
196C4×D3—122no—
197C4×D3—124no—
198C4×D3—122no—
199C4×D3—122no—
200C4×D3—124no—
201C2×C32—41no—
202C2×C32C2×C22⋊Q811yescharacteristic
203C2×C32#53312yes—
204C3⋊D3—41no—
205C3⋊D3#52511yescharacteristic
206C3⋊D3#52912yes—
207C3⋊D3#53211yescharacteristic
208C24—91no—
209C2×Q8—182no—
210C2×Q8—184no—
211C22×C4—91no—
212C22×C4—92no—
213C22×C4—91no—
214C22×C4—92no—
215C22×C4—91no—
216C22×C4—91no—
217C22×C4—91no—
218C22×C4—91no—
219C22×C4—91no—
220C22×C4—92no—
221C22×C4—91no—
222C22×C4—91no—
223C22×C4—92no—
224C22×C4—182no—
225C22×C4—182no—
226C22×C4PSU(3,𝔽2)11yescharacteristic, complemented
227C4⋊C4—182no—
228C4⋊C4—182no—
229C4⋊C4—184no—
230C4⋊C4—182no—
231C4⋊C4—184no—
232C4⋊C4—184no—
233C4⋊C4—184no—
234C22⋊C4—182no—
235C22⋊C4—182no—
236C22⋊C4—184no—
237C22⋊C4—184no—
238C42—94no—
239C42—182no—
240C42—184no—
241C42—184no—
242C22×C3—42no—
243C22×C3—44no—
244C22×C3—41no—
245C2×D3—121no—
246C2×D3—121no—
247C2×D3—122no—
248C2×D3—122no—
249C2×D3—122no—
250C2×D3—124no—
251C2×D3—124no—
252C2×D3—128no—
253C2×D3—124no—
254C3×C4—42no—
255C3×C4—42no—
256Dic3—122no—
257Dic3—122no—
258C32#239611yescharacteristic, complemented, Sylow
259C23—91no—
260C23—91no—
261C23—181no—
262C23—182no—
263C23—182no—
264C23—181no—
265C23C2×PSU(3,𝔽2)11yescharacteristic
266Q8—188no—
267C2×C4—22no—
268C2×C4—91no—
269C2×C4—92no—
270C2×C4—92no—
271C2×C4—92no—
272C2×C4—92no—
273C2×C4—94no—
274C2×C4—94no—
275C2×C4—91no—
276C2×C4—182no—
277C2×C4—181no—
278C2×C4—181no—
279C2×C4—182no—
280C2×C4—182no—
281C2×C4—182no—
282C2×C4—182no—
283C2×C4—182no—
284C2×C4—182no—
285C2×C4—182no—
286C2×C4—182no—
287C2×C4—181no—
288C2×C4—181no—
289C2×C4—182no—
290C2×C4—182no—
291C2×C4—182no—
292C2×C4—182no—
293C2×C4—182no—
294C2×C4—182no—
295C2×C4—182no—
296C2×C4—182no—
297C2×C4C2×PSU(3,𝔽2)11yescharacteristic
298C2×C4C2×PSU(3,𝔽2)11yescharacteristic
299C2×C3—44no—
300C2×C3—42no—
301C2×C3—41no—
302D3—121no—
303D3—122no—
304D3—121no—
305D3—124no—
306C22—22no—
307C22—21no—
308C22—92no—
309C22—92no—
310C22—91no—
311C22—91no—
312C22—182no—
313C22—181no—
314C22—361no—
315C22—361no—
316C22—362no—
317C22C22×PSU(3,𝔽2)11yescharacteristic
318C4—21no—
319C4—92no—
320C4—182no—
321C4—184no—
322C4—182no—
323C4—184no—
324C4—181no—
325C4—361no—
326C4—361no—
327C4—362no—
328C4—362no—
329C4C22⋊PSU(3,𝔽2)12yes—
330C3—41no—
331C2—41no—
332C2—91no—
333C2—91no—
334C2—92no—
335C2—361no—
336C2#1059654212yes—
337C2C2×C22⋊PSU(3,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C4C2×PSU(3,𝔽2)
commutator, cosocleC22×C3⋊D3C24
FrattiniC22C22×PSU(3,𝔽2)
FittingC22×C32×C4Q8
socleC22×C32C22×Q8
prime radicalC23×C3⋊D3C23
hypercenterC22×C4PSU(3,𝔽2)
hypocenterC32#2396
LCSG ▷ C22×C3⋊D3 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×PSU(3,𝔽2)C2semidirect
7. #10593556C2semidirect
8. #10593552C2semidirect
9. #10593551C2semidirect
10. C4×#3777C2semidirect
11. C4×#3777C2semidirect
48. C22×C32×C4Q8semidirect
49. C22×C3⋊Dic3Q8semidirect
52. C2×C4×C3⋊D3Q8semidirect
53. C2×C4×C3⋊D3Q8semidirect
226. C22×C4PSU(3,𝔽2)semidirect
258. C32#2396semidirect

Maximal subgroups

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

subgroupindex
C2×#642862
C2×C4×PSU(3,𝔽2)2
C4×C22⋊#1572
#105935702
#105935672
#105935662
#105935562
#105935522
#105935512
C4×#37772
#23969