Finite Group Databasebeta

← #80261

Subgroups — #80261

Order 384.

398 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5834C211yesmaximal, characteristic
2#5635C211yesmaximal, characteristic, complemented
3C3×#452C211yesmaximal, characteristic, complemented
4#5232C211yesmaximal, characteristic, complemented
5#5187C211yesmaximal, characteristic, complemented
6Dic3×Q16C211yesmaximal, characteristic, complemented
7#5174C211yesmaximal, characteristic, complemented
8#5163C211yesmaximal, characteristic, complemented
9#5049C211yesmaximal, characteristic, complemented
10#4803C211yesmaximal, characteristic, complemented
11#4793C211yesmaximal, characteristic, complemented
12#4774C211yesmaximal, characteristic, complemented
13#4760C211yesmaximal, characteristic, complemented
14#4736C211yesmaximal, characteristic, complemented
15#4729C211yesmaximal, characteristic, complemented
16#3333—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C3×C22⋊Q8C2211yescharacteristic
22#946C2211yescharacteristic
23Q8×Dic3C2211yescharacteristic
24Q8×Dic3C2211yescharacteristic
25C2×#215C2211yescharacteristic
26#940—21no—
27#940C2211yescharacteristic
28#938—21no—
29#938C2211yescharacteristic
30#930—21no—
31#930C2211yescharacteristic
32C4×#67—21no—
33C4×#67C2211yescharacteristic
34C2×C4⋊Dic3C2211yescharacteristic
35C2×#208C2211yescharacteristic
36C2×ΓL(1,𝔽25)C2211yescharacteristic
37#897—21no—
38D3×C4⋊C4C2211yescharacteristic
39#889—21no—
40#888C2211yescharacteristic
41#67⋊C4—21no—
42#878—21no—
43C3×#103C2211yescharacteristic
44C3×#102C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51#821C2211yescharacteristic
52#821C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54#818C2211yescharacteristic
55#818C2211yescharacteristic
56Dic3⋊C8C2211yescharacteristic
57C8×Dic3C2211yescharacteristic
58#809C2211yescharacteristic
59#807C2211yescharacteristic
60#554—31no—
61#550—31no—
62#486—31no—
63#480—31no—
64#480—31no—
65#477—31no—
66#468—31no—
67#468—31no—
68#460—31no—
69#452—31no—
70#452D311yescharacteristic, complemented
71C4×Q16—31no—
72#436—31no—
73#417—31no—
74#417—31no—
75C3×#41—22no—
76C3×#41—41no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4C2311yescharacteristic
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67D411yescharacteristic
88C22×Dic3—21no—
89#239—22no—
90#239—41no—
91C2×D12—21no—
92C2×D12C2311yescharacteristic
93C2×C4×D3—21no—
94C2×C4×D3—21no—
95C2×C4×D3C2311yescharacteristic
96C2×Dic6—21no—
97C3×Q16—22no—
98C3×SD16—41no—
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C3×C4⋊C4—21no—
102C3×C4⋊C4C2311yescharacteristic
103C3×C22⋊C4—21no—
104#217—21no—
105#217D411yescharacteristic
106#215—41no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4D411yescharacteristic
110C4⋊Dic3—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3—21no—
113C4⋊Dic3C2311yescharacteristic
114C4⋊Dic3C2311yescharacteristic
115C4⋊Dic3C2311yescharacteristic
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4D411yescharacteristic
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3C2311yescharacteristic
123#208—41no—
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126ΓL(1,𝔽25)—41no—
127C2×#41—31no—
128C2×#41C2×D311yescharacteristic
129C2×Q16C2×D311yescharacteristic
130C2×SD16—31no—
131C2×SD16C2×D311yescharacteristic
132C2×M16—31no—
133C2×M16—31no—
134C4⋊Q8—31no—
135#127—31no—
136#127—61no—
137#125—61no—
138C22⋊Q8—31no—
139C22⋊Q8—61no—
140C22⋊Q8—61no—
141C22⋊Q8C2×D311yescharacteristic
142#121—31no—
143#121—31no—
144#121—61no—
145#121—61no—
146C4×Q8—31no—
147C4×Q8—31no—
148C4×D4—31no—
149C4×D4—61no—
150C4×D4—61no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153#107—31no—
154#107—31no—
155#107—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103C2×D311yescharacteristic
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8C2×D311yescharacteristic
169C4×C8—31no—
170C23×C3—21no—
171C23×C3C2×D411yescharacteristic
172C22×D3—21no—
173C22×D3C2×D411yescharacteristic
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×Q8#4112yes—
178C3×D4—21no—
179C3×D4—21no—
180C3×D4—42no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—42no—
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2411yescharacteristic
189#67—22no—
190#67—42no—
191#67—81no—
192#67—81no—
193#67—81no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—81no—
198C2×Dic3C2×D411yescharacteristic
199C2×Dic3C2×D411yescharacteristic
200C2×Dic3C2×D411yescharacteristic
201D12—21no—
202D12—21no—
203D12—42no—
204C4×D3—21no—
205C4×D3—21no—
206C4×D3—42no—
207C4×D3—81no—
208Dic6—81no—
209C3×C8—21no—
210C3×C8—21no—
211C3⋊C8—21no—
212C3⋊C8—21no—
213#41—22no—
214#41—41no—
215#41—62no—
216#41—121no—
217C2×Q8—61no—
218C2×Q8C22×D311yescharacteristic
219C2×Q8C22×D311yescharacteristic
220C2×D4—21no—
221C2×D4—31no—
222C2×D4—31no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4C22×D311yescharacteristic
229C22×C4—21no—
230C22×C4—31no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4C22×D311yescharacteristic
235Q16—22no—
236SD16—41no—
237SD16—121no—
238M16—121no—
239M16—121no—
240C2×C8—31no—
241C2×C8—31no—
242C2×C8C22×D311yescharacteristic
243C2×C8C22×D311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—31no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4C22×D311yescharacteristic
256C22⋊C4—21no—
257C22⋊C4—31no—
258C22⋊C4—31no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C42—31no—
263C42—61no—
264C42—61no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
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—22no—
279C3×C4—21no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4C2×#4111yescharacteristic
283C3×C4H(5,𝔽2)11yescharacteristic
284Dic3—21no—
285Dic3—21no—
286Dic3—41no—
287Dic3—41no—
288Dic3—41no—
289Dic3#13712yes—
290C23—21no—
291C23—31no—
292C23—61no—
293C23D3×D411yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—241no—
298Q8#23712yes—
299D4—21no—
300D4—21no—
301D4—42no—
302D4—62no—
303D4—61no—
304D4—61no—
305D4—122no—
306D4—122no—
307D4—241no—
308D4—241no—
309D4—241no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—31no—
316C2×C4—31no—
317C2×C4—31no—
318C2×C4—42no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—122no—
325C2×C4—241no—
326C2×C4—241no—
327C2×C4D3×D411yescharacteristic
328C2×C4C23×D311yescharacteristic
329C8—21no—
330C8—21no—
331C8—61no—
332C8—61no—
333C2×C3—41no—
334C2×C3—81no—
335C2×C3#54611yescharacteristic
336C2×C3C2×#13711yescharacteristic
337C2×C3#57611yescharacteristic
338D3—41no—
339D3—81no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—61no—
347C22—61no—
348C22—61no—
349C22—121no—
350C22—121no—
351C22—121no—
352C22C2×D3×D411yescharacteristic
353C4—21no—
354C4—22no—
355C4—32no—
356C4—41no—
357C4—41no—
358C4—61no—
359C4—61no—
360C4—121no—
361C4—121no—
362C4—121no—
363C4C2×#23711yescharacteristic
364C4#100411yescharacteristic
365C3#333311yescharacteristic, complemented, Sylow
366C2—41no—
367C2—81no—
368C2—121no—
369C2—241no—
370C2#559111yescharacteristic
371C2#576911yescharacteristic
372C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#452C2
socleC22×C3C22×D4
prime radical#5834C2
hypercenter#452D3
hypocenterC3#3333
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. #5635C2semidirect
3. C3×#452C2semidirect
4. #5232C2semidirect
5. #5187C2semidirect
6. Dic3×Q16C2semidirect
7. #5174C2semidirect
8. #5163C2semidirect
9. #5049C2semidirect
10. #4803C2semidirect
11. #4793C2semidirect
12. #4774C2semidirect
13. #4760C2semidirect
14. #4736C2semidirect
15. #4729C2semidirect
70. #452D3semidirect
365. C3#3333semidirect

Maximal subgroups

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

subgroupindex
#58342
#56352
C3×#4522
#52322
#51872
Dic3×Q162
#51742
#51632
#50492
#48032
#47932
#47742
#47602
#47362
#47292
#33333