Finite Group Databasebeta

← #78973

Subgroups — #78973

Order 384.

386 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
1Q8×#67C211yesmaximal, characteristic, complemented
2#5532C211yesmaximal, characteristic
3C3×#419C211yesmaximal, characteristic, complemented
4#5227C211yesmaximal, characteristic, complemented
5#5134C211yesmaximal, characteristic, complemented
6#5128C211yesmaximal, characteristic, complemented
7#4996C211yesmaximal, characteristic, complemented
8#4818C211yesmaximal, characteristic, complemented
9#4814C211yesmaximal, characteristic, complemented
10#4812C211yesmaximal, characteristic, complemented
11#4806C211yesmaximal, characteristic, complemented
12#4802C211yesmaximal, characteristic, complemented
13#4793C211yesmaximal, characteristic, complemented
14#4792C211yesmaximal, characteristic, complemented
15#4788C211yesmaximal, characteristic, complemented
16#3329—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×C3×M16C2211yescharacteristic
20C3×#117C2211yescharacteristic
21#946—21no—
22#946C2211yescharacteristic
23Q8×Dic3C2211yescharacteristic
24#944—21no—
25#944C2211yescharacteristic
26C2×#215C2211yescharacteristic
27C2×#215C2211yescharacteristic
28#930C2211yescharacteristic
29C4×#67—21no—
30C4×#67C2211yescharacteristic
31#924—21no—
32#924C2211yescharacteristic
33C2×#204C2211yescharacteristic
34C2×ΓL(1,𝔽25)C2211yescharacteristic
35#895C2211yescharacteristic
36#893C2211yescharacteristic
37#892C2211yescharacteristic
38#888C2211yescharacteristic
39#885—21no—
40#884—21no—
41#875—21no—
42#872—21no—
43C3×#103C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#103C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#830C2211yescharacteristic
50#821C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#818C2211yescharacteristic
53#816C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55C8×Dic3C2211yescharacteristic
56#810C2211yescharacteristic
57#810C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#555—31no—
61D4×Q8—31no—
62#487—31no—
63#485—31no—
64#485—31no—
65#477—31no—
66#468—31no—
67#468—31no—
68#461—31no—
69#448—31no—
70#448—31no—
71C4×SD16—31no—
72#436—31no—
73#419—31no—
74#419D311yescharacteristic, complemented
75C2×C3×Q8—21no—
76C2×C3×Q8—22no—
77C2×C3×Q8—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C22×C3×C4—21no—
81C22×C3×C4C2311yescharacteristic
82C2×#67—21no—
83C2×#67D411yescharacteristic
84D3×Q8—22no—
85D3×Q8—41no—
86C2×D12—21no—
87C2×D12C2311yescharacteristic
88C2×C4×D3—21no—
89C2×C4×D3—21no—
90C2×C4×D3C2311yescharacteristic
91C2×Dic6—21no—
92C2×Dic6—21no—
93C2×Dic6C2311yescharacteristic
94C3×M16—41no—
95C2×C3×C8C2311yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C3×C4⋊C4C2311yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C22⋊C4—21no—
100C3×C42—21no—
101#217—21no—
102#217D411yescharacteristic
103#215—22no—
104#215—41no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4D411yescharacteristic
111C4⋊Dic3—21no—
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4D411yescharacteristic
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125#204—41no—
126ΓL(1,𝔽25)—41no—
127C22×Q8—31no—
128C22×Q8C2×D311yescharacteristic
129C2×SD16—31no—
130C2×SD16—31no—
131C2×SD16—31no—
132C2×M16—31no—
133C2×M16C2×D311yescharacteristic
134C4⋊Q8—31no—
135C4⋊Q8—31no—
136C4⋊Q8—61no—
137#124—61no—
138#124—61no—
139C22⋊Q8—31no—
140C22⋊Q8—31no—
141C22⋊Q8—61no—
142C22⋊Q8—61no—
143#121—31no—
144#121—31no—
145#121—61no—
146C4×Q8—31no—
147C4×D4—31no—
148C4×D4—31no—
149C4×D4—61no—
150#117—31no—
151#117—61no—
152#117C2×D311yescharacteristic
153#107—31no—
154#107—31no—
155#106—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103C2×D311yescharacteristic
163#103C2×D311yescharacteristic
164#103C2×D311yescharacteristic
165#103C2×D311yescharacteristic
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8—31no—
169C4×C8—31no—
170C23×C3C2×D411yescharacteristic
171C22×D3—21no—
172C22×D3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—42no—
177C3×Q8#4112yes—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—42no—
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2411yescharacteristic
187#67—22no—
188#67—81no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3C2×D411yescharacteristic
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195D12—21no—
196D12—21no—
197D12—81no—
198C4×D3—21no—
199C4×D3—21no—
200C4×D3—42no—
201C4×D3—42no—
202Dic6—21no—
203Dic6—21no—
204Dic6—42no—
205Dic6—81no—
206C3×C8—21no—
207C3×C8—21no—
208C3⋊C8—21no—
209C3⋊C8—21no—
210C2×Q8—21no—
211C2×Q8—22no—
212C2×Q8—31no—
213C2×Q8—41no—
214C2×Q8—61no—
215C2×Q8—61no—
216C2×Q8—62no—
217C2×Q8—121no—
218C2×Q8C22×D311yescharacteristic
219C2×Q8C22×D311yescharacteristic
220C2×D4—31no—
221C2×D4—31no—
222C2×D4—61no—
223C2×D4—61no—
224C22×C4—21no—
225C22×C4—31no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4C22×D311yescharacteristic
229SD16—62no—
230SD16—121no—
231SD16—121no—
232M16—41no—
233M16—121no—
234C2×C8—31no—
235C2×C8—31no—
236C2×C8C22×D311yescharacteristic
237C2×C8C22×D311yescharacteristic
238C4⋊C4—31no—
239C4⋊C4—31no—
240C4⋊C4—31no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4C22×D311yescharacteristic
248C4⋊C4C22×D311yescharacteristic
249C22⋊C4—21no—
250C22⋊C4—31no—
251C22⋊C4—31no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C42—21no—
259C42—31no—
260C42—61no—
261C42—61no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3C22×D411yescharacteristic
266C2×D3—21no—
267C2×D3—21no—
268C2×D3—21no—
269C2×D3—41no—
270C2×D3—41no—
271C2×D3—41no—
272C3×C4—22no—
273C3×C4—21no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4C2×#4111yescharacteristic
278C3×C4H(5,𝔽2)11yescharacteristic
279Dic3—21no—
280Dic3—21no—
281Dic3—41no—
282Dic3—41no—
283Dic3—41no—
284Dic3#13712yes—
285C23—31no—
286C23—61no—
287C23D3×D411yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—42no—
292Q8—61no—
293Q8—61no—
294Q8—122no—
295Q8—241no—
296Q8#23512yes—
297D4—62no—
298D4—61no—
299D4—61no—
300D4—241no—
301D4—241no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—42no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4D3×D411yescharacteristic
320C2×C4C23×D311yescharacteristic
321C8—21no—
322C8—21no—
323C8—61no—
324C8—61no—
325C2×C3—41no—
326C2×C3#54611yescharacteristic
327C2×C3C2×#13711yescharacteristic
328C2×C3#57711yescharacteristic
329D3—41no—
330D3—81no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—61no—
335C22—61no—
336C22—61no—
337C22—121no—
338C22—121no—
339C22—121no—
340C22C2×D3×D411yescharacteristic
341C4—21no—
342C4—22no—
343C4—32no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—61no—
348C4—61no—
349C4—121no—
350C4—121no—
351C4—121no—
352C4C2×#23511yescharacteristic
353C4#100411yescharacteristic
354C3#332911yescharacteristic, complemented, Sylow
355C2—41no—
356C2—121no—
357C2—241no—
358C2#549011yescharacteristic
359C2D3×#13711yescharacteristic
360C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#419C2
socleC22×C3C22×D4
prime radical#5532C2
hypercenter#419D3
hypocenterC3#3329
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. Q8×#67C2semidirect
3. C3×#419C2semidirect
4. #5227C2semidirect
5. #5134C2semidirect
6. #5128C2semidirect
7. #4996C2semidirect
8. #4818C2semidirect
9. #4814C2semidirect
10. #4812C2semidirect
11. #4806C2semidirect
12. #4802C2semidirect
13. #4793C2semidirect
14. #4792C2semidirect
15. #4788C2semidirect
74. #419D3semidirect
354. C3#3329semidirect

Maximal subgroups

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

subgroupindex
Q8×#672
#55322
C3×#4192
#52272
#51342
#51282
#49962
#48182
#48142
#48122
#48062
#48022
#47932
#47922
#47882
#33293