Finite Group Databasebeta

← #80049

Subgroups — #80049

Order 384.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#67C211yesmaximal, characteristic, complemented
2#5611C211yesmaximal, characteristic
3C3×#448C211yesmaximal, characteristic, complemented
4C2×#835C211yesmaximal, characteristic, complemented
5#5171C211yesmaximal, characteristic, complemented
6#5170C211yesmaximal, characteristic, complemented
7#5162C211yesmaximal, characteristic, complemented
8Dic3×SD16C211yesmaximal, characteristic, complemented
9#5041C211yesmaximal, characteristic, complemented
10#4805C211yesmaximal, characteristic, complemented
11D3×#103C211yesmaximal, characteristic, complemented
12#4792C211yesmaximal, characteristic, complemented
13#4791C211yesmaximal, characteristic, complemented
14#67⋊C8C211yesmaximal, characteristic, complemented
15#4721C211yesmaximal, characteristic, complemented
16#3298—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×#237—21no—
20C2×C3×SD16C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C3×#121C2211yescharacteristic
23#946—21no—
24#946C2211yescharacteristic
25Q8×Dic3C2211yescharacteristic
26#944—21no—
27#944C2211yescharacteristic
28#938C2211yescharacteristic
29#935—21no—
30D4×Dic3C2211yescharacteristic
31C2×#214C2211yescharacteristic
32C4×#67—21no—
33C4×#67C2211yescharacteristic
34C2×C4⋊Dic3C2211yescharacteristic
35#924—21no—
36#924C2211yescharacteristic
37C22×C3⋊C8C2211yescharacteristic
38C2×D3×C8C2211yescharacteristic
39D3×C4⋊C4C2211yescharacteristic
40#888C2211yescharacteristic
41#883—21no—
42#878—21no—
43C3×#103C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#835C2211yescharacteristic
50#832C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#817C2211yescharacteristic
53#817C2211yescharacteristic
54#816C2211yescharacteristic
55#816C2211yescharacteristic
56Dic3⋊C8C2211yescharacteristic
57C8×Dic3C2211yescharacteristic
58#810C2211yescharacteristic
59#808C2211yescharacteristic
60D4×Q8—31no—
61#547—31no—
62#487—31no—
63#483—31no—
64#483—31no—
65#475—31no—
66#465—31no—
67#465—31no—
68#461—31no—
69#448—31no—
70#448D311yescharacteristic, complemented
71C4×SD16—31no—
72C8×D4—31no—
73C2×#103—31no—
74C2×#103—31no—
75C2×C3×Q8—21no—
76C2×C3×Q8—22no—
77C2×C3×Q8—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C2×C3×D4—21no—
81C2×C3×D4C2311yescharacteristic
82C22×C3×C4—21no—
83C22×C3×C4C2311yescharacteristic
84C2×#67—21no—
85C2×#67D411yescharacteristic
86C22×Dic3—21no—
87C22×Dic3—21no—
88D3×Q8—22no—
89D3×Q8—41no—
90#237—44no—
91C2×C4×D3—21no—
92C2×C4×D3—21no—
93C2×C4×D3C2311yescharacteristic
94C2×Dic6—21no—
95C2×Dic6—21no—
96C2×Dic6C2311yescharacteristic
97C3×SD16—22no—
98C3×SD16—41no—
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C22⋊C4—21no—
103#217—21no—
104#217—21no—
105#217—21no—
106#217D411yescharacteristic
107#214—41no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4D411yescharacteristic
110C4⋊Dic3—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114C4⋊Dic3C2311yescharacteristic
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4D411yescharacteristic
122C4×Dic3—21no—
123C4×Dic3C2311yescharacteristic
124C2×C3⋊C8—41no—
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127D3×C8—41no—
128C2×#41—61no—
129C22×Q8—31no—
130C22×Q8C2×D311yescharacteristic
131C2×SD16—31no—
132C2×SD16C2×D311yescharacteristic
133C2×SD16C2×D311yescharacteristic
134C22×C8—31no—
135C22×C8—31no—
136C4⋊Q8—31no—
137C4⋊Q8—31no—
138C4⋊Q8—61no—
139#123—61no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—61no—
144C22⋊Q8—61no—
145C22⋊Q8—61no—
146#121—31no—
147#121C2×D311yescharacteristic
148C4×Q8—31no—
149C4×D4—31no—
150C4×D4—31no—
151C4×D4—61no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154#106—31no—
155#106—31no—
156#106—31no—
157C4⋊C8—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#103—31no—
165#103C2×D311yescharacteristic
166#103C2×D311yescharacteristic
167#102—31no—
168C22⋊C8—31no—
169C22⋊C8C2×D311yescharacteristic
170C4×C8—31no—
171C23×C3—21no—
172C23×C3C2×D411yescharacteristic
173C22×D3C2×D411yescharacteristic
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×Q8—42no—
178C3×Q8#4112yescomplemented
179C3×D4—21no—
180C3×D4—21no—
181C3×D4—42no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—42no—
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2411yescharacteristic
190#67—42no—
191#67SD1614yescomplemented
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—42no—
197C2×Dic3—42no—
198C2×Dic3C2×D411yescharacteristic
199C2×Dic3C2×D411yescharacteristic
200C2×Dic3C2×D411yescharacteristic
201C4×D3—21no—
202C4×D3—21no—
203C4×D3—42no—
204C4×D3—42no—
205Dic6—21no—
206Dic6—21no—
207Dic6—42no—
208Dic6—42no—
209C3×C8—21no—
210C3×C8—21no—
211C3⋊C8—21no—
212C3⋊C8—21no—
213#41—124no—
214C2×Q8—21no—
215C2×Q8—22no—
216C2×Q8—31no—
217C2×Q8—41no—
218C2×Q8—61no—
219C2×Q8—61no—
220C2×Q8—62no—
221C2×Q8—121no—
222C2×Q8C22×D311yescharacteristic
223C2×Q8C22×D311yescharacteristic
224C2×D4—21no—
225C2×D4—31no—
226C2×D4—61no—
227C2×D4C22×D311yescharacteristic
228C22×C4—21no—
229C22×C4—31no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4C22×D311yescharacteristic
235SD16—22no—
236SD16—41no—
237SD16—121no—
238C2×C8—31no—
239C2×C8—31no—
240C2×C8—121no—
241C2×C8—121no—
242C2×C8C22×D311yescharacteristic
243C2×C8C22×D311yescharacteristic
244C4⋊C4—31no—
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4C22×D311yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—31no—
259C22⋊C4—31no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C42—31no—
265C42—61no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3C2×SD1612yes—
272C22×C3C22×D411yescharacteristic
273C2×D3—21no—
274C2×D3—21no—
275C2×D3C2×SD1612yes—
276C3×C4—22no—
277C3×C4—21no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4C2×#4111yescharacteristic
281C3×C4H(5,𝔽2)11yescharacteristic
282Dic3—21no—
283Dic3—21no—
284Dic3—41no—
285Dic3—41no—
286Dic3—41no—
287Dic3—41no—
288Dic3C2×SD1612yes—
289C23—21no—
290C23—31no—
291C23D3×D411yescharacteristic
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—61no—
297Q8—61no—
298Q8—122no—
299Q8—122no—
300Q8#23712yescomplemented
301D4—21no—
302D4—21no—
303D4—34no—
304D4—42no—
305D4—122no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—42no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—122no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4D3×D411yescharacteristic
326C2×C4C23×D311yescharacteristic
327C8—21no—
328C8—21no—
329C8—61no—
330C8—61no—
331C2×C3—22no—
332C2×C3—81no—
333C2×C3#54611yescharacteristic
334C2×C3C22×SD1611yescharacteristic
335C2×C3#57811yescharacteristic
336D3—22no—
337C22—21no—
338C22—21no—
339C22—32no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—61no—
344C22—61no—
345C22D3×SD1612yes—
346C22C2×D3×D411yescharacteristic
347C4—21no—
348C4—22no—
349C4—32no—
350C4—41no—
351C4—41no—
352C4—61no—
353C4—61no—
354C4—121no—
355C4—121no—
356C4—121no—
357C4—121no—
358C4C2×#23711yescharacteristic
359C4#100411yescharacteristic
360C3#329811yescharacteristic, complemented, Sylow
361C2—22no—
362C2—62no—
363C2—81no—
364C2#559111yescharacteristic
365C2C2×D3×SD1611yescharacteristic
366C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#448C2
socleC22×C3C22×D4
prime radical#5611C2
hypercenter#448D3
hypocenterC3#3298
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. Q8×#67C2semidirect
3. C3×#448C2semidirect
4. C2×#835C2semidirect
5. #5171C2semidirect
6. #5170C2semidirect
7. #5162C2semidirect
8. Dic3×SD16C2semidirect
9. #5041C2semidirect
10. #4805C2semidirect
11. D3×#103C2semidirect
12. #4792C2semidirect
13. #4791C2semidirect
14. #67⋊C8C2semidirect
15. #4721C2semidirect
70. #448D3semidirect
178. C3×Q8#41semidirect
191. #67SD16semidirect
300. Q8#237semidirect
360. C3#3298semidirect

Maximal subgroups

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

subgroupindex
Q8×#672
#56112
C3×#4482
C2×#8352
#51712
#51702
#51622
Dic3×SD162
#50412
#48052
D3×#1032
#47922
#47912
#67⋊C82
#47212
#32983