Finite Group Databasebeta

← #80048

Subgroups — #80048

Order 384.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5816C211yesmaximal, characteristic, complemented
2#5608C211yesmaximal, characteristic
3C3×#448C211yesmaximal, characteristic, complemented
4#5228C211yesmaximal, characteristic, complemented
5#5174C211yesmaximal, characteristic, complemented
6#5173C211yesmaximal, characteristic, complemented
7#5173C211yesmaximal, characteristic, complemented
8#5171C211yesmaximal, characteristic, complemented
9#5035C211yesmaximal, characteristic, complemented
10#4816C211yesmaximal, characteristic, complemented
11#4813C211yesmaximal, characteristic, complemented
12#4811C211yesmaximal, characteristic, complemented
13#4805C211yesmaximal, characteristic, complemented
14#4727C211yesmaximal, characteristic, complemented
15#4723C211yesmaximal, characteristic, complemented
16#3201—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C3×#121C2211yescharacteristic
23#947—21no—
24#946C2211yescharacteristic
25#946C2211yescharacteristic
26#946C2211yescharacteristic
27#946C2211yescharacteristic
28#944—21no—
29C2×#216C2211yescharacteristic
30C2×#216C2211yescharacteristic
31#939—21no—
32#938—21no—
33#938C2211yescharacteristic
34#929—21no—
35#929—21no—
36#926C2211yescharacteristic
37#924C2211yescharacteristic
38C2×ΓL(1,𝔽25)C2211yescharacteristic
39C2×D3×C8C2211yescharacteristic
40#897C2211yescharacteristic
41D3×C4⋊C4C2211yescharacteristic
42#884—21no—
43#878—21no—
44C3×#103C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×C22⋊C8C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51#830C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54#818C2211yescharacteristic
55#817C2211yescharacteristic
56#816C2211yescharacteristic
57#816C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#537—31no—
61#536—31no—
62#485—31no—
63#484—31no—
64#483—31no—
65#482—31no—
66#470—31no—
67#468—31no—
68#467—31no—
69#465—31no—
70#452—31no—
71#452—31no—
72#451—31no—
73#448D311yescharacteristic, complemented
74#408—31no—
75C2×C3×Q8—21no—
76C2×C3×Q8—41no—
77C2×C3×Q8—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C2×C3×D4—21no—
81C2×C3×D4C2311yescharacteristic
82C22×C3×C4—21no—
83C22×C3×C4C2311yescharacteristic
84C2×#67—21no—
85C2×#67—21no—
86C2×#67D411yescharacteristic
87C2×#67D411yescharacteristic
88C22×Dic3—21no—
89#237—41no—
90#237—41no—
91#235—41no—
92#235—41no—
93C2×D12D411yescharacteristic
94C2×C4×D3—21no—
95C2×C4×D3C2311yescharacteristic
96C2×C4×D3C2311yescharacteristic
97C2×Dic6D411yescharacteristic
98C2×Dic6C2311yescharacteristic
99C3×SD16—41no—
100C3×SD16—41no—
101C2×C3×C8C2311yescharacteristic
102C2×C3×C8C2311yescharacteristic
103C3×C4⋊C4C2311yescharacteristic
104C3×C22⋊C4—21no—
105#217—21no—
106#217—21no—
107#217—21no—
108#216—41no—
109#216—41no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115C4⋊Dic3—21no—
116C4⋊Dic3C2311yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123Dic3⋊C4—21no—
124Dic3⋊C4—21no—
125C4×Dic3—21no—
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128ΓL(1,𝔽25)—41no—
129D3×C8—41no—
130C2×#41—31no—
131C2×#41—31no—
132C22×Q8C2×D311yescharacteristic
133C2×Q16—31no—
134C2×Q16—31no—
135C2×SD16C2×D311yescharacteristic
136C2×SD16C2×D311yescharacteristic
137C2×M16—31no—
138C22×C8—31no—
139C4⋊Q8—61no—
140#124—61no—
141#123—61no—
142#123—61no—
143C22⋊Q8—31no—
144C22⋊Q8—31no—
145C22⋊Q8—31no—
146C22⋊Q8—31no—
147C22⋊Q8—31no—
148C22⋊Q8—31no—
149C22⋊Q8—61no—
150#121—31no—
151#121—61no—
152#121—61no—
153#121—61no—
154#121C2×D311yescharacteristic
155#117—31no—
156C2×C4⋊C4—31no—
157#107—31no—
158#107—31no—
159#106—31no—
160#106—31no—
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—
171C22⋊C8—31no—
172C22⋊C8C2×D311yescharacteristic
173C23×C3—21no—
174C23×C3C2×D411yescharacteristic
175C22×D3C2×D411yescharacteristic
176C22×D3C2×D411yescharacteristic
177C3×Q8—21no—
178C3×Q8—21no—
179C3×Q8—21no—
180C3×Q8—21no—
181C3×Q8—42no—
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—81no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—42no—
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2411yescharacteristic
193#67—41no—
194#67—41no—
195#67—42no—
196#67—81no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—42no—
202C2×Dic3C2×D411yescharacteristic
203C2×Dic3C2×D411yescharacteristic
204D12—41no—
205C4×D3—21no—
206C4×D3—21no—
207C4×D3—21no—
208C4×D3—21no—
209C4×D3—81no—
210Dic6—21no—
211Dic6—21no—
212Dic6—41no—
213C3×C8—21no—
214C3×C8—21no—
215C3⋊C8—21no—
216C3⋊C8—21no—
217#41—121no—
218#41—121no—
219#41—121no—
220#41—121no—
221C2×Q8—21no—
222C2×Q8—31no—
223C2×Q8—31no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8C22×D311yescharacteristic
227C2×Q8C22×D311yescharacteristic
228C2×D4—21no—
229C2×D4—31no—
230C2×D4—31no—
231C2×D4—31no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4C22×D311yescharacteristic
235C22×C4—21no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4C22×D311yescharacteristic
241Q16—121no—
242Q16—121no—
243SD16—41no—
244SD16—41no—
245M16—121no—
246C2×C8—31no—
247C2×C8—31no—
248C2×C8—121no—
249C2×C8C22×D311yescharacteristic
250C2×C8C22×D311yescharacteristic
251C4⋊C4—31no—
252C4⋊C4—31no—
253C4⋊C4—31no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4C22×D311yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C42—61no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3C22×D411yescharacteristic
279C2×D3—21no—
280C2×D3—21no—
281C2×D3—21no—
282C2×D3—21no—
283C2×D3—21no—
284C2×D3—21no—
285C3×C4—21no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4H(5,𝔽2)11yescharacteristic
290C3×C4H(5,𝔽2)11yescharacteristic
291Dic3—21no—
292Dic3—21no—
293Dic3—41no—
294Dic3—41no—
295Dic3—41no—
296Dic3—41no—
297C23—21no—
298C23—31no—
299C23—31no—
300C23D3×D411yescharacteristic
301Q8—21no—
302Q8—21no—
303Q8—21no—
304Q8—21no—
305Q8—42no—
306Q8—61no—
307Q8—61no—
308Q8—121no—
309D4—21no—
310D4—21no—
311D4—81no—
312D4—121no—
313D4—121no—
314D4—121no—
315D4—122no—
316D4—241no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—31no—
323C2×C4—31no—
324C2×C4—42no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—122no—
334C2×C4—241no—
335C2×C4D3×D411yescharacteristic
336C2×C4C23×D311yescharacteristic
337C8—21no—
338C8—21no—
339C8—61no—
340C8—61no—
341C2×C3—41no—
342C2×C3—81no—
343C2×C3#53411yescharacteristic
344C2×C3#57711yescharacteristic
345C2×C3#57811yescharacteristic
346D3—41no—
347D3—41no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—61no—
355C22—61no—
356C22—61no—
357C22—61no—
358C22—61no—
359C22—61no—
360C22C2×D3×D411yescharacteristic
361C4—21no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—61no—
366C4—61no—
367C4—121no—
368C4—121no—
369C4—121no—
370C4—121no—
371C4#100411yescharacteristic
372C4#100411yescharacteristic
373C3#320111yescharacteristic, complemented, Sylow
374C2—41no—
375C2—81no—
376C2—121no—
377C2—121no—
378C2#558911yescharacteristic
379C2#576211yescharacteristic
380C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#448C2
socleC22×C3C22×D4
prime radical#5608C2
hypercenter#448D3
hypocenterC3#3201
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. #5816C2semidirect
3. C3×#448C2semidirect
4. #5228C2semidirect
5. #5174C2semidirect
6. #5173C2semidirect
7. #5173C2semidirect
8. #5171C2semidirect
9. #5035C2semidirect
10. #4816C2semidirect
11. #4813C2semidirect
12. #4811C2semidirect
13. #4805C2semidirect
14. #4727C2semidirect
15. #4723C2semidirect
73. #448D3semidirect
373. C3#3201semidirect

Maximal subgroups

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

subgroupindex
#58162
#56082
C3×#4482
#52282
#51742
#51732
#51712
#50352
#48162
#48132
#48112
#48052
#47272
#47232
#32013