Finite Group Databasebeta

← #80917

Subgroups — #80917

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5635C211yesmaximal, characteristic, complemented
2#5633C211yesmaximal, characteristic
3C3×#470C211yesmaximal, characteristic, complemented
4#5186C211yesmaximal, characteristic, complemented
5#5185C211yesmaximal, characteristic, complemented
6#5133C211yesmaximal, characteristic, complemented
7#5127C211yesmaximal, characteristic, complemented
8#5047C211yesmaximal, characteristic, complemented
9#5043C211yesmaximal, characteristic, complemented
10#4863C211yesmaximal, characteristic, complemented
11#4854C211yesmaximal, characteristic, complemented
12#4807C211yesmaximal, characteristic, complemented
13D3×#103C211yesmaximal, characteristic, complemented
14#4794C211yesmaximal, characteristic, complemented
15#4789C211yesmaximal, characteristic, complemented
16#3316—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C2×C3×M16C2211yescharacteristic
21C3×C22⋊Q8C2211yescharacteristic
22C3×C22⋊Q8C2211yescharacteristic
23#947C2211yescharacteristic
24#946C2211yescharacteristic
25Q8×Dic3C2211yescharacteristic
26C2×#216C2211yescharacteristic
27C2×#215C2211yescharacteristic
28C4×#67C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×D3×C8C2211yescharacteristic
31#897—21no—
32#896—21no—
33#895C2211yescharacteristic
34#893—21no—
35D3×C4⋊C4C2211yescharacteristic
36#889—21no—
37#888C2211yescharacteristic
38#887C2211yescharacteristic
39#885—21no—
40#884—21no—
41#67⋊C4—21no—
42#878—21no—
43C3×#106C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#103C2211yescharacteristic
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#808C2211yescharacteristic
60#554—31no—
61#548—31no—
62#489—31no—
63#483—31no—
64#481—31no—
65#475—31no—
66#470D311yescharacteristic, complemented
67#465—31no—
68#464—31no—
69#452—31no—
70#451—31no—
71#441—31no—
72#435—31no—
73#417—31no—
74C2×#103—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×Q8C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C2×#67—21no—
79C2×#67D411yescharacteristic
80C22×Dic3—21no—
81D3×Q8—22no—
82D3×Q8—41no—
83#235—22no—
84#235—41no—
85C2×D12—21no—
86C2×D12C2311yescharacteristic
87C2×C4×D3—21no—
88C2×C4×D3—21no—
89C2×C4×D3—21no—
90C2×C4×D3C2311yescharacteristic
91C2×Dic6—21no—
92C2×Dic6—21no—
93C2×Dic6C2311yescharacteristic
94C3×Q16—41no—
95C3×M16—41no—
96C2×C3×C8C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C3×C4⋊C4—21no—
99C3×C4⋊C4—21no—
100C3×C4⋊C4C2311yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C22⋊C4—21no—
103C3×C22⋊C4—21no—
104#217D411yescharacteristic
105#216—22no—
106#215—41no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4D411yescharacteristic
112C4⋊Dic3—21no—
113C4⋊Dic3—21no—
114C4⋊Dic3C2311yescharacteristic
115C4⋊Dic3C2311yescharacteristic
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4D411yescharacteristic
121C4×Dic3—21no—
122C4×Dic3—21no—
123C4×Dic3C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126D3×C8—41no—
127C2×#41—31no—
128C22×Q8—31no—
129C2×Q16—31no—
130C2×Q16C2×D311yescharacteristic
131C2×SD16—31no—
132C2×M16C2×D311yescharacteristic
133C22×C8—31no—
134C4⋊Q8—31no—
135#125—61no—
136#124—31no—
137#124—61no—
138C22⋊Q8—31no—
139C22⋊Q8—61no—
140C22⋊Q8—61no—
141C22⋊Q8—61no—
142C22⋊Q8C2×D311yescharacteristic
143C22⋊Q8C2×D311yescharacteristic
144#121—31no—
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#106—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#103C2×D311yescharacteristic
164#102—31no—
165#102—31no—
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8—31no—
169#97—31no—
170C23×C3C2×D411yescharacteristic
171C22×D3—21no—
172C22×D3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2411yescharacteristic
185#67—22no—
186#67—42no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—81no—
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193C2×Dic3C2×D411yescharacteristic
194D12—21no—
195D12—21no—
196D12—42no—
197C4×D3—21no—
198C4×D3—21no—
199C4×D3—42no—
200C4×D3—42no—
201C4×D3—81no—
202Dic6—21no—
203Dic6—42no—
204Dic6—81no—
205Dic6#4112yes—
206C3×C8—21no—
207C3×C8—21no—
208C3⋊C8—21no—
209C3⋊C8—21no—
210#41—62no—
211#41—121no—
212C2×Q8—31no—
213C2×Q8—61no—
214C2×Q8—61no—
215C2×Q8—62no—
216C2×Q8—121no—
217C2×Q8C22×D311yescharacteristic
218C2×Q8C22×D311yescharacteristic
219C2×D4—31no—
220C2×D4—31no—
221C2×D4—61no—
222C2×D4—61no—
223C22×C4—31no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4C22×D311yescharacteristic
229Q16—41no—
230Q16—62no—
231SD16—121no—
232M16—41no—
233C2×C8—31no—
234C2×C8—31no—
235C2×C8—121no—
236C2×C8C22×D311yescharacteristic
237C2×C8C22×D311yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—31no—
241C4⋊C4—31no—
242C4⋊C4—31no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4C22×D311yescharacteristic
250C4⋊C4C22×D311yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—31no—
254C22⋊C4—31no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C42—31no—
260C42—61no—
261C42—61no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3C22×D411yescharacteristic
266C2×D3—21no—
267C2×D3—21no—
268C2×D3—41no—
269C2×D3—41no—
270C2×D3—41no—
271C2×D3#13712yes—
272C3×C4—21no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4C2×#4111yescharacteristic
278C3×C4H(5,𝔽2)11yescharacteristic
279Dic3—22no—
280Dic3—21no—
281Dic3—21no—
282Dic3—21no—
283Dic3—41no—
284Dic3—41no—
285C23—31no—
286C23—61no—
287C23D3×D411yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—21no—
292Q8—32no—
293Q8—61no—
294Q8—122no—
295Q8—241no—
296D4—62no—
297D4—61no—
298D4—61no—
299D4—122no—
300D4—122no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—122no—
316C2×C4—122no—
317C2×C4—241no—
318C2×C4—241no—
319C2×C4D3×D411yescharacteristic
320C2×C4C23×D311yescharacteristic
321C8—21no—
322C8—21no—
323C8—61no—
324C8—61no—
325C2×C3—41no—
326C2×C3#54611yescharacteristic
327C2×C3C2×#13711yescharacteristic
328C2×C3#57711yescharacteristic
329D3—22no—
330D3—81no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—32no—
335C22—61no—
336C22—61no—
337C22—121no—
338C22—121no—
339C22—121no—
340C22C2×D3×D411yescharacteristic
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—61no—
347C4—61no—
348C4—61no—
349C4—62no—
350C4—121no—
351C4—121no—
352C4#100411yescharacteristic
353C4D3×#4111yescharacteristic
354C3#331611yescharacteristic, complemented, Sylow
355C2—41no—
356C2—62no—
357C2—241no—
358C2#561011yescharacteristic
359C2D3×#13711yescharacteristic
360C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#470C2
socleC22×C3C22×D4
prime radical#5633C2
hypercenter#470D3
hypocenterC3#3316
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #5635C2semidirect
3. C3×#470C2semidirect
4. #5186C2semidirect
5. #5185C2semidirect
6. #5133C2semidirect
7. #5127C2semidirect
8. #5047C2semidirect
9. #5043C2semidirect
10. #4863C2semidirect
11. #4854C2semidirect
12. #4807C2semidirect
13. D3×#103C2semidirect
14. #4794C2semidirect
15. #4789C2semidirect
66. #470D3semidirect
354. C3#3316semidirect

Maximal subgroups

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

subgroupindex
#56352
#56332
C3×#4702
#51862
#51852
#51332
#51272
#50472
#50432
#48632
#48542
#48072
D3×#1032
#47942
#47892
#33163