Finite Group Databasebeta

← #83552

Subgroups — #83552

Order 384.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D4×Q8C211yesmaximal, characteristic
2#5541C211yesmaximal, characteristic
3#5228C211yesmaximal, characteristic, complemented
4#5228C211yesmaximal, characteristic, complemented
5#5225C211yesmaximal, characteristic, complemented
6C2×#835C211yesmaximal, characteristic, complemented
7#5090C211yesmaximal, characteristic, complemented
8#5082C211yesmaximal, characteristic, complemented
9#5052C211yesmaximal, characteristic, complemented
10#5051C211yesmaximal, characteristic, complemented
11#5045C211yesmaximal, characteristic, complemented
12#5043C211yesmaximal, characteristic, complemented
13#5030C211yesmaximal, characteristic, complemented
14#5022C211yesmaximal, characteristic, complemented
15#5011C211yesmaximal, characteristic
16#3312—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic
19C3×C4⋊Q8—21no—
20C3×C4⋊Q8C2211yescharacteristic
21C3×C22⋊Q8—21no—
22C3×C22⋊Q8—21no—
23C3×C22⋊Q8C2211yescharacteristic
24C3×C22⋊Q8C2211yescharacteristic
25C3×C4×Q8C2211yescharacteristic
26C3×C4×D4—21no—
27C3×C4×D4C2211yescharacteristic
28C2×#216C2211yescharacteristic
29C2×#216C2211yescharacteristic
30C2×#216C2211yescharacteristic
31#935—21no—
32#926C2211yescharacteristic
33C2×C4⋊Dic3C2211yescharacteristic
34#924C2211yescharacteristic
35#924C2211yescharacteristic
36C2×#208C2211yescharacteristic
37C22×C3⋊C8C2211yescharacteristic
38#889—21no—
39#879—21no—
40#877—21no—
41#870C2211yescharacteristic
42C4×Dic6C2211yescharacteristic
43#835C2211yescharacteristic
44#835C2211yescharacteristic
45#835C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#835C2211yescharacteristic
50#835C2211yescharacteristic
51#835C2211yescharacteristic
52#830C2211yescharacteristic
53#830C2211yescharacteristic
54#810C2211yescharacteristic
55#808C2211yescharacteristic
56#808C2211yescharacteristic
57#807C2211yescharacteristic
58C4⋊(C3⋊C8)C2211yescharacteristic
59#803C2211yescharacteristic
60#552—31no—
61D4×Q8D311yescharacteristic
62#491—31no—
63#485—31no—
64#483—31no—
65#479—31no—
66#470—31no—
67#467—31no—
68#462—31no—
69#451—31no—
70#451—31no—
71#441—31no—
72#435—31no—
73#419—31no—
74C2×#103—31no—
75C2×C3×Q8—21no—
76C2×C3×Q8—21no—
77C2×C3×Q8—22no—
78C2×C3×Q8—22no—
79C2×C3×Q8—41no—
80C2×C3×Q8—41no—
81C2×C3×Q8C2311yescharacteristic
82C2×C3×Q8C2311yescharacteristic
83C2×C3×Q8C2311yescharacteristic
84C2×C3×D4D411yescharacteristic
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C22×Dic3—21no—
90C2×Dic6C2311yescharacteristic
91C3×C4⋊C4—21no—
92C3×C4⋊C4—21no—
93C3×C4⋊C4—21no—
94C3×C4⋊C4—21no—
95C3×C4⋊C4—21no—
96C3×C4⋊C4D411yescharacteristic
97C3×C4⋊C4C2311yescharacteristic
98C3×C22⋊C4—21no—
99C3×C22⋊C4—21no—
100C3×C22⋊C4D411yescharacteristic
101C3×C22⋊C4D411yescharacteristic
102C3×C42—21no—
103C3×C42C2311yescharacteristic
104#217—21no—
105#217—21no—
106#217—21no—
107#216—22no—
108#216—41no—
109#216—41no—
110C4⋊Dic3—21no—
111C4⋊Dic3C2311yescharacteristic
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119C4×Dic3—21no—
120C4×Dic3—21no—
121#208—41no—
122C2×C3⋊C8—41no—
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C22×Q8C2×D311yescharacteristic
128C22×Q8C2×D311yescharacteristic
129C2×Q16—31no—
130C2×Q16—31no—
131C2×Q16—31no—
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8—21no—
135C4⋊Q8C2×D311yescharacteristic
136#126—61no—
137#125—31no—
138#125—61no—
139#123—61no—
140C22⋊Q8—21no—
141C22⋊Q8—21no—
142C22⋊Q8—31no—
143C22⋊Q8—31no—
144C22⋊Q8C2×D311yescharacteristic
145C22⋊Q8C2×D311yescharacteristic
146C4×Q8—31no—
147C4×Q8C2×D311yescharacteristic
148C4×D4—21no—
149C4×D4C2×D311yescharacteristic
150#117—31no—
151#117—61no—
152C2×C4⋊C4—31no—
153#107—31no—
154#106—31no—
155#106—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#103—31no—
165#103—31no—
166#103—31no—
167C22⋊C8—31no—
168C22⋊C8—31no—
169#97—31no—
170C23×C3C2×D411yescharacteristic
171C23×C3C2×D411yescharacteristic
172C3×Q8—21no—
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×Q8—42no—
178C3×Q8—42no—
179C3×Q8#4112yes—
180C3×D4—22no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2×D411yescharacteristic
194C2×C3×C4C2×D411yescharacteristic
195C2×C3×C4C2411yescharacteristic
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—42no—
201Dic6—21no—
202Dic6—21no—
203C3⋊C8—21no—
204C3⋊C8—21no—
205C3⋊C8—21no—
206C3⋊C8—21no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—31no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8C22×D311yescharacteristic
215C2×Q8C22×D311yescharacteristic
216C2×Q8C22×D311yescharacteristic
217C2×D4#6711yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—61no—
221C22×C4C22×D311yescharacteristic
222C22×C4C22×D311yescharacteristic
223Q16—62no—
224Q16—121no—
225Q16—121no—
226M16—121no—
227C2×C8—31no—
228C2×C8—31no—
229C2×C8—31no—
230C2×C8—31no—
231C2×C8—121no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—31no—
238C4⋊C4—31no—
239C4⋊C4—31no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4#6711yescharacteristic
247C4⋊C4C22×D311yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—21no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4#6711yescharacteristic
254C22⋊C4#6711yescharacteristic
255C42—21no—
256C42—61no—
257C42—61no—
258C42C22×D311yescharacteristic
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3#13712yes—
265C22×C3C22×D411yescharacteristic
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—22no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4C2×#4111yescharacteristic
275C3×C4H(5,𝔽2)11yescharacteristic
276Dic3—41no—
277Dic3—41no—
278Dic3—41no—
279Dic3—41no—
280C23C2×#6711yescharacteristic
281C23C2×#6711yescharacteristic
282Q8—21no—
283Q8—21no—
284Q8—21no—
285Q8—21no—
286Q8—21no—
287Q8—42no—
288Q8—42no—
289Q8—61no—
290Q8—61no—
291Q8#23712yes—
292D4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—61no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—122no—
308C2×C4C2×#6711yescharacteristic
309C2×C4C2×#6711yescharacteristic
310C2×C4C2×#6711yescharacteristic
311C2×C4C2×#6711yescharacteristic
312C2×C4C23×D311yescharacteristic
313C8—61no—
314C8—61no—
315C8—61no—
316C8—61no—
317C2×C3—22no—
318C2×C3—41no—
319C2×C3#54611yescharacteristic
320C2×C3C2×#13711yescharacteristic
321C2×C3#57811yescharacteristic
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22#94212yes—
328C22C22×#6711yescharacteristic
329C4—21no—
330C4—21no—
331C4—22no—
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—121no—
338C4—121no—
339C4—121no—
340C4—121no—
341C4C2×#23711yescharacteristic
342C4#100411yescharacteristic
343C3#331211yescharacteristic, complemented, Sylow
344C2—22no—
345C2—41no—
346C2#580611yescharacteristic
347C2C2×#94211yescharacteristic
348C2#583811yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×D4×Q8C2
socleC22×C3C22×D4
prime radicalC2×C3×D4D4
hypercenterD4×Q8D3
hypocenterC3#3312
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
3. #5228C2semidirect
4. #5228C2semidirect
5. #5225C2semidirect
6. C2×#835C2semidirect
7. #5090C2semidirect
8. #5082C2semidirect
9. #5052C2semidirect
10. #5051C2semidirect
11. #5045C2semidirect
12. #5043C2semidirect
13. #5030C2semidirect
14. #5022C2semidirect
343. C3#3312semidirect

Maximal subgroups

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

subgroupindex
C3×D4×Q82
#55412
#52282
#52252
C2×#8352
#50902
#50822
#50522
#50512
#50452
#50432
#50302
#50222
#50112
#33123