Finite Group Databasebeta

← #80595

Subgroups — #80595

Order 384.

380 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#5681C211yesmaximal, characteristic, complemented
2#5585C211yesmaximal, characteristic, complemented
3C3×#464C211yesmaximal, characteristic, complemented
4#5191C211yesmaximal, characteristic, complemented
5#5185C211yesmaximal, characteristic, complemented
6#5176C211yesmaximal, characteristic, complemented
7#5165C211yesmaximal, characteristic, complemented
8#5058C211yesmaximal, characteristic, complemented
9#5026C211yesmaximal, characteristic, complemented
10#4843C211yesmaximal, characteristic, complemented
11#4828C211yesmaximal, characteristic, complemented
12#4819C211yesmaximal, characteristic, complemented
13#4817C211yesmaximal, characteristic, complemented
14#4786C211yesmaximal, characteristic, complemented
15#4756C211yesmaximal, characteristic, complemented
16#3403—31nomaximal, Sylow
17C2×C3×Q16C2211yescharacteristic
18C2×C3×SD16C2211yescharacteristic
19C3×#124C2211yescharacteristic
20C3×C4×Q8C2211yescharacteristic
21#947C2211yescharacteristic
22#947C2211yescharacteristic
23#947C2211yescharacteristic
24Q8×Dic3C2211yescharacteristic
25C2×#215C2211yescharacteristic
26C2×#215C2211yescharacteristic
27C2×#215C2211yescharacteristic
28#940C2211yescharacteristic
29D4×Dic3C2211yescharacteristic
30C2×#213C2211yescharacteristic
31C2×#213C2211yescharacteristic
32C2×D24C2211yescharacteristic
33C2×D24C2211yescharacteristic
34#894—21no—
35#894—21no—
36#893C2211yescharacteristic
37#893C2211yescharacteristic
38#886—21no—
39#885—21no—
40#883—21no—
41#67⋊C4—21no—
42#875—21no—
43#874C2211yescharacteristic
44C4×D12—21no—
45C4×D12C2211yescharacteristic
46C4×Dic6C2211yescharacteristic
47C3×C4⋊C8C2211yescharacteristic
48C3×#103C2211yescharacteristic
49C3×#102C2211yescharacteristic
50#835C2211yescharacteristic
51#821C2211yescharacteristic
52#821C2211yescharacteristic
53#815C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56C8×Dic3C2211yescharacteristic
57#809C2211yescharacteristic
58#807C2211yescharacteristic
59#803C2211yescharacteristic
60#559—31no—
61#559—31no—
62#496—31no—
63#496—31no—
64#495—31no—
65#464—31no—
66#464D311yescharacteristic, complemented
67#463—31no—
68#463—31no—
69#459—31no—
70#459—31no—
71#446—31no—
72#440—31no—
73#440—31no—
74C4×D8—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C2×#67—21no—
79C2×#67—21no—
80C22×Dic3—21no—
81C2×D12—21no—
82C2×D12—21no—
83C2×D12C2311yescharacteristic
84C2×D12C2311yescharacteristic
85C2×D12C2311yescharacteristic
86C2×C4×D3—21no—
87C2×C4×D3—21no—
88C2×C4×D3—21no—
89C2×Dic6D411yescharacteristic
90C3×Q16—41no—
91C3×SD16—41no—
92C2×C3×C8C2311yescharacteristic
93C2×C3×C8C2311yescharacteristic
94C3×C4⋊C4—21no—
95C3×C4⋊C4C2311yescharacteristic
96C3×C22⋊C4—21no—
97C3×C22⋊C4—21no—
98C3×C42—21no—
99C3×C42C2311yescharacteristic
100#217—21no—
101#215—22no—
102#215—22no—
103#215—41no—
104#213—22no—
105#213—41no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
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)⋊C4—21no—
115C4⋊Dic3—21no—
116C4⋊Dic3D411yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118Dic3⋊C4D411yescharacteristic
119Dic3⋊C4D411yescharacteristic
120C4×Dic3—21no—
121C4×Dic3C2311yescharacteristic
122C4×Dic3C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125D24—41no—
126D24—41no—
127C2×Q16C2×D311yescharacteristic
128C2×SD16—31no—
129C2×SD16—31no—
130C2×SD16—31no—
131C2×SD16C2×D311yescharacteristic
132C2×D8—31no—
133C2×D8—31no—
134C2×D8—31no—
135C2×D8—31no—
136#127—31no—
137#127—31no—
138#124—31no—
139#124—31no—
140#124—31no—
141#124—61no—
142#124—61no—
143#124C2×D311yescharacteristic
144#123—61no—
145#123—61no—
146#123—61no—
147#123—61no—
148C4×Q8—31no—
149C4×Q8—31no—
150C4×Q8C2×D311yescharacteristic
151C4×D4—31no—
152C4×D4—31no—
153C4×D4—31no—
154C4×D4—31no—
155C4×D4—61no—
156C4×D4—61no—
157#107—31no—
158C4⋊C8—31no—
159C4⋊C8—31no—
160C4⋊C8C2×D311yescharacteristic
161#103—31no—
162#103C2×D311yescharacteristic
163#102—31no—
164#102—31no—
165#102—31no—
166#102C2×D311yescharacteristic
167#97—31no—
168#97—31no—
169C4×C8—31no—
170C23×C3—21no—
171C22×D3—21no—
172C22×D3—21no—
173C22×D3—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×Q8—21no—
178C3×D4—21no—
179C3×D4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2411yescharacteristic
186#67—42no—
187#67—81no—
188C2×Dic3—21no—
189C2×Dic3—81no—
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193C2×Dic3C2×D411yescharacteristic
194D12—21no—
195D12—21no—
196D12—21no—
197D12—21no—
198D12—21no—
199D12—21no—
200D12—42no—
201D12—81no—
202C4×D3—81no—
203C4×D3—81no—
204C4×D3—81no—
205Dic6—41no—
206C3×C8—21no—
207C3×C8—21no—
208C3⋊C8—21no—
209C3⋊C8#4112yes—
210C2×Q8—31no—
211C2×Q8C22×D311yescharacteristic
212C2×Q8C22×D311yescharacteristic
213C2×D4—31no—
214C2×D4—31no—
215C2×D4—31no—
216C2×D4—61no—
217C2×D4—61no—
218C2×D4—61no—
219C2×D4—61no—
220C2×D4C22×D311yescharacteristic
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—61no—
225Q16—41no—
226SD16—41no—
227SD16—62no—
228SD16—62no—
229SD16—121no—
230D8—62no—
231D8—121no—
232D8—121no—
233D8—121no—
234C2×C8—31no—
235C2×C8—31no—
236C2×C8C22×D311yescharacteristic
237C2×C8C22×D311yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—31no—
240C4⋊C4—31no—
241C4⋊C4—31no—
242C4⋊C4—31no—
243C4⋊C4—61no—
244C4⋊C4C22×D311yescharacteristic
245C22⋊C4—21no—
246C22⋊C4—21no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C42—21no—
258C42—31no—
259C42—31no—
260C42—61no—
261C42C22×D311yescharacteristic
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3C22×D411yescharacteristic
266C2×D3—41no—
267C2×D3—41no—
268C2×D3—41no—
269C2×D3—41no—
270C2×D3—41no—
271C2×D3—41no—
272C2×D3—41no—
273C2×D3—41no—
274C2×D3—41no—
275C3×C4—21no—
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—21no—
286Dic3—41no—
287C23—21no—
288C23—61no—
289C23—61no—
290C23—61no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—121no—
296D4—21no—
297D4—21no—
298D4—61no—
299D4—61no—
300D4—61no—
301D4—61no—
302D4—61no—
303D4—61no—
304D4—122no—
305D4—122no—
306D4—241no—
307D4—241no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—61no—
316C2×C4—241no—
317C2×C4—241no—
318C2×C4—241no—
319C2×C4—241no—
320C2×C4D3×D411yescharacteristic
321C2×C4D3×D411yescharacteristic
322C2×C4C23×D311yescharacteristic
323C8—21no—
324C8—21no—
325C8—32no—
326C8—61no—
327C2×C3—81no—
328C2×C3#55011yescharacteristic
329C2×C3#57511yescharacteristic
330C2×C3#57611yescharacteristic
331D3—81no—
332D3—81no—
333D3—81no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—121no—
338C22—121no—
339C22—121no—
340C22—121no—
341C22—121no—
342C22—121no—
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—61no—
356C4—121no—
357C4#100411yescharacteristic
358C4D3×#4111yescharacteristic
359C3#340311yescharacteristic, complemented, Sylow
360C2—81no—
361C2—241no—
362C2—241no—
363C2—241no—
364C2#559911yescharacteristic
365C2#576911yescharacteristic
366C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#464C2
socleC22×C3C22×D4
hypercenter#464D3
hypocenterC3#3403
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #5681C2semidirect
2. #5585C2semidirect
3. C3×#464C2semidirect
4. #5191C2semidirect
5. #5185C2semidirect
6. #5176C2semidirect
7. #5165C2semidirect
8. #5058C2semidirect
9. #5026C2semidirect
10. #4843C2semidirect
11. #4828C2semidirect
12. #4819C2semidirect
13. #4817C2semidirect
14. #4786C2semidirect
15. #4756C2semidirect
66. #464D3semidirect
359. C3#3403semidirect

Maximal subgroups

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

subgroupindex
#56812
#55852
C3×#4642
#51912
#51852
#51762
#51652
#50582
#50262
#48432
#48282
#48192
#48172
#47862
#47562
#34033