Finite Group Databasebeta

← #80646

Subgroups — #80646

Order 384.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5625C211yesmaximal, characteristic, complemented
2#5619C211yesmaximal, characteristic
3C3×#465C211yesmaximal, characteristic, complemented
4#5163C211yesmaximal, characteristic, complemented
5#5162C211yesmaximal, characteristic, complemented
6#5114C211yesmaximal, characteristic, complemented
7#5102C211yesmaximal, characteristic, complemented
8#5049C211yesmaximal, characteristic, complemented
9#5042C211yesmaximal, characteristic, complemented
10#4871C211yesmaximal, characteristic, complemented
11#4857C211yesmaximal, characteristic, complemented
12#4818C211yesmaximal, characteristic, complemented
13#4799C211yesmaximal, characteristic, complemented
14#4763C211yesmaximal, characteristic, complemented
15#4762C211yesmaximal, characteristic, complemented
16#3260—31nomaximal, Sylow
17C2×C3×SD16C2211yescharacteristic
18C22×C3×C8C2211yescharacteristic
19C3×C22⋊Q8C2211yescharacteristic
20C3×#121C2211yescharacteristic
21Q8×Dic3C2211yescharacteristic
22#944C2211yescharacteristic
23C2×#215C2211yescharacteristic
24#940C2211yescharacteristic
25#939—21no—
26#938—21no—
27D4×Dic3C2211yescharacteristic
28C2×#214C2211yescharacteristic
29#930C2211yescharacteristic
30#926C2211yescharacteristic
31#924C2211yescharacteristic
32C2×#208C2211yescharacteristic
33C2×#204C2211yescharacteristic
34#893C2211yescharacteristic
35#890—21no—
36#890C2211yescharacteristic
37#890C2211yescharacteristic
38#889—21no—
39#887C2211yescharacteristic
40#886—21no—
41#883—21no—
42#879—21no—
43#879—21no—
44C3×#106C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#102C2211yescharacteristic
47#835C2211yescharacteristic
48#832C2211yescharacteristic
49#821C2211yescharacteristic
50#817C2211yescharacteristic
51#816C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53Dic3⋊C8C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56#810C2211yescharacteristic
57#809C2211yescharacteristic
58#807C2211yescharacteristic
59#807C2211yescharacteristic
60#541—31no—
61#540—31no—
62#479—31no—
63#479—31no—
64#478—31no—
65#478—31no—
66#468—31no—
67#468—31no—
68#465—31no—
69#465D311yescharacteristic, complemented
70#461—31no—
71#461—31no—
72#460—31no—
73#460—31no—
74#424—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4—21no—
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82C22×Dic3—21no—
83C2×D12C2311yescharacteristic
84C2×C4×D3—21no—
85C2×Dic6C2311yescharacteristic
86C3×SD16—41no—
87C2×C3×C8—41no—
88C2×C3×C8C2311yescharacteristic
89C2×C3×C8C2311yescharacteristic
90C3×C4⋊C4—21no—
91C3×C4⋊C4C2311yescharacteristic
92C3×C4⋊C4C2311yescharacteristic
93C3×C22⋊C4—21no—
94C3×C22⋊C4—21no—
95#217—21no—
96#217—21no—
97#217D411yescharacteristic
98#217D411yescharacteristic
99#215—41no—
100#214—41no—
101(C2×D3)⋊C4—21no—
102(C2×D3)⋊C4—21no—
103C4⋊Dic3—21no—
104C4⋊Dic3—21no—
105C4⋊Dic3—21no—
106C4⋊Dic3D411yescharacteristic
107C4⋊Dic3D411yescharacteristic
108C4⋊Dic3C2311yescharacteristic
109C4⋊Dic3C2311yescharacteristic
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116C4×Dic3—21no—
117C4×Dic3—21no—
118C4×Dic3C2311yescharacteristic
119C4×Dic3C2311yescharacteristic
120#208—41no—
121C2×C3⋊C8C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123#204—41no—
124C2×SD16—31no—
125C2×SD16—31no—
126C2×SD16—31no—
127C2×SD16C2×D311yescharacteristic
128C2×M16—31no—
129C22×C8C2×D311yescharacteristic
130C4⋊Q8—31no—
131#127—31no—
132#126—61no—
133#126—61no—
134#125—31no—
135#125—31no—
136#125—61no—
137#125—61no—
138#123—61no—
139#123—61no—
140C22⋊Q8—31no—
141C22⋊Q8C2×D311yescharacteristic
142#121—31no—
143#121—61no—
144#121—61no—
145#121C2×D311yescharacteristic
146C4×Q8—31no—
147C4×Q8—31no—
148C4×D4—31no—
149C4×D4—31no—
150#117—31no—
151#107—31no—
152#107—31no—
153#106—31no—
154#106C2×D311yescharacteristic
155C4⋊C8—31no—
156C4⋊C8—31no—
157C4⋊C8—31no—
158C4⋊C8—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103C2×D311yescharacteristic
163#102—31no—
164#102—31no—
165#102—31no—
166#102C2×D311yescharacteristic
167C23×C3—21no—
168C23×C3C2×D411yescharacteristic
169C22×D3—21no—
170C3×Q8—21no—
171C3×Q8—21no—
172C3×D4—21no—
173C3×D4—21no—
174C3×D4—81no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4C2×D411yescharacteristic
181C2×C3×C4C2411yescharacteristic
182#67—81no—
183#67—81no—
184#67—81no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—81no—
189C2×Dic3C2×D411yescharacteristic
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193D12—21no—
194D12—21no—
195C4×D3—81no—
196Dic6—21no—
197Dic6—21no—
198C3×C8—21no—
199C3×C8—21no—
200C3⋊C8—21no—
201C3⋊C8—21no—
202C2×Q8—31no—
203C2×Q8C22×D311yescharacteristic
204C2×D4—21no—
205C2×D4—31no—
206C2×D4—61no—
207C2×D4—61no—
208C2×D4—61no—
209C2×D4C22×D311yescharacteristic
210C22×C4—61no—
211C22×C4—61no—
212C22×C4C22×D311yescharacteristic
213SD16—41no—
214SD16—121no—
215SD16—121no—
216SD16—121no—
217M16—121no—
218C2×C8—31no—
219C2×C8—31no—
220C2×C8—41no—
221C2×C8C22×D311yescharacteristic
222C2×C8C22×D311yescharacteristic
223C4⋊C4—21no—
224C4⋊C4—31no—
225C4⋊C4—31no—
226C4⋊C4—31no—
227C4⋊C4—31no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4C22×D311yescharacteristic
238C4⋊C4C22×D311yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—21no—
241C22⋊C4—31no—
242C22⋊C4—31no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C42—31no—
248C42—31no—
249C42—61no—
250C42—61no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3C22×D411yescharacteristic
258C2×D3—41no—
259C2×D3—41no—
260C2×D3—41no—
261C3×C4—21no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4H(5,𝔽2)11yescharacteristic
266C3×C42−1+411yescharacteristic
267Dic3—21no—
268Dic3—21no—
269Dic3—21no—
270Dic3—21no—
271Dic3—41no—
272Dic3—41no—
273Dic3—41no—
274C23—21no—
275C23—61no—
276C23D3×D411yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—61no—
280Q8—61no—
281D4—21no—
282D4—21no—
283D4—61no—
284D4—61no—
285D4—81no—
286D4—241no—
287D4—241no—
288D4—241no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—61no—
299C2×C4—61no—
300C2×C4—61no—
301C2×C4—241no—
302C2×C4—241no—
303C2×C4D3×D411yescharacteristic
304C2×C4C23×D311yescharacteristic
305C8—21no—
306C8—21no—
307C8—61no—
308C8—61no—
309C2×C3—41no—
310C2×C3—81no—
311C2×C3#53711yescharacteristic
312C2×C3#57711yescharacteristic
313C2×C3#57711yescharacteristic
314D3—81no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—121no—
322C22—121no—
323C22—121no—
324C22C2×D3×D411yescharacteristic
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—61no—
330C4—61no—
331C4—61no—
332C4—61no—
333C4—121no—
334C4—121no—
335C4—121no—
336C4#100411yescharacteristic
337C4#101011yescharacteristic
338C3#326011yescharacteristic, complemented, Sylow
339C2—41no—
340C2—81no—
341C2—241no—
342C2#559711yescharacteristic
343C2#576211yescharacteristic
344C2#577011yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#465C2
socleC22×C3C22×D4
prime radical#5619C2
hypercenter#465D3
hypocenterC3#3260
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #5625C2semidirect
3. C3×#465C2semidirect
4. #5163C2semidirect
5. #5162C2semidirect
6. #5114C2semidirect
7. #5102C2semidirect
8. #5049C2semidirect
9. #5042C2semidirect
10. #4871C2semidirect
11. #4857C2semidirect
12. #4818C2semidirect
13. #4799C2semidirect
14. #4763C2semidirect
15. #4762C2semidirect
69. #465D3semidirect
338. C3#3260semidirect

Maximal subgroups

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

subgroupindex
#56252
#56192
C3×#4652
#51632
#51622
#51142
#51022
#50492
#50422
#48712
#48572
#48182
#47992
#47632
#47622
#32603