Finite Group Databasebeta

← #79228

Subgroups — #79228

Order 384.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5532C211yesmaximal, characteristic
2#5513C211yesmaximal, characteristic, complemented
3C3×#428C211yesmaximal, characteristic, complemented
4#5134C211yesmaximal, characteristic, complemented
5#5127C211yesmaximal, characteristic, complemented
6#4996C211yesmaximal, characteristic, complemented
7#4968C211yesmaximal, characteristic, complemented
8#4891C211yesmaximal, characteristic, complemented
9#4889C211yesmaximal, characteristic, complemented
10#4888C211yesmaximal, characteristic, complemented
11#4880C211yesmaximal, characteristic, complemented
12#4866C211yesmaximal, characteristic, complemented
13#4864C211yesmaximal, characteristic, complemented
14#4856C211yesmaximal, characteristic, complemented
15#4854C211yesmaximal, characteristic, complemented
16#3348—31nomaximal, Sylow
17C2×C3×M16C2211yescharacteristic
18C3×#117C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20#930C2211yescharacteristic
21#929—21no—
22C4×#67C2211yescharacteristic
23#926—21no—
24#924C2211yescharacteristic
25C2×#204C2211yescharacteristic
26C2×D3×C8C2211yescharacteristic
27#898—21no—
28#897C2211yescharacteristic
29#895C2211yescharacteristic
30#893C2211yescharacteristic
31#892C2211yescharacteristic
32#892C2211yescharacteristic
33#890C2211yescharacteristic
34#889—21no—
35#888C2211yescharacteristic
36#887C2211yescharacteristic
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(C2×D3)⋊C8C2211yescharacteristic
48#818C2211yescharacteristic
49#816C2211yescharacteristic
50#815C2211yescharacteristic
51Dic3⋊C8C2211yescharacteristic
52#810C2211yescharacteristic
53#810C2211yescharacteristic
54#809C2211yescharacteristic
55#809C2211yescharacteristic
56#808C2211yescharacteristic
57#808C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#555—31no—
61#552—31no—
62#501—31no—
63#485—31no—
64#485—31no—
65#482—31no—
66#481—31no—
67#478—31no—
68#475—31no—
69#468—31no—
70#465—31no—
71#440—31no—
72#435—31no—
73#428D311yescharacteristic, complemented
74#427—31no—
75C22×C3×C4—21no—
76C22×C3×C4C2311yescharacteristic
77C2×#67—21no—
78C2×#67D411yescharacteristic
79C2×D12—21no—
80C2×D12C2311yescharacteristic
81C2×C4×D3—21no—
82C2×C4×D3C2311yescharacteristic
83C2×Dic6—21no—
84C2×Dic6C2311yescharacteristic
85C3×M16—22no—
86C2×C3×C8C2311yescharacteristic
87C2×C3×C8C2311yescharacteristic
88C3×C4⋊C4—21no—
89C3×C4⋊C4C2311yescharacteristic
90C3×C4⋊C4C2311yescharacteristic
91C3×C4⋊C4C2311yescharacteristic
92C3×C4⋊C4C2311yescharacteristic
93C3×C22⋊C4—21no—
94C3×C42—21no—
95#217—21no—
96#217D411yescharacteristic
97(C2×D3)⋊C4—21no—
98(C2×D3)⋊C4—21no—
99(C2×D3)⋊C4—21no—
100(C2×D3)⋊C4—21no—
101(C2×D3)⋊C4—21no—
102(C2×D3)⋊C4—21no—
103(C2×D3)⋊C4D411yescharacteristic
104C4⋊Dic3—21no—
105C4⋊Dic3—21no—
106C4⋊Dic3C2311yescharacteristic
107C4⋊Dic3C2311yescharacteristic
108Dic3⋊C4—21no—
109Dic3⋊C4—21no—
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4D411yescharacteristic
113C4×Dic3—21no—
114C4×Dic3—21no—
115C4×Dic3—21no—
116C4×Dic3C2311yescharacteristic
117C2×C3⋊C8C2311yescharacteristic
118C2×C3⋊C8C2311yescharacteristic
119#204—22no—
120D3×C8—22no—
121C2×SD16—31no—
122C2×M16C2×D311yescharacteristic
123C22×C8—31no—
124C4⋊Q8—31no—
125#126—61no—
126#125—31no—
127#125—61no—
128#124—61no—
129#124—61no—
130#123—61no—
131C22⋊Q8—31no—
132C22⋊Q8—31no—
133#121—31no—
134#121—31no—
135#121—61no—
136C4×Q8—31no—
137C4×D4—31no—
138C4×D4—31no—
139#117—31no—
140#117—31no—
141#117—61no—
142#117—61no—
143#117C2×D311yescharacteristic
144C2×C4⋊C4C2×D311yescharacteristic
145#107—31no—
146#107—31no—
147#107C2×D311yescharacteristic
148#107C2×D311yescharacteristic
149#106—31no—
150#106—31no—
151#106—31no—
152#106C2×D311yescharacteristic
153#106C2×D311yescharacteristic
154C4⋊C8—31no—
155#103—31no—
156#103—31no—
157#103—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×D3C2×D411yescharacteristic
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—42no—
174C2×C3×C4C2×D411yescharacteristic
175C2×C3×C4C2411yescharacteristic
176#67—41no—
177#67—81no—
178C2×Dic3—21no—
179C2×Dic3—21no—
180C2×Dic3—21no—
181C2×Dic3C2×D411yescharacteristic
182C2×Dic3C2×D411yescharacteristic
183C2×Dic3C2×D411yescharacteristic
184D12—21no—
185D12—21no—
186D12—81no—
187C4×D3—21no—
188C4×D3—21no—
189C4×D3—42no—
190Dic6—21no—
191Dic6—21no—
192Dic6—81no—
193C3×C8—21no—
194C3×C8#4112yes—
195C3⋊C8—21no—
196C3⋊C8—21no—
197C2×Q8—31no—
198C2×Q8—61no—
199C2×D4—31no—
200C2×D4—31no—
201C2×D4—61no—
202C2×D4—61no—
203C22×C4—21no—
204C22×C4—31no—
205C22×C4—61no—
206C22×C4C22×D311yescharacteristic
207SD16—62no—
208M16—22no—
209C2×C8—31no—
210C2×C8—31no—
211C2×C8—62no—
212C2×C8C22×D311yescharacteristic
213C2×C8C22×D311yescharacteristic
214C4⋊C4—21no—
215C4⋊C4—31no—
216C4⋊C4—31no—
217C4⋊C4—31no—
218C4⋊C4—61no—
219C4⋊C4—61no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4C22×D311yescharacteristic
225C4⋊C4C22×D311yescharacteristic
226C4⋊C4C22×D311yescharacteristic
227C4⋊C4C22×D311yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—31no—
230C22⋊C4—31no—
231C22⋊C4—61no—
232C22⋊C4—61no—
233C22⋊C4—61no—
234C22⋊C4—61no—
235C22⋊C4—61no—
236C22⋊C4—61no—
237C22⋊C4—61no—
238C42—21no—
239C42—31no—
240C42—61no—
241C42—61no—
242C42—61no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3C22×D411yescharacteristic
247C2×D3—21no—
248C2×D3—21no—
249C2×D3—21no—
250C2×D3—41no—
251C2×D3—41no—
252C2×D3—41no—
253C3×C4—21no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—41no—
258C3×C4C2×#4111yescharacteristic
259C3×C42−1+411yescharacteristic
260Dic3—21no—
261Dic3—21no—
262Dic3—21no—
263Dic3—41no—
264Dic3—41no—
265Dic3—41no—
266C23—31no—
267C23—61no—
268C23D3×D411yescharacteristic
269Q8—61no—
270Q8—61no—
271Q8—241no—
272D4—61no—
273D4—61no—
274D4—121no—
275D4—241no—
276D4—241no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—42no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—122no—
293C2×C4D3×D411yescharacteristic
294C2×C4C23×D311yescharacteristic
295C8—21no—
296C8—61no—
297C8—61no—
298C8#23512yes—
299C2×C3—41no—
300C2×C3#54711yescharacteristic
301C2×C3#57511yescharacteristic
302C2×C3#57711yescharacteristic
303D3—41no—
304D3—81no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—61no—
309C22—61no—
310C22—61no—
311C22—121no—
312C22—121no—
313C22—121no—
314C22C2×D3×D411yescharacteristic
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—61no—
321C4—61no—
322C4—61no—
323C4—121no—
324C4—121no—
325C4—121no—
326C4C2×#23511yescharacteristic
327C4#100711yescharacteristic
328C3#334811yescharacteristic, complemented, Sylow
329C2—41no—
330C2—121no—
331C2—241no—
332C2#550711yescharacteristic
333C2#577311yescharacteristic
334C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#428C2
socleC22×C3C22×D4
prime radical#5532C2
hypercenter#428D3
hypocenterC3#3348
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
2. #5513C2semidirect
3. C3×#428C2semidirect
4. #5134C2semidirect
5. #5127C2semidirect
6. #4996C2semidirect
7. #4968C2semidirect
8. #4891C2semidirect
9. #4889C2semidirect
10. #4888C2semidirect
11. #4880C2semidirect
12. #4866C2semidirect
13. #4864C2semidirect
14. #4856C2semidirect
15. #4854C2semidirect
73. #428D3semidirect
328. C3#3348semidirect

Maximal subgroups

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

subgroupindex
#55322
#55132
C3×#4282
#51342
#51272
#49962
#49682
#48912
#48892
#48882
#48802
#48662
#48642
#48562
#48542
#33483