Finite Group Databasebeta

← #83539

Subgroups — #83539

Order 384.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#548C211yesmaximal, characteristic, complemented
2#5563C211yesmaximal, characteristic, complemented
3#5238C211yesmaximal, characteristic, complemented
4#5233C211yesmaximal, characteristic, complemented
5#5227C211yesmaximal, characteristic, complemented
6#5225C211yesmaximal, characteristic, complemented
7#5057C211yesmaximal, characteristic, complemented
8#5055C211yesmaximal, characteristic, complemented
9#5048C211yesmaximal, characteristic, complemented
10#5045C211yesmaximal, characteristic, complemented
11#5039C211yesmaximal, characteristic, complemented
12#5035C211yesmaximal, characteristic, complemented
13#5026C211yesmaximal, characteristic, complemented
14#5020C211yesmaximal, characteristic, complemented
15#5011C211yesmaximal, characteristic, complemented
16#3321—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic
19C3×#124—21no—
20C3×#124C2211yescharacteristic
21C3×C22⋊Q8—21no—
22C3×C22⋊Q8C2211yescharacteristic
23C3×#121—21no—
24C3×#121C2211yescharacteristic
25C3×C4×Q8C2211yescharacteristic
26C3×C4×D4—21no—
27C3×C4×D4C2211yescharacteristic
28C2×#215C2211yescharacteristic
29C2×#215C2211yescharacteristic
30#938—21no—
31C2×#213C2211yescharacteristic
32#930C2211yescharacteristic
33#930C2211yescharacteristic
34#926C2211yescharacteristic
35#926C2211yescharacteristic
36C2×#208C2211yescharacteristic
37C22×C3⋊C8C2211yescharacteristic
38#892—21no—
39#885—21no—
40#885—21no—
41C4×D12C2211yescharacteristic
42#869C2211yescharacteristic
43#835C2211yescharacteristic
44#835C2211yescharacteristic
45#835C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51#832C2211yescharacteristic
52#830C2211yescharacteristic
53#830C2211yescharacteristic
54#809C2211yescharacteristic
55#808C2211yescharacteristic
56#808C2211yescharacteristic
57#807C2211yescharacteristic
58C4⋊(C3⋊C8)C2211yescharacteristic
59#803C2211yescharacteristic
60#555—31no—
61#548D311yescharacteristic, complemented
62#489—31no—
63#485—31no—
64#482—31no—
65#475—31no—
66#469—31no—
67#465—31no—
68#463—31no—
69#449—31no—
70#448—31no—
71#440—31no—
72#435—31no—
73#419—31no—
74#416—31no—
75C3×#41—22no—
76C3×#41—41no—
77C2×C3×Q8—21no—
78C2×C3×Q8—22no—
79C2×C3×Q8—41no—
80C2×C3×Q8C2311yescharacteristic
81C2×C3×Q8C2311yescharacteristic
82C2×C3×D4—21no—
83C2×C3×D4—21no—
84C2×C3×D4D411yescharacteristic
85C2×C3×D4C2311yescharacteristic
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C2×#67—21no—
91C2×#67—21no—
92C2×D12C2311yescharacteristic
93C2×C4×D3—21no—
94C2×Dic6—21no—
95C3×C4⋊C4—21no—
96C3×C4⋊C4—21no—
97C3×C4⋊C4D411yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C22⋊C4—21no—
100C3×C22⋊C4—21no—
101C3×C22⋊C4—21no—
102C3×C22⋊C4—21no—
103C3×C22⋊C4D411yescharacteristic
104C3×C22⋊C4D411yescharacteristic
105C3×C42—21no—
106C3×C42C2311yescharacteristic
107#217—21no—
108#217—21no—
109#215—22no—
110#215—41no—
111#213—41no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115C4⋊Dic3—21no—
116C4⋊Dic3C2311yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119C4×Dic3—21no—
120C4×Dic3—21no—
121#208—41no—
122C2×C3⋊C8—41no—
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×#41C2×D311yescharacteristic
128C22×Q8C2×D311yescharacteristic
129C2×SD16—31no—
130C2×SD16—31no—
131C2×D8—31no—
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8—31no—
135#124—21no—
136#124—61no—
137#124—61no—
138#124C2×D311yescharacteristic
139C22⋊Q8—21no—
140C22⋊Q8C2×D311yescharacteristic
141#121—21no—
142#121—31no—
143#121—31no—
144#121—61no—
145#121C2×D311yescharacteristic
146C4×Q8C2×D311yescharacteristic
147C4×D4—21no—
148C4×D4—31no—
149C4×D4C2×D311yescharacteristic
150#117—31no—
151#117—31no—
152#117—61no—
153#107—31no—
154#106—31no—
155#106—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8—31no—
169#97—31no—
170C23×C3—21no—
171C23×C3C2×D411yescharacteristic
172C23×C3C2×D411yescharacteristic
173C22×D3—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×Q8—42no—
178C3×Q8#4112yes—
179C3×D4—21no—
180C3×D4—21no—
181C3×D4—41no—
182C3×D4—42no—
183C3×D4—42no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4C2×D411yescharacteristic
194C2×C3×C4C2×D411yescharacteristic
195C2×C3×C4C2×D411yescharacteristic
196C2×C3×C4C2×D411yescharacteristic
197C2×C3×C4C2411yescharacteristic
198#67—81no—
199#67—81no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203D12—21no—
204D12—21no—
205C4×D3—42no—
206Dic6—81no—
207C3⋊C8—21no—
208C3⋊C8—21no—
209C3⋊C8—21no—
210C3⋊C8—21no—
211#41—22no—
212#41—41no—
213C2×Q8—21no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×Q8—61no—
217C2×Q8C22×D311yescharacteristic
218C2×Q8C22×D311yescharacteristic
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—31no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4#6711yescharacteristic
225C2×D4C22×D311yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—61no—
229C22×C4C22×D311yescharacteristic
230C22×C4C22×D311yescharacteristic
231SD16—62no—
232SD16—121no—
233D8—121no—
234M16—121no—
235C2×C8—31no—
236C2×C8—31no—
237C2×C8—31no—
238C2×C8—31no—
239C2×C8—121no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—31no—
243C4⋊C4—31no—
244C4⋊C4—31no—
245C4⋊C4—61no—
246C4⋊C4#6711yescharacteristic
247C4⋊C4C22×D311yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—21no—
250C22⋊C4—21no—
251C22⋊C4—21no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4#6711yescharacteristic
258C22⋊C4#6711yescharacteristic
259C42—21no—
260C42—61no—
261C42—61no—
262C42C22×D311yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3C22×D411yescharacteristic
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—41no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—22no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—41no—
282C3×C4—41no—
283C3×C4C2×#4111yescharacteristic
284C3×C4H(5,𝔽2)11yescharacteristic
285Dic3—41no—
286Dic3—41no—
287Dic3—41no—
288C23—21no—
289C23—61no—
290C23C2×#6711yescharacteristic
291C23C2×#6711yescharacteristic
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—241no—
297Q8#23712yes—
298D4—21no—
299D4—21no—
300D4—41no—
301D4—42no—
302D4—42no—
303D4—61no—
304D4—61no—
305D4—241no—
306D4—241no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—122no—
320C2×C4C2×#6711yescharacteristic
321C2×C4C2×#6711yescharacteristic
322C2×C4C2×#6711yescharacteristic
323C2×C4C2×#6711yescharacteristic
324C2×C4C23×D311yescharacteristic
325C8—61no—
326C8—61no—
327C8—61no—
328C8—61no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—81no—
332C2×C3#54611yescharacteristic
333C2×C3#57511yescharacteristic
334C2×C3#57711yescharacteristic
335D3—81no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—121no—
346C22—121no—
347C22—121no—
348C22C22×#6711yescharacteristic
349C4—21no—
350C4—21no—
351C4—22no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—121no—
357C4—121no—
358C4—121no—
359C4C2×#23711yescharacteristic
360C4#100411yescharacteristic
361C3#332111yescharacteristic, complemented, Sylow
362C2—41no—
363C2—41no—
364C2—81no—
365C2—241no—
366C2#580611yescharacteristic
367C2#582211yescharacteristic
368C2#583711yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#548C2
socleC22×C3C22×D4
hypercenter#548D3
hypocenterC3#3321
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C3×#548C2semidirect
2. #5563C2semidirect
3. #5238C2semidirect
4. #5233C2semidirect
5. #5227C2semidirect
6. #5225C2semidirect
7. #5057C2semidirect
8. #5055C2semidirect
9. #5048C2semidirect
10. #5045C2semidirect
11. #5039C2semidirect
12. #5035C2semidirect
13. #5026C2semidirect
14. #5020C2semidirect
15. #5011C2semidirect
61. #548D3semidirect
361. C3#3321semidirect

Maximal subgroups

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

subgroupindex
C3×#5482
#55632
#52382
#52332
#52272
#52252
#50572
#50552
#50482
#50452
#50392
#50352
#50262
#50202
#50112
#33213