Finite Group Databasebeta

← #80684

Subgroups — #80684

Order 384.

394 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
1#5631C211yesmaximal, characteristic, complemented
2#5614C211yesmaximal, characteristic, complemented
3C3×#465C211yesmaximal, characteristic, complemented
4#5172C211yesmaximal, characteristic, complemented
5#5162C211yesmaximal, characteristic, complemented
6#5113C211yesmaximal, characteristic, complemented
7#5111C211yesmaximal, characteristic, complemented
8#5045C211yesmaximal, characteristic, complemented
9#5037C211yesmaximal, characteristic, complemented
10#4871C211yesmaximal, characteristic, complemented
11#4862C211yesmaximal, characteristic, complemented
12#4803C211yesmaximal, characteristic, complemented
13#4797C211yesmaximal, characteristic, complemented
14#4785C211yesmaximal, characteristic, complemented
15#4780C211yesmaximal, characteristic, complemented
16#3307—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C22×C3×C8C2211yescharacteristic
21C3×C22⋊Q8C2211yescharacteristic
22C3×#121C2211yescharacteristic
23#944C2211yescharacteristic
24#938—21no—
25#938C2211yescharacteristic
26#935—21no—
27D4×Dic3C2211yescharacteristic
28C2×#214C2211yescharacteristic
29C2×#213C2211yescharacteristic
30C4×#67C2211yescharacteristic
31#926C2211yescharacteristic
32C2×ΓL(1,𝔽25)C2211yescharacteristic
33#897C2211yescharacteristic
34#894—21no—
35#893C2211yescharacteristic
36D3×C4⋊C4—21no—
37D3×C4⋊C4C2211yescharacteristic
38#890C2211yescharacteristic
39#884—21no—
40#883—21no—
41#67⋊C4—21no—
42#879—21no—
43C3×#106C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#102C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#830C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#816C2211yescharacteristic
53#816C2211yescharacteristic
54#815C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56#810C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#553—31no—
61#547—31no—
62#491—31no—
63#485—31no—
64#483—31no—
65#478—31no—
66#468—31no—
67#465D311yescharacteristic, complemented
68#461—31no—
69#449—31no—
70#449—31no—
71#442—31no—
72#435—31no—
73#417—31no—
74#416—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4—21no—
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C2×#67—21no—
80C2×#67—21no—
81C2×#67D411yescharacteristic
82C22×Dic3—21no—
83#239—22no—
84#239—41no—
85#235—22no—
86#235—41no—
87C2×D12—21no—
88C2×D12C2311yescharacteristic
89C2×C4×D3—21no—
90C2×C4×D3—21no—
91C2×C4×D3—21no—
92C2×C4×D3—21no—
93C2×C4×D3C2311yescharacteristic
94C2×Dic6C2311yescharacteristic
95C3×SD16—41no—
96C2×C3×C8—41no—
97C2×C3×C8C2311yescharacteristic
98C2×C3×C8C2311yescharacteristic
99C3×C4⋊C4—21no—
100C3×C4⋊C4C2311yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C22⋊C4—21no—
103C3×C22⋊C4—21no—
104#217—21no—
105#217—21no—
106#217—21no—
107#217D411yescharacteristic
108#214—41no—
109#213—22no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4D411yescharacteristic
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3C2311yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4D411yescharacteristic
122C4×Dic3—21no—
123C4×Dic3C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126ΓL(1,𝔽25)—41no—
127C2×#41—31no—
128C2×#41—31no—
129C2×SD16—31no—
130C2×SD16C2×D311yescharacteristic
131C2×D8—31no—
132C2×M16—31no—
133C22×C8C2×D311yescharacteristic
134C4⋊Q8—31no—
135#126—61no—
136#125—31no—
137#123—61no—
138#123—61no—
139#123—61no—
140C22⋊Q8—31no—
141C22⋊Q8C2×D311yescharacteristic
142#121—31no—
143#121—61no—
144#121—61no—
145#121C2×D311yescharacteristic
146C4×D4—31no—
147C4×D4—31no—
148C4×D4—31no—
149C4×D4—61no—
150#117—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—61no—
153#107—31no—
154#106—31no—
155#106C2×D311yescharacteristic
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103C2×D311yescharacteristic
163#102—31no—
164#102—31no—
165#102—31no—
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8—31no—
169#97—31no—
170C23×C3—21no—
171C23×C3C2×D411yescharacteristic
172C22×D3—21no—
173C22×D3C2×D411yescharacteristic
174C3×Q8—21no—
175C3×Q8—21no—
176C3×D4—21no—
177C3×D4—21no—
178C3×D4—81no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2411yescharacteristic
186#67—41no—
187#67—42no—
188#67—42no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—81no—
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196D12—21no—
197D12—42no—
198D12#4112yes—
199C4×D3—21no—
200C4×D3—21no—
201C4×D3—42no—
202C4×D3—42no—
203C4×D3—42no—
204C4×D3—81no—
205Dic6—21no—
206Dic6—21no—
207C3×C8—21no—
208C3×C8—21no—
209C3⋊C8—21no—
210C3⋊C8—21no—
211#41—62no—
212#41—62no—
213#41—121no—
214#41—121no—
215C2×Q8—31no—
216C2×Q8C22×D311yescharacteristic
217C2×D4—21no—
218C2×D4—31no—
219C2×D4—31no—
220C2×D4—61no—
221C2×D4—61no—
222C2×D4—61no—
223C2×D4C22×D311yescharacteristic
224C22×C4—31no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4C22×D311yescharacteristic
231SD16—41no—
232SD16—121no—
233D8—62no—
234M16—121no—
235C2×C8—31no—
236C2×C8—31no—
237C2×C8—41no—
238C2×C8C22×D311yescharacteristic
239C2×C8C22×D311yescharacteristic
240C4⋊C4—21no—
241C4⋊C4—31no—
242C4⋊C4—31no—
243C4⋊C4—31no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4C22×D311yescharacteristic
251C4⋊C4C22×D311yescharacteristic
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—31no—
255C22⋊C4—31no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C42—31no—
262C42—61no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3C22×D411yescharacteristic
270C2×D3—21no—
271C2×D3—22no—
272C2×D3—21no—
273C2×D3—21no—
274C2×D3—41no—
275C2×D3—41no—
276C3×C4—21no—
277C3×C4—41no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4C2×#4111yescharacteristic
281C3×C4H(5,𝔽2)11yescharacteristic
282Dic3—21no—
283Dic3—21no—
284Dic3—21no—
285Dic3—41no—
286Dic3—41no—
287Dic3—41no—
288C23—21no—
289C23—31no—
290C23—61no—
291C23D3×D411yescharacteristic
292Q8—21no—
293Q8—21no—
294Q8—61no—
295Q8—61no—
296D4—21no—
297D4—21no—
298D4—32no—
299D4—61no—
300D4—81no—
301D4—122no—
302D4—121no—
303D4—122no—
304D4—122no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4—122no—
321C2×C4—241no—
322C2×C4—241no—
323C2×C4D3×D411yescharacteristic
324C2×C4C23×D311yescharacteristic
325C8—21no—
326C8—21no—
327C8—61no—
328C8—61no—
329C2×C3—41no—
330C2×C3—81no—
331C2×C3#54611yescharacteristic
332C2×C3#57511yescharacteristic
333C2×C3#57711yescharacteristic
334D3—41no—
335D3—42no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—61no—
343C22—61no—
344C22—61no—
345C22—62no—
346C22—121no—
347C22—121no—
348C22C2×D3×D411yescharacteristic
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—61no—
354C4—61no—
355C4—61no—
356C4—121no—
357C4—121no—
358C4—121no—
359C4#100411yescharacteristic
360C4D3×#4111yescharacteristic
361C3#330711yescharacteristic, complemented, Sylow
362C2—41no—
363C2—81no—
364C2—121no—
365C2—122no—
366C2#561011yescharacteristic
367C2#576211yescharacteristic
368C2#576811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#465C2
socleC22×C3C22×D4
hypercenter#465D3
hypocenterC3#3307
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #5631C2semidirect
2. #5614C2semidirect
3. C3×#465C2semidirect
4. #5172C2semidirect
5. #5162C2semidirect
6. #5113C2semidirect
7. #5111C2semidirect
8. #5045C2semidirect
9. #5037C2semidirect
10. #4871C2semidirect
11. #4862C2semidirect
12. #4803C2semidirect
13. #4797C2semidirect
14. #4785C2semidirect
15. #4780C2semidirect
67. #465D3semidirect
361. C3#3307semidirect

Maximal subgroups

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

subgroupindex
#56312
#56142
C3×#4652
#51722
#51622
#51132
#51112
#50452
#50372
#48712
#48622
#48032
#47972
#47852
#47802
#33073