Finite Group Databasebeta

← #85817

Subgroups — #85817

Order 384.

1094 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×Q8C211yesmaximal, characteristic, complemented
2#5817C211yesmaximal, characteristic, complemented
3Q8×#67C211yesmaximal, characteristic, complemented
4#5581C213yesmaximal, complemented
5#5580C213yesmaximal, complemented
6#5578C213yesmaximal, complemented
7#5576C211yesmaximal, characteristic, complemented
8C4×#239C211yesmaximal, characteristic, complemented
9C4×D3×Q8C211yesmaximal, characteristic, complemented
10#5569C211yesmaximal, characteristic, complemented
11#5568C213yesmaximal, complemented
12#5513C213yesmaximal, complemented
13#5507C213yesmaximal, complemented
14#5479C213yesmaximal, complemented
15C4×#235C213yesmaximal, complemented
16#3482—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×D3×Q8C2211yescharacteristic
20C2×#235C2213yes—
21C3×C4×Q8C2213yes—
22C3×C4×Q8C2211yescharacteristic
23C3×C4×Q8C2211yescharacteristic
24C3×C4×Q8C2213yes—
25C2×C3×C4⋊C4C2213yes—
26C2×C3×C42C2213yes—
27#947C2213yes—
28#946C2213yes—
29#946C2213yes—
30Q8×Dic3C2211yescharacteristic
31Q8×Dic3C2211yescharacteristic
32#944C2213yes—
33#930C2213yes—
34#929C2216yes—
35C4×#67C2211yescharacteristic
36C4×#67C2213yes—
37C4×#67C2213yes—
38C4×#67C2213yes—
39#926C2213yes—
40#924C2213yes—
41#898C2216yes—
42#897C2213yes—
43#896C2216yes—
44#895C2213yes—
45#894C2216yes—
46#893C2213yes—
47#892C2213yes—
48D3×C4⋊C4C2213yes—
49#890C2213yes—
50#889C2216yes—
51#888C2213yes—
52#887C2213yes—
53#876C2216yes—
54#875C2213yes—
55C4×D12C2213yes—
56C4×D12C2213yes—
57C4×D12C2213yes—
58#872C2213yes—
59#872C2213yes—
60#872C2216yes—
61C42×D3C2213yes—
62C42×D3C2213yes—
63#870C2213yes—
64C4×Dic6C2213yes—
65C4×Dic6C2213yes—
66C4×Dic6C2213yes—
67#557—31no—
68#556—33no—
69#552—33no—
70#552—33no—
71D4×Q8—31no—
72#548—31no—
73#548—31no—
74#547—33no—
75#533—33no—
76#529—33no—
77#529—33no—
78C4×#41—31no—
79C4×#41—33no—
80C2×C4×Q8—31no—
81C2×C4×Q8D311yescharacteristic, complemented
82C2×C3×Q8—24no—
83C2×C3×Q8C2313yes—
84C2×C3×Q8C2311yescharacteristic
85C22×C3×C4C2313yes—
86C22×C3×C4C2313yes—
87C22×C3×C4C2311yescharacteristic
88C2×#67C2313yes—
89C2×#67C2311yescharacteristic
90#239—24no—
91D3×Q8—24no—
92#235—212no—
93C2×D12C2313yes—
94C2×D12C2313yes—
95C2×C4×D3C2311yescharacteristic
96C2×C4×D3C2313yes—
97C2×C4×D3C2313yes—
98C2×C4×D3C2313yes—
99C2×C4×D3C2313yes—
100C2×C4×D3C2313yes—
101C2×Dic6C2313yes—
102C2×Dic6C2313yes—
103C3×C4⋊C4C2313yes—
104C3×C4⋊C4C2313yes—
105C3×C4⋊C4C2316yes—
106C3×C42C2313yes—
107C3×C42C2313yes—
108C3×C42C2313yes—
109C3×C42C2313yes—
110#217C2313yes—
111#217C2311yescharacteristic
112(C2×D3)⋊C4C2311yescharacteristic
113(C2×D3)⋊C4C2313yes—
114(C2×D3)⋊C4C2313yes—
115(C2×D3)⋊C4C2313yes—
116(C2×D3)⋊C4C2316yes—
117(C2×D3)⋊C4C2316yes—
118(C2×D3)⋊C4C2316yes—
119C4⋊Dic3C2313yes—
120C4⋊Dic3C2313yes—
121C4⋊Dic3C2313yes—
122C4⋊Dic3C2313yes—
123Dic3⋊C4C2311yescharacteristic
124Dic3⋊C4C2313yes—
125Dic3⋊C4C2313yes—
126Dic3⋊C4C2313yes—
127Dic3⋊C4C2316yes—
128Dic3⋊C4C2316yes—
129Dic3⋊C4C2316yes—
130C4×Dic3C2311yescharacteristic
131C4×Dic3C2313yes—
132C4×Dic3C2313yes—
133C4×Dic3C2313yes—
134C4×Dic3C2313yes—
135C4×Dic3C2313yes—
136C2×#41—31no—
137C2×#41—33no—
138C22×Q8—31no—
139C22×Q8C2×D311yescharacteristic
140C4⋊Q8—33no—
141C4⋊Q8—33no—
142#126—36no—
143#126—36no—
144#125—33no—
145#125—36no—
146#125—33no—
147#124—33no—
148#124—33no—
149#123—36no—
150#123—36no—
151C22⋊Q8—33no—
152C22⋊Q8—33no—
153C22⋊Q8—33no—
154C22⋊Q8—36no—
155C22⋊Q8—33no—
156#121—33no—
157#121—33no—
158C4×Q8—31no—
159C4×Q8—31no—
160C4×Q8—33no—
161C4×Q8—33no—
162C4×Q8—33no—
163C4×Q8—33no—
164C4×Q8C2×D311yescharacteristic
165C4×Q8C2×D313yes—
166C4×Q8C2×D311yescharacteristic
167C4×Q8C2×D313yes—
168C4×D4—31no—
169C4×D4—33no—
170C4×D4—33no—
171C4×D4—33no—
172C4×D4—33no—
173C4×D4—33no—
174C4×D4—33no—
175C4×D4—33no—
176#117—33no—
177#117—33no—
178#117—33no—
179#117—33no—
180#117—36no—
181C2×C4⋊C4—33no—
182C2×C4⋊C4C2×D313yes—
183C2×C42—33no—
184C2×C42—33no—
185C2×C42C2×D313yes—
186C23×C3C2411yescharacteristic
187C22×D3C2413yes—
188C22×D3C2411yescharacteristic
189C3×Q8—26no—
190C3×Q8#4114yescomplemented
191C2×C3×C4—26no—
192C2×C3×C4—26no—
193C2×C3×C4—22no—
194C2×C3×C4C2413yes—
195C2×C3×C4C2411yescharacteristic
196C2×C3×C4C2413yes—
197C2×C3×C4C2413yes—
198C2×C3×C4C2413yes—
199C2×C3×C4C2411yescharacteristic
200#67—26no—
201#67#4114yescomplemented
202C2×Dic3C2411yescharacteristic
203C2×Dic3C2411yescharacteristic
204C2×Dic3C2413yes—
205C2×Dic3C2413yes—
206C2×Dic3C2413yes—
207C2×Dic3C2411yescharacteristic
208D12—26no—
209D12—26no—
210C4×D3—26no—
211C4×D3—26no—
212C4×D3—26no—
213C4×D3—22no—
214C4×D3—26no—
215C4×D3—26no—
216Dic6—26no—
217Dic6—26no—
218#41—612no—
219#41—64no—
220C2×Q8—24no—
221C2×Q8—33no—
222C2×Q8—33no—
223C2×Q8—64no—
224C2×Q8C22×D313yes—
225C2×Q8C22×D311yescharacteristic
226C2×D4—31no—
227C2×D4—33no—
228C2×D4—33no—
229C2×D4—33no—
230C22×C4—33no—
231C22×C4—33no—
232C22×C4—33no—
233C22×C4—33no—
234C22×C4—31no—
235C22×C4—33no—
236C22×C4C22×D313yes—
237C22×C4C22×D313yes—
238C22×C4C22×D311yescharacteristic
239C4⋊C4—33no—
240C4⋊C4—36no—
241C4⋊C4—31no—
242C4⋊C4—33no—
243C4⋊C4—33no—
244C4⋊C4—33no—
245C4⋊C4—33no—
246C4⋊C4—33no—
247C4⋊C4—33no—
248C4⋊C4—36no—
249C4⋊C4—36no—
250C4⋊C4C22×D313yes—
251C4⋊C4C22×D313yes—
252C4⋊C4C22×D316yes—
253C22⋊C4—31no—
254C22⋊C4—33no—
255C22⋊C4—31no—
256C22⋊C4—33no—
257C22⋊C4—36no—
258C22⋊C4—33no—
259C22⋊C4—33no—
260C22⋊C4—36no—
261C22⋊C4—36no—
262C42—33no—
263C42—33no—
264C42—33no—
265C42—33no—
266C42—31no—
267C42—33no—
268C42C22×D313yes—
269C42C22×D313yes—
270C42C22×D313yes—
271C42C22×D313yes—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3C2×#4112yes—
275C22×C3C2511yescharacteristic
276C2×D3—21no—
277C2×D3—21no—
278C2×D3—23no—
279C2×D3—23no—
280C2×D3—23no—
281C2×D3C2×#4112yes—
282C3×C4—23no—
283C3×C4—23no—
284C3×C4—23no—
285C3×C4—21no—
286C3×C4C2×#4116yes—
287C3×C42−1+412yes—
288Dic3—23no—
289Dic3—23no—
290Dic3—21no—
291Dic3—21no—
292Dic3—23no—
293Dic3C2×#4112yes—
294C23—31no—
295C23—33no—
296C23C23×D311yescharacteristic
297Q8—26no—
298Q8—66no—
299Q8—66no—
300Q8#23514yescomplemented
301D4—34no—
302D4—66no—
303D4—66no—
304D4—66no—
305C2×C4—26no—
306C2×C4—26no—
307C2×C4—22no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—33no—
312C2×C4—33no—
313C2×C4—33no—
314C2×C4—66no—
315C2×C4—66no—
316C2×C4—66no—
317C2×C4—66no—
318C2×C4—62no—
319C2×C4—66no—
320C2×C4C23×D313yes—
321C2×C4C23×D311yescharacteristic
322C2×C4C23×D313yes—
323C2×C4C23×D313yes—
324C2×C4C23×D313yes—
325C2×C4C23×D311yescharacteristic
326C2×C3—22no—
327C2×C3C22×#4111yescharacteristic
328C2×C3C22×#4111yescharacteristic
329C2×C3C2×2−1+411yescharacteristic
330D3—22no—
331D3—43no—
332C22—21no—
333C22—21no—
334C22—32no—
335C22—61no—
336C22—61no—
337C22—63no—
338C22—63no—
339C22—63no—
340C22D3×#4112yes—
341C22C24×D311yescharacteristic
342C4—23no—
343C4—21no—
344C4—23no—
345C4—23no—
346C4—32no—
347C4—61no—
348C4—63no—
349C4—61no—
350C4—63no—
351C4—63no—
352C4C2×#23516yes—
353C4#100712yes—
354C3#348211yescharacteristic, complemented, Sylow
355C2—22no—
356C2—62no—
357C2—123no—
358C2C22×#23511yescharacteristic
359C2C2×#100711yescharacteristic
360C2C2×D3×#4111yescharacteristic
subgroupG/H
centerC2×C4C23×D3
commutator, socle, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×C4×Q8C2
hypercenterC2×C4×Q8D3
hypocenterC3#3482
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×Q8C2semidirect
2. #5817C2semidirect
3. Q8×#67C2semidirect
4. #5581C2semidirect
5. #5580C2semidirect
6. #5578C2semidirect
7. #5576C2semidirect
8. C4×#239C2semidirect
9. C4×D3×Q8C2semidirect
10. #5569C2semidirect
11. #5568C2semidirect
12. #5513C2semidirect
13. #5507C2semidirect
14. #5479C2semidirect
15. C4×#235C2semidirect
81. C2×C4×Q8D3semidirect
190. C3×Q8#41semidirect
201. #67#41semidirect
300. Q8#235semidirect
354. C3#3482semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×Q82
#58172
Q8×#672
#55812
#55802
#55782
#55762
C4×#2392
C4×D3×Q82
#55692
#55682
#55132
#55072
#54792
C4×#2352
#34823