Finite Group Databasebeta

← #80437

Subgroups — #80437

Order 384.

376 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5724C211yesmaximal, characteristic, complemented
2#5561C211yesmaximal, characteristic, complemented
3C3×#461C211yesmaximal, characteristic, complemented
4#5172C211yesmaximal, characteristic
5#5170C211yesmaximal, characteristic, complemented
6#5168C211yesmaximal, characteristic, complemented
7#5167C211yesmaximal, characteristic, complemented
8#5090C211yesmaximal, characteristic, complemented
9#5016C211yesmaximal, characteristic, complemented
10#4845C211yesmaximal, characteristic, complemented
11#4840C211yesmaximal, characteristic, complemented
12#4809C211yesmaximal, characteristic, complemented
13#4808C211yesmaximal, characteristic, complemented
14#4794C211yesmaximal, characteristic, complemented
15#4792C211yesmaximal, characteristic, complemented
16#3225—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C3×C4⋊Q8C2211yescharacteristic
22C3×C4×D4C2211yescharacteristic
23#947C2211yescharacteristic
24#944C2211yescharacteristic
25C2×#216C2211yescharacteristic
26C2×#216C2211yescharacteristic
27#938C2211yescharacteristic
28#936C2211yescharacteristic
29C2×#214C2211yescharacteristic
30C2×#213C2211yescharacteristic
31#929—21no—
32#924—21no—
33C2×Dic12C2211yescharacteristic
34C2×#204C2211yescharacteristic
35#897C2211yescharacteristic
36#896—21no—
37#896—21no—
38#894—21no—
39#894—21no—
40#888C2211yescharacteristic
41#885—21no—
42#879—21no—
43#875C2211yescharacteristic
44#872C2211yescharacteristic
45C4×Dic6C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic
47C3×#103C2211yescharacteristic
48C3×#103C2211yescharacteristic
49#835C2211yescharacteristic
50#835C2211yescharacteristic
51#821C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54#816C2211yescharacteristic
55#815C2211yescharacteristic
56C8×Dic3C2211yescharacteristic
57#810C2211yescharacteristic
58#809C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#542—31no—
61#536—31no—
62#497—31no—
63#495—31no—
64#489—31no—
65#487—31no—
66#464—31no—
67#463—31no—
68#462—31no—
69#461D311yescharacteristic, complemented
70#452—31no—
71#451—31no—
72#449—31no—
73#448—31no—
74#433—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4—21no—
79C2×#67—21no—
80#239—41no—
81#239—41no—
82D3×Q8—41no—
83D3×Q8—41no—
84C2×D12—21no—
85C2×D12C2311yescharacteristic
86C2×C4×D3—21no—
87C2×C4×D3—21no—
88C2×C4×D3C2311yescharacteristic
89C2×Dic6—21no—
90C2×Dic6—21no—
91C2×Dic6C2311yescharacteristic
92C2×Dic6C2311yescharacteristic
93C3×SD16—41no—
94C3×SD16—41no—
95C2×C3×C8C2311yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C3×C4⋊C4—21no—
98C3×C4⋊C4—21no—
99C3×C4⋊C4C2311yescharacteristic
100C3×C22⋊C4—21no—
101C3×C42C2311yescharacteristic
102#217—21no—
103#217—21no—
104#217—21no—
105#216—41no—
106#216—41no—
107#214—41no—
108#213—41no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4D411yescharacteristic
115(C2×D3)⋊C4D411yescharacteristic
116C4⋊Dic3—21no—
117C4⋊Dic3C2311yescharacteristic
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4D411yescharacteristic
123Dic3⋊C4D411yescharacteristic
124C4×Dic3—21no—
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128Dic12—41no—
129#204—41no—
130C2×#41—31no—
131C22×Q8—31no—
132C2×Q16—31no—
133C2×Q16—31no—
134C2×Q16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×SD16C2×D311yescharacteristic
138C2×SD16C2×D311yescharacteristic
139C2×D8—31no—
140C4⋊Q8—31no—
141C4⋊Q8—31no—
142C4⋊Q8C2×D311yescharacteristic
143#126—61no—
144#124—31no—
145#124—31no—
146#124—31no—
147#124—61no—
148#123—61no—
149#123—61no—
150#123—61no—
151C22⋊Q8—31no—
152C22⋊Q8—61no—
153C22⋊Q8—61no—
154C22⋊Q8—61no—
155#121—31no—
156C4×Q8—31no—
157C4×D4C2×D311yescharacteristic
158#117—31no—
159C4⋊C8—31no—
160C4⋊C8C2×D311yescharacteristic
161#103—31no—
162#103—31no—
163#103—31no—
164#103—31no—
165#103C2×D311yescharacteristic
166#103C2×D311yescharacteristic
167#102—31no—
168#102—31no—
169C22⋊C8—31no—
170C22⋊C8—31no—
171#97—31no—
172C4×C8—31no—
173C23×C3—21no—
174C22×D3—21no—
175C22×D3C2×D411yescharacteristic
176C3×Q8—21no—
177C3×Q8—21no—
178C3×Q8—21no—
179C3×Q8—21no—
180C3×D4—21no—
181C3×D4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—81no—
186C2×C3×C4C2×D411yescharacteristic
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2411yescharacteristic
189#67—81no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196D12—21no—
197D12—21no—
198D12—42no—
199C4×D3—21no—
200C4×D3—21no—
201C4×D3—42no—
202C4×D3—42no—
203Dic6—21no—
204Dic6—21no—
205Dic6—21no—
206Dic6—21no—
207Dic6—42no—
208Dic6—81no—
209C3×C8—21no—
210C3×C8—21no—
211C3⋊C8—21no—
212C3⋊C8—21no—
213#41—121no—
214#41—121no—
215C2×Q8—31no—
216C2×Q8—31no—
217C2×Q8—61no—
218C2×Q8—61no—
219C2×Q8—121no—
220C2×Q8—121no—
221C2×Q8C22×D311yescharacteristic
222C2×Q8C22×D311yescharacteristic
223C2×D4—31no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4C22×D311yescharacteristic
227C22×C4—21no—
228C22×C4—31no—
229C22×C4—61no—
230C22×C4—61no—
231Q16—121no—
232Q16—121no—
233Q16—121no—
234SD16—41no—
235SD16—41no—
236SD16—121no—
237SD16—121no—
238D8—121no—
239C2×C8—31no—
240C2×C8—31no—
241C2×C8C22×D311yescharacteristic
242C2×C8C22×D311yescharacteristic
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4C22×D311yescharacteristic
254C22⋊C4—21no—
255C22⋊C4—31no—
256C22⋊C4—31no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C42—31no—
266C42—61no—
267C42C22×D311yescharacteristic
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3C22×D411yescharacteristic
272C2×D3—21no—
273C2×D3—21no—
274C2×D3—21no—
275C2×D3—41no—
276C2×D3—41no—
277C2×D3—41no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4—41no—
283C3×C4H(5,𝔽2)11yescharacteristic
284C3×C4H(5,𝔽2)11yescharacteristic
285Dic3—21no—
286Dic3—21no—
287Dic3—21no—
288Dic3—41no—
289Dic3—41no—
290Dic3—41no—
291C23—21no—
292C23—31no—
293C23—61no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—61no—
299Q8—61no—
300Q8—61no—
301Q8—61no—
302Q8—122no—
303Q8—241no—
304D4—21no—
305D4—21no—
306D4—61no—
307D4—61no—
308D4—122no—
309D4—241no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—31no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—81no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4D3×D411yescharacteristic
325C2×C4D3×D411yescharacteristic
326C2×C4C23×D311yescharacteristic
327C8—21no—
328C8—21no—
329C8—61no—
330C8—61no—
331C2×C3—81no—
332C2×C3#53511yescharacteristic
333C2×C3#57711yescharacteristic
334C2×C3#57811yescharacteristic
335D3—41no—
336D3—81no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—61no—
341C22—61no—
342C22—61no—
343C22—121no—
344C22—121no—
345C22—121no—
346C22C2×D3×D411yescharacteristic
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—61no—
353C4—61no—
354C4—61no—
355C4—121no—
356C4—121no—
357C4—121no—
358C4#100411yescharacteristic
359C4#100911yescharacteristic
360C3#322511yescharacteristic, complemented, Sylow
361C2—81no—
362C2—121no—
363C2—241no—
364C2#560711yescharacteristic
365C2#576211yescharacteristic
366C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#461C2
socleC22×C3C22×D4
prime radical#5172C2
hypercenter#461D3
hypocenterC3#3225
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #5724C2semidirect
2. #5561C2semidirect
3. C3×#461C2semidirect
5. #5170C2semidirect
6. #5168C2semidirect
7. #5167C2semidirect
8. #5090C2semidirect
9. #5016C2semidirect
10. #4845C2semidirect
11. #4840C2semidirect
12. #4809C2semidirect
13. #4808C2semidirect
14. #4794C2semidirect
15. #4792C2semidirect
69. #461D3semidirect
360. C3#3225semidirect

Maximal subgroups

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

subgroupindex
#57242
#55612
C3×#4612
#51722
#51702
#51682
#51672
#50902
#50162
#48452
#48402
#48092
#48082
#47942
#47922
#32253