Finite Group Databasebeta

← #11824517

Subgroups — #11824517

Order 1152.

1565 conjugacy classes of subgroups; only the 252 normal subgroups are shown.

normal subgroups only

This group has 1565 conjugacy classes of subgroups — too many to draw. Only its 252 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×#64102C211yesmaximal, characteristic
2C2×#63919C211yesmaximal, characteristic, complemented
3C2×#63865C214yesmaximal, complemented
4C3×#4975C212yesmaximal, complemented
5C4×#3724C211yesmaximal, characteristic, complemented
6#10591320C212yesmaximal
7#10591203C214yesmaximal, complemented
8#74334—32nomaximal
9C22×C4⋊(C3⋊D3)C2211yescharacteristic
10C22×C4×C3⋊D3C2212yes—
11C22×#627C2212yes—
12C2×C32×C4⋊C4C2211yescharacteristic
13C2×#3752C2212yes—
14C2×C4×C3⋊Dic3C2211yescharacteristic
15#64102—24no—
16C2×C3×C4⋊Dic3C2212yes—
17C2×C3×Dic3⋊C4C2218yes—
18C2×C3×C4×Dic3C2212yes—
19C2×#3724C2212yes—
20C2×#3724C2212yes—
21C2×#3722C2218yes—
22#63919—24no—
23#63865—216no—
24#63579C2212yes—
25C2×#896—38no—
26C2×#893—32no—
27C2×#893—61no—
28C2×#888—32no—
29C3×#384—32no—
30#4993—38no—
31#4988—32no—
32#4975D312yescomplemented
33#4939—32no—
34C4×Dic3⋊C4—32no—
35C23×C3⋊D3C2312yes—
36C22×C32×C4C2312yes—
37C22×C32×C4C2311yescharacteristic
38C22×C3⋊Dic3C2311yescharacteristic
39C22×C3⋊Dic3C2311yescharacteristic
40C2×C4⋊(C3⋊D3)—22no—
41C2×C4⋊(C3⋊D3)—22no—
42C2×C4⋊(C3⋊D3)Q814yescomplemented
43C2×C4×C3⋊D3—24no—
44C2×C4×C3⋊D3—28no—
45C22×C3×Dic3C2314yes—
46C22×C3×Dic3C2314yes—
47C2×#627—28no—
48C2×#627—24no—
49C32×C4⋊C4—22no—
50#3752—24no—
51C4×C3⋊Dic3D414yes—
52C3×C4⋊Dic3—24no—
53C3×Dic3⋊C4—216no—
54C3×C4×Dic3—24no—
55#3724—24no—
56#3724D418yescomplemented
57#3722—216no—
58#2542—91noSylow
59C22×D12—32no—
60C22×D12—61no—
61C22×C4×D3—34no—
62C22×C4×D3—62no—
63C22×Dic6—34no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—38no—
66C2×C3×C4⋊C4—32no—
67C2×C3×C4⋊C4C2×D312yes—
68C2×C3×C42—32no—
69C2×(C2×D3)⋊C4—34no—
70C2×(C2×D3)⋊C4—38no—
71C2×(C2×D3)⋊C4—38no—
72C2×(C2×D3)⋊C4—62no—
73C2×C4⋊Dic3C2×D312yes—
74C2×Dic3⋊C4—34no—
75C2×Dic3⋊C4—34no—
76C2×Dic3⋊C4C2×D318yes—
77C2×C4×Dic3—32no—
78C2×C4×Dic3—61no—
79C2×C4×Dic3C2×D312yes—
80#896—632no—
81#893—68no—
82#893—68no—
83#888—68no—
84C3⋊#95—34no—
85C23×C32C2411yescharacteristic
86C22×C3⋊D3—22no—
87C22×C3⋊D3—22no—
88C22×C3⋊D3—24no—
89C22×C3⋊D3—24no—
90C22×C3⋊D3C2×Q814yes—
91C2×C32×C4—22no—
92C2×C32×C4—24no—
93C2×C32×C4C2×D412yes—
94C2×C32×C4C2×Q812yes—
95C2×C32×C4#4112yes—
96C2×C3⋊Dic3—22no—
97C2×C3⋊Dic3C2×D412yes—
98C2×C3⋊Dic3C2×D414yes—
99C2×C3⋊Dic3C2×D412yes—
100C4⋊(C3⋊D3)—28no—
101C4×C3⋊D3—48no—
102C2×C3×Dic3—24no—
103C2×C3×Dic3—24no—
104C2×C3×Dic3—28no—
105C2×C3×Dic3—24no—
106C2×C3×Dic3C2×D418yes—
107#627—48no—
108C2×C4⋊Q8—91no—
109C2×C22⋊Q8—94no—
110C2×C4×D4—91no—
111#399—94no—
112#390—92no—
113#384—32no—
114C4×C4⋊C4—91no—
115C23×D3—34no—
116C23×D3—62no—
117C22×C3×C4—21no—
118C22×C3×C4—22no—
119C22×C3×C4—34no—
120C22×C3×C4—34no—
121C22×C3×C4C22×D314yes—
122C22×C3×C4C22×D312yes—
123C22×Dic3—32no—
124C22×Dic3—32no—
125C22×Dic3—61no—
126C22×Dic3—61no—
127C22×Dic3C22×D314yes—
128C22×Dic3C22×D314yes—
129C2×D12—38no—
130C2×D12—64no—
131C2×D12—64no—
132C2×D12—64no—
133C2×D12—64no—
134C2×D12—64no—
135C2×C4×D3—68no—
136C2×C4×D3—616no—
137C2×C4×D3—68no—
138C2×C4×D3—128no—
139C2×Dic6—68no—
140C2×Dic6—616no—
141C3×C4⋊C4—24no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4—616no—
144C3×C4⋊C4—64no—
145C3×C42—64no—
146(C2×D3)⋊C4—616no—
147(C2×D3)⋊C4—68no—
148(C2×D3)⋊C4—68no—
149(C2×D3)⋊C4—616no—
150C4⋊Dic3—24no—
151Dic3⋊C4—216no—
152Dic3⋊C4—316no—
153Dic3⋊C4—68no—
154C4×Dic3—24no—
155C4×Dic3—38no—
156C4×Dic3—64no—
157C22×C32C22×D411yescharacteristic
158C22×C32C22×D412yes—
159C22×C32C22×Q811yescharacteristic
160C22×C32C2×#4111yescharacteristic
161C22×C322−1+412yes—
162C2×C3⋊D3—24no—
163C2×C3⋊D3—28no—
164C2×C3⋊D3—42no—
165C2×C3⋊D3—42no—
166C2×C3⋊D3—44no—
167C32×C4—42no—
168C32×C4C22⋊Q814yescomplemented
169C3⋊Dic3—22no—
170C3⋊Dic3#12714yes—
171C3×Dic3—28no—
172C3×Dic3—44no—
173C22×Q8—92no—
174C22×D4—91no—
175C23×C4—92no—
176C4⋊Q8—184no—
177C22⋊Q8—1816no—
178C4×D4—184no—
179C2×C4⋊C4—38no—
180C2×C4⋊C4—32no—
181C2×C4⋊C4—92no—
182C2×C4⋊C4—92no—
183C2×C4⋊C4D3211yescharacteristic
184C2×C22⋊C4—98no—
185C2×C22⋊C4—92no—
186C2×C42—32no—
187C2×C42—91no—
188#95—92no—
189C23×C3—21no—
190C23×C3C23×D312yes—
191C22×D3—38no—
192C22×D3—64no—
193C22×D3—64no—
194C22×D3—64no—
195C22×D3—64no—
196C22×D3—68no—
197C22×D3—64no—
198C22×D3—68no—
199C22×D3—124no—
200C22×D3—124no—
201C2×C3×C4—24no—
202C2×C3×C4—28no—
203C2×C3×C4—24no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—38no—
208C2×C3×C4—44no—
209C2×C3×C4—64no—
210C2×C3×C4—64no—
211C2×C3×C4—64no—
212C2×C3×C4—68no—
213C2×C3×C4#23514yes—
214C2×C3×C4D3×D414yes—
215C2×C3×C4D3×Q814yes—
216C2×Dic3—24no—
217C2×Dic3—24no—
218C2×Dic3—28no—
219C2×Dic3—24no—
220C2×Dic3—34no—
221C2×Dic3—38no—
222C2×Dic3—34no—
223C2×Dic3—62no—
224C2×Dic3—64no—
225C2×Dic3—62no—
226C2×Dic3—64no—
227C2×Dic3—122no—
228C2×Dic3D3×D418yes—
229D12—616no—
230D12—616no—
231C4×D3—1216no—
232C4×D3—1216no—
233Dic6—1216no—
234C2×C32C2×C22⋊Q811yescharacteristic
235C2×C32C2×#12711yescharacteristic
236C2×C32#53611yescharacteristic
237C2×C32#54712yes—
238C2×C32D4×Q812yes—
239C3⋊D3—44no—
240C24—92no—
241C2×Q8—188no—
242C2×Q8—184no—
243C2×D4—94no—
244C2×D4—182no—
245C2×D4—182no—
246C22×C4—34no—
247C22×C4—34no—
248C22×C4—91no—
249C22×C4—91no—
250C22×C4—184no—
251C22×C4—188no—
252C22×C4C2×D3212yes—
253C22×C4C2×D3211yescharacteristic
254C4⋊C4—22no—
255C4⋊C4—616no—
256C4⋊C4—64no—
257C4⋊C4—98no—
258C4⋊C4—184no—
259C22⋊C4—184no—
260C22⋊C4—1816no—
261C42—64no—
262C42—94no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—22no—
266C22×C3—22no—
267C22×C3—21no—
268C22×C3C2×#23512yes—
269C22×C3C2×D3×D414yes—
270C22×C3C2×D3×D412yes—
271C22×C3C2×D3×Q812yes—
272C22×C3#100714yes—
273C2×D3—68no—
274C2×D3—616no—
275C2×D3—68no—
276C2×D3—128no—
277C2×D3—124no—
278C2×D3—124no—
279C2×D3—128no—
280C2×D3—124no—
281C2×D3—124no—
282C2×D3—128no—
283C3×C4—24no—
284C3×C4—44no—
285C3×C4—44no—
286C3×C4—68no—
287C3×C4—124no—
288C3×C4#89618yescomplemented
289Dic3—28no—
290Dic3—38no—
291Dic3—44no—
292Dic3—64no—
293Dic3—64no—
294Dic3—122no—
295C32#254211yescharacteristic, complemented, Sylow
296C23—94no—
297C23—182no—
298C23—184no—
299C23—184no—
300C23—182no—
301C23C22×D3211yescharacteristic
302Q8—368no—
303D4—188no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—38no—
307C2×C4—64no—
308C2×C4—64no—
309C2×C4—68no—
310C2×C4—64no—
311C2×C4—94no—
312C2×C4—92no—
313C2×C4—92no—
314C2×C4—182no—
315C2×C4—368no—
316C2×C4#379712yes—
317C2×C4#379812yes—
318C2×C4#381112yes—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—22no—
322C2×C3—22no—
323C2×C3—21no—
324C2×C3#550714yes—
325C2×C3C2×#89612yes—
326C2×C3D3×#12712yes—
327C2×C3#572412yes—
328C2×C3#572714yes—
329D3—128no—
330D3—128no—
331C22—184no—
332C22—188no—
333C22—362no—
334C22—364no—
335C22—362no—
336C22C2×#379711yescharacteristic
337C22C2×#379811yescharacteristic
338C22D32×D412yes—
339C22#6431612yes—
340C22C2×#381111yescharacteristic
341C4—42no—
342C4—68no—
343C4—94no—
344C4—124no—
345C4—182no—
346C4#6386514yescomplemented
347C3—21no—
348C3#7433412yescomplemented
349C2—364no—
350C2#1059437912yes—
351C2#1059451212yes—
352C2C2×#6386511yescharacteristic
353C2#1059530011yescharacteristic
354C2#1059535011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutator, socle, cosocleC23×C32C24
FittingC2×C32×C4⋊C4C22
prime radicalC22×C4⋊(C3⋊D3)C22
hypercenterC2×C4⋊C4D32
hypocenterC32#2542
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#63919C2semidirect
3. C2×#63865C2semidirect
4. C3×#4975C2semidirect
5. C4×#3724C2semidirect
7. #10591203C2semidirect
32. #4975D3semidirect
42. C2×C4⋊(C3⋊D3)Q8semidirect
56. #3724D4semidirect
168. C32×C4C22⋊Q8semidirect
288. C3×C4#896semidirect
295. C32#2542semidirect
346. C4#63865semidirect
348. C3#74334semidirect

Maximal subgroups

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

subgroupindex
C2×#641022
C2×#639192
C2×#638652
C3×#49752
C4×#37242
#105913202
#105912032
#743343