Finite Group Databasebeta

← #79227

Subgroups — #79227

Order 384.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5532C211yesmaximal, characteristic, complemented
2#5514C211yesmaximal, characteristic, complemented
3C3×#428C211yesmaximal, characteristic, complemented
4#5135C211yesmaximal, characteristic, complemented
5#5127C211yesmaximal, characteristic, complemented
6#5003C211yesmaximal, characteristic, complemented
7#4967C211yesmaximal, characteristic, complemented
8#4890C211yesmaximal, characteristic, complemented
9#4885C211yesmaximal, characteristic, complemented
10#4883C211yesmaximal, characteristic, complemented
11#4876C211yesmaximal, characteristic, complemented
12#4871C211yesmaximal, characteristic, complemented
13#4869C211yesmaximal, characteristic, complemented
14#4866C211yesmaximal, characteristic, complemented
15#4860C211yesmaximal, characteristic, complemented
16#3347—31nomaximal, Sylow
17C2×C3×M16C2211yescharacteristic
18C3×#117C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20#930—21no—
21#930C2211yescharacteristic
22#930C2211yescharacteristic
23C4×#67—21no—
24C4×#67C2211yescharacteristic
25C2×D24C2211yescharacteristic
26C2×D3×C8C2211yescharacteristic
27#898—21no—
28#895C2211yescharacteristic
29#895C2211yescharacteristic
30#894—21no—
31#893C2211yescharacteristic
32#893C2211yescharacteristic
33#892C2211yescharacteristic
34#892C2211yescharacteristic
35#890C2211yescharacteristic
36#888C2211yescharacteristic
37#885—21no—
38#884—21no—
39#875—21no—
40#872—21no—
41C3×#107C2211yescharacteristic
42C3×#107C2211yescharacteristic
43C3×#106C2211yescharacteristic
44C3×#106C2211yescharacteristic
45#830C2211yescharacteristic
46#821C2211yescharacteristic
47#821C2211yescharacteristic
48(C2×D3)⋊C8C2211yescharacteristic
49#817C2211yescharacteristic
50#815C2211yescharacteristic
51Dic3⋊C8C2211yescharacteristic
52#809C2211yescharacteristic
53#809C2211yescharacteristic
54#809C2211yescharacteristic
55#809C2211yescharacteristic
56#808C2211yescharacteristic
57#807C2211yescharacteristic
58#807C2211yescharacteristic
59#807C2211yescharacteristic
60#555—31no—
61#553—31no—
62#501—31no—
63#482—31no—
64#482—31no—
65#482—31no—
66#480—31no—
67#478—31no—
68#474—31no—
69#469—31no—
70#466—31no—
71#442—31no—
72#435—31no—
73#428D311yescharacteristic, complemented
74#427—31no—
75C22×C3×C4—21no—
76C22×C3×C4C2311yescharacteristic
77C2×#67—21no—
78C2×#67—21no—
79C2×#67D411yescharacteristic
80C2×D12—21no—
81C2×D12—21no—
82C2×D12C2311yescharacteristic
83C2×D12C2311yescharacteristic
84C2×C4×D3—21no—
85C2×C4×D3—21no—
86C2×C4×D3C2311yescharacteristic
87C2×Dic6—21no—
88C3×M16—22no—
89C2×C3×C8C2311yescharacteristic
90C2×C3×C8C2311yescharacteristic
91C3×C4⋊C4—21no—
92C3×C4⋊C4C2311yescharacteristic
93C3×C4⋊C4C2311yescharacteristic
94C3×C4⋊C4C2311yescharacteristic
95C3×C4⋊C4C2311yescharacteristic
96C3×C22⋊C4—21no—
97C3×C42—21no—
98#217D411yescharacteristic
99(C2×D3)⋊C4—21no—
100(C2×D3)⋊C4—21no—
101(C2×D3)⋊C4—21no—
102(C2×D3)⋊C4—21no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4D411yescharacteristic
108C4⋊Dic3—21no—
109C4⋊Dic3C2311yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4D411yescharacteristic
114C4×Dic3—21no—
115C4×Dic3—21no—
116C4×Dic3C2311yescharacteristic
117C2×C3⋊C8C2311yescharacteristic
118C2×C3⋊C8C2311yescharacteristic
119D24—22no—
120D3×C8—22no—
121C2×D8—31no—
122C2×M16C2×D311yescharacteristic
123C22×C8—31no—
124C4⋊Q8—31no—
125#126—61no—
126#125—31no—
127#124—61no—
128#124—61no—
129#123—61no—
130#121—31no—
131#121—31no—
132#121—31no—
133#121—31no—
134#121—61no—
135#121—61no—
136C4×D4—31no—
137C4×D4—31no—
138C4×D4—31no—
139C4×D4—61no—
140#117—31no—
141#117—31no—
142#117—61no—
143#117C2×D311yescharacteristic
144C2×C4⋊C4C2×D311yescharacteristic
145#107—31no—
146#107—31no—
147#107—31no—
148#107C2×D311yescharacteristic
149#107C2×D311yescharacteristic
150#106—31no—
151#106—31no—
152#106C2×D311yescharacteristic
153#106C2×D311yescharacteristic
154C4⋊C8—31no—
155#102—31no—
156#102—31no—
157#102—31no—
158#102—31no—
159#102—31no—
160#102—31no—
161C22⋊C8—31no—
162C22⋊C8—31no—
163#97—31no—
164C23×C3C2×D411yescharacteristic
165C22×D3—21no—
166C22×D3—21no—
167C22×D3C2×D411yescharacteristic
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—81no—
175C2×C3×C4C2×D411yescharacteristic
176C2×C3×C4C2411yescharacteristic
177#67—41no—
178#67—42no—
179#67—81no—
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3C2×D411yescharacteristic
183C2×Dic3C2×D411yescharacteristic
184C2×Dic3C2×D411yescharacteristic
185D12—21no—
186D12—21no—
187D12—21no—
188D12—21no—
189D12—81no—
190D12—81no—
191C4×D3—21no—
192C4×D3—21no—
193C4×D3—42no—
194C4×D3—81no—
195Dic6—81no—
196C3×C8—21no—
197C3×C8#4112yes—
198C3⋊C8—21no—
199C3⋊C8—21no—
200C2×Q8—61no—
201C2×D4—31no—
202C2×D4—31no—
203C2×D4—31no—
204C2×D4—61no—
205C2×D4—61no—
206C2×D4—61no—
207C2×D4—61no—
208C22×C4—21no—
209C22×C4—31no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4C22×D311yescharacteristic
213D8—62no—
214M16—22no—
215C2×C8—31no—
216C2×C8—31no—
217C2×C8—62no—
218C2×C8C22×D311yescharacteristic
219C2×C8C22×D311yescharacteristic
220C4⋊C4—21no—
221C4⋊C4—31no—
222C4⋊C4—31no—
223C4⋊C4—31no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4C22×D311yescharacteristic
228C4⋊C4C22×D311yescharacteristic
229C4⋊C4C22×D311yescharacteristic
230C4⋊C4C22×D311yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—31no—
233C22⋊C4—31no—
234C22⋊C4—61no—
235C22⋊C4—61no—
236C22⋊C4—61no—
237C22⋊C4—61no—
238C22⋊C4—61no—
239C22⋊C4—61no—
240C22⋊C4—61no—
241C22⋊C4—61no—
242C42—21no—
243C42—31no—
244C42—61no—
245C42—61no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3C22×D411yescharacteristic
250C2×D3—21no—
251C2×D3—21no—
252C2×D3—21no—
253C2×D3—41no—
254C2×D3—41no—
255C2×D3—41no—
256C2×D3—41no—
257C2×D3—41no—
258C2×D3—41no—
259C3×C4—21no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4C2×#4111yescharacteristic
265C3×C42−1+411yescharacteristic
266Dic3—21no—
267Dic3—21no—
268Dic3—21no—
269Dic3—41no—
270Dic3—41no—
271C23—31no—
272C23—61no—
273C23—61no—
274C23D3×D411yescharacteristic
275Q8—241no—
276D4—61no—
277D4—61no—
278D4—61no—
279D4—61no—
280D4—121no—
281D4—122no—
282D4—241no—
283D4—241no—
284D4—241no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—31no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4—81no—
299C2×C4—122no—
300C2×C4—241no—
301C2×C4D3×D411yescharacteristic
302C2×C4C23×D311yescharacteristic
303C8—21no—
304C8—61no—
305C8—61no—
306C8#23512yes—
307C2×C3—41no—
308C2×C3#54711yescharacteristic
309C2×C3#57511yescharacteristic
310C2×C3#57611yescharacteristic
311D3—41no—
312D3—81no—
313D3—81no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—61no—
318C22—61no—
319C22—61no—
320C22—121no—
321C22—121no—
322C22—121no—
323C22—121no—
324C22—121no—
325C22—121no—
326C22C2×D3×D411yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—61no—
333C4—61no—
334C4—61no—
335C4—121no—
336C4—121no—
337C4C2×#23511yescharacteristic
338C4#100711yescharacteristic
339C3#334711yescharacteristic, complemented, Sylow
340C2—41no—
341C2—121no—
342C2—241no—
343C2—241no—
344C2#550711yescharacteristic
345C2#577411yescharacteristic
346C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#428C2
socleC22×C3C22×D4
hypercenter#428D3
hypocenterC3#3347
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #5532C2semidirect
2. #5514C2semidirect
3. C3×#428C2semidirect
4. #5135C2semidirect
5. #5127C2semidirect
6. #5003C2semidirect
7. #4967C2semidirect
8. #4890C2semidirect
9. #4885C2semidirect
10. #4883C2semidirect
11. #4876C2semidirect
12. #4871C2semidirect
13. #4869C2semidirect
14. #4866C2semidirect
15. #4860C2semidirect
73. #428D3semidirect
339. C3#3347semidirect

Maximal subgroups

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

subgroupindex
#55322
#55142
C3×#4282
#51352
#51272
#50032
#49672
#48902
#48852
#48832
#48762
#48712
#48692
#48662
#48602
#33473