Finite Group Databasebeta

← #79858

Subgroups — #79858

Order 384.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5559C211yesmaximal, characteristic, complemented
2#5542C211yesmaximal, characteristic, complemented
3C3×#442C211yesmaximal, characteristic, complemented
4#5155C211yesmaximal, characteristic, complemented
5#5154C211yesmaximal, characteristic
6#5019C211yesmaximal, characteristic, complemented
7#5016C211yesmaximal, characteristic, complemented
8#4853C211yesmaximal, characteristic, complemented
9#4852C211yesmaximal, characteristic, complemented
10#4787C211yesmaximal, characteristic, complemented
11#4767C211yesmaximal, characteristic, complemented
12#4767C211yesmaximal, characteristic, complemented
13#4762C211yesmaximal, characteristic, complemented
14#4715C211yesmaximal, characteristic, complemented
15#4701C211yesmaximal, characteristic, complemented
16#3401—31nomaximal, Sylow
17C2×C3×D8C2211yescharacteristic
18C3×C4×D4C2211yescharacteristic
19C3×C4×D4C2211yescharacteristic
20#940C2211yescharacteristic
21#936C2211yescharacteristic
22#936C2211yescharacteristic
23#936C2211yescharacteristic
24C2×#214C2211yescharacteristic
25C2×#214C2211yescharacteristic
26C2×#214C2211yescharacteristic
27C2×#213C2211yescharacteristic
28#929—21no—
29C4×#67—21no—
30#926—21no—
31#924—21no—
32C2×Dic12C2211yescharacteristic
33C2×Dic12C2211yescharacteristic
34C2×#204C2211yescharacteristic
35C2×#204C2211yescharacteristic
36#893C2211yescharacteristic
37#887C2211yescharacteristic
38#887C2211yescharacteristic
39#887C2211yescharacteristic
40#885—21no—
41#883—21no—
42#879—21no—
43#879—21no—
44#875C2211yescharacteristic
45#869C2211yescharacteristic
46C4×Dic6C2211yescharacteristic
47C3×#106C2211yescharacteristic
48C3×#102C2211yescharacteristic
49C3×#102C2211yescharacteristic
50C3×#97C2211yescharacteristic
51#815C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53Dic3⋊C8C2211yescharacteristic
54C8×Dic3C2211yescharacteristic
55#810C2211yescharacteristic
56#810C2211yescharacteristic
57#810C2211yescharacteristic
58#809C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#559—31no—
61#556—31no—
62#497—31no—
63#496—31no—
64#495—31no—
65#464—31no—
66#464—31no—
67#463—31no—
68#463—31no—
69#461—31no—
70#460—31no—
71#446—31no—
72#442D311yescharacteristic, complemented
73#441—31no—
74C4×SD16—31no—
75C2×C3×D4C2311yescharacteristic
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C2×#67—21no—
80C2×#67—21no—
81C2×D12C2311yescharacteristic
82C2×C4×D3—21no—
83C2×Dic6D411yescharacteristic
84C2×Dic6C2311yescharacteristic
85C2×Dic6C2311yescharacteristic
86C2×Dic6C2311yescharacteristic
87C3×D8—22no—
88C2×C3×C8C2311yescharacteristic
89C2×C3×C8C2311yescharacteristic
90C3×C4⋊C4C2311yescharacteristic
91C3×C4⋊C4C2311yescharacteristic
92C3×C22⋊C4—21no—
93C3×C22⋊C4—21no—
94C3×C42C2311yescharacteristic
95#217—21no—
96#217—21no—
97#217—21no—
98#217—21no—
99#217—21no—
100#217—21no—
101#214—41no—
102#214—41no—
103#214—41no—
104#213—41no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108C4⋊Dic3—21no—
109C4⋊Dic3—21no—
110C4⋊Dic3D411yescharacteristic
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4D411yescharacteristic
115Dic3⋊C4D411yescharacteristic
116C4×Dic3—21no—
117C4×Dic3—21no—
118C4×Dic3—21no—
119C4×Dic3C2311yescharacteristic
120C4×Dic3C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123Dic12—22no—
124Dic12—41no—
125#204—22no—
126#204—41no—
127C2×Q16—31no—
128C2×Q16—31no—
129C2×SD16—31no—
130C2×SD16—31no—
131C2×SD16—31no—
132C2×SD16—31no—
133C2×SD16—31no—
134C2×D8—31no—
135C2×D8C2×D311yescharacteristic
136C4⋊Q8—31no—
137#127—31no—
138#126—61no—
139#126—61no—
140#124—31no—
141#124—31no—
142#124—31no—
143#124—31no—
144#124—61no—
145#123—61no—
146#123—61no—
147C22⋊Q8—61no—
148C4×Q8—31no—
149C4×Q8—31no—
150C4×Q8—31no—
151C4×Q8—31no—
152C4×D4—31no—
153C4×D4—61no—
154C4×D4C2×D311yescharacteristic
155C4×D4C2×D311yescharacteristic
156#117—61no—
157#106C2×D311yescharacteristic
158C4⋊C8—31no—
159C4⋊C8—31no—
160C4⋊C8—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#102—31no—
165#102C2×D311yescharacteristic
166#102C2×D311yescharacteristic
167#97—31no—
168#97C2×D311yescharacteristic
169C4×C8—31no—
170C23×C3—21no—
171C23×C3—21no—
172C22×D3—21no—
173C3×D4—21no—
174C3×D4—21no—
175C3×D4—21no—
176C3×D4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—42no—
180C2×C3×C4—81no—
181C2×C3×C4C2×D411yescharacteristic
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2411yescharacteristic
184#67—42no—
185#67—81no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3C2×D411yescharacteristic
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193D12—21no—
194D12—21no—
195C4×D3—81no—
196Dic6—21no—
197Dic6—21no—
198Dic6—21no—
199Dic6—21no—
200Dic6—21no—
201Dic6—21no—
202Dic6—41no—
203C3×C8—21no—
204C3×C8#4112yes—
205C3⋊C8—21no—
206C3⋊C8—21no—
207C2×Q8—31no—
208C2×Q8—31no—
209C2×Q8—31no—
210C2×Q8—31no—
211C2×D4—31no—
212C2×D4—61no—
213C2×D4—61no—
214C2×D4C22×D311yescharacteristic
215C2×D4C22×D311yescharacteristic
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—61no—
219Q16—62no—
220Q16—121no—
221SD16—62no—
222SD16—121no—
223SD16—121no—
224SD16—121no—
225SD16—121no—
226D8—22no—
227D8—121no—
228C2×C8—31no—
229C2×C8—31no—
230C2×C8C22×D311yescharacteristic
231C2×C8C22×D311yescharacteristic
232C4⋊C4—31no—
233C4⋊C4—31no—
234C4⋊C4—31no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4C22×D311yescharacteristic
241C4⋊C4C22×D311yescharacteristic
242C22⋊C4—21no—
243C22⋊C4—21no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C42—31no—
254C42—31no—
255C42—61no—
256C42—61no—
257C42—61no—
258C42C22×D311yescharacteristic
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3C22×D411yescharacteristic
266C2×D3—41no—
267C2×D3—41no—
268C2×D3—41no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4C2×#4111yescharacteristic
274C3×C4H(5,𝔽2)11yescharacteristic
275Dic3—21no—
276Dic3—21no—
277Dic3—21no—
278Dic3—21no—
279Dic3—41no—
280Dic3—41no—
281Dic3—41no—
282C23—21no—
283C23—21no—
284C23—61no—
285Q8—61no—
286Q8—61no—
287Q8—61no—
288Q8—61no—
289Q8—61no—
290Q8—61no—
291Q8—121no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—21no—
296D4—61no—
297D4—61no—
298D4—122no—
299D4—241no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—42no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—81no—
311C2×C4—241no—
312C2×C4D3×D411yescharacteristic
313C2×C4D3×D411yescharacteristic
314C2×C4C23×D311yescharacteristic
315C8—21no—
316C8—61no—
317C8—61no—
318C8#23512yes—
319C2×C3—81no—
320C2×C3—81no—
321C2×C3#55011yescharacteristic
322C2×C3#57511yescharacteristic
323C2×C3#57711yescharacteristic
324D3—81no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—121no—
332C22—121no—
333C22—121no—
334C22C2×D3×D411yescharacteristic
335C4—21no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—61no—
340C4—61no—
341C4—61no—
342C4—61no—
343C4—121no—
344C4—121no—
345C4—121no—
346C4C2×#23511yescharacteristic
347C4#100911yescharacteristic
348C3#340111yescharacteristic, complemented, Sylow
349C2—81no—
350C2—81no—
351C2—241no—
352C2#555311yescharacteristic
353C2#577311yescharacteristic
354C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#442C2
socleC22×C3C22×D4
prime radical#5154C2
hypercenter#442D3
hypocenterC3#3401
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #5559C2semidirect
2. #5542C2semidirect
3. C3×#442C2semidirect
4. #5155C2semidirect
6. #5019C2semidirect
7. #5016C2semidirect
8. #4853C2semidirect
9. #4852C2semidirect
10. #4787C2semidirect
11. #4767C2semidirect
12. #4767C2semidirect
13. #4762C2semidirect
14. #4715C2semidirect
15. #4701C2semidirect
72. #442D3semidirect
348. C3#3401semidirect

Maximal subgroups

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

subgroupindex
#55592
#55422
C3×#4422
#51552
#51542
#50192
#50162
#48532
#48522
#47872
#47672
#47622
#47152
#47012
#34013