Finite Group Databasebeta

← #79876

Subgroups — #79876

Order 384.

410 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5557C211yesmaximal, characteristic, complemented
2#5548C211yesmaximal, characteristic, complemented
3C3×#442C211yesmaximal, characteristic, complemented
4#5157C211yesmaximal, characteristic, complemented
5#5154C211yesmaximal, characteristic, complemented
6#5018C211yesmaximal, characteristic, complemented
7#5014C211yesmaximal, characteristic, complemented
8#4864C211yesmaximal, characteristic, complemented
9#4856C211yesmaximal, characteristic, complemented
10#4778C211yesmaximal, characteristic, complemented
11#4775C211yesmaximal, characteristic, complemented
12#4765C211yesmaximal, characteristic, complemented
13#4764C211yesmaximal, characteristic, complemented
14#4710C211yesmaximal, characteristic, complemented
15D3×#97C211yesmaximal, characteristic, complemented
16#3180—31nomaximal, Sylow
17C2×#235—21no—
18C2×C3×D8C2211yescharacteristic
19C3×C4×D4C2211yescharacteristic
20C3×C4×D4C2211yescharacteristic
21#940C2211yescharacteristic
22#938C2211yescharacteristic
23#938C2211yescharacteristic
24#936C2211yescharacteristic
25C2×#214C2211yescharacteristic
26C2×#213C2211yescharacteristic
27C4×#67—21no—
28C4×#67—21no—
29#926—21no—
30C2×#204C2211yescharacteristic
31C2×D3×C8C2211yescharacteristic
32C2×D3×C8C2211yescharacteristic
33#897C2211yescharacteristic
34#895C2211yescharacteristic
35#890C2211yescharacteristic
36#888C2211yescharacteristic
37#885—21no—
38#884—21no—
39#883—21no—
40#879—21no—
41C4×D12C2211yescharacteristic
42C42×D3C2211yescharacteristic
43C4×Dic6C2211yescharacteristic
44C3×#106C2211yescharacteristic
45C3×#102C2211yescharacteristic
46C3×#102C2211yescharacteristic
47C3×#97C2211yescharacteristic
48#832C2211yescharacteristic
49#832C2211yescharacteristic
50#821C2211yescharacteristic
51#818C2211yescharacteristic
52#816C2211yescharacteristic
53#815C2211yescharacteristic
54#815C2211yescharacteristic
55#810C2211yescharacteristic
56#809C2211yescharacteristic
57#808C2211yescharacteristic
58#807C2211yescharacteristic
59#803C2211yescharacteristic
60#532—31no—
61#529—31no—
62#501—31no—
63#496—31no—
64#490—31no—
65#489—31no—
66#466—31no—
67#466—31no—
68#465—31no—
69#465—31no—
70#442—31no—
71#442D311yescharacteristic, complemented
72#440—31no—
73#440—31no—
74C2×#97—31no—
75C2×C3×D4C2311yescharacteristic
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83#235—44no—
84C2×D12—21no—
85C2×D12C2311yescharacteristic
86C2×C4×D3—21no—
87C2×C4×D3D411yescharacteristic
88C2×C4×D3D411yescharacteristic
89C2×C4×D3C2311yescharacteristic
90C2×Dic6—21no—
91C2×Dic6C2311yescharacteristic
92C3×D8—22no—
93C2×C3×C8C2311yescharacteristic
94C2×C3×C8C2311yescharacteristic
95C3×C4⋊C4C2311yescharacteristic
96C3×C4⋊C4C2311yescharacteristic
97C3×C22⋊C4—21no—
98C3×C22⋊C4—21no—
99C3×C42C2311yescharacteristic
100#217—21no—
101#217—21no—
102#217—21no—
103#217—21no—
104#214—22no—
105#213—22no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109C4⋊Dic3—21no—
110C4⋊Dic3—21no—
111C4⋊Dic3C2311yescharacteristic
112C4⋊Dic3C2311yescharacteristic
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116C4×Dic3—21no—
117C4×Dic3D411yescharacteristic
118C4×Dic3D411yescharacteristic
119C4×Dic3C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122#204—22no—
123D3×C8—22no—
124D3×C8—22no—
125C2×#41—61no—
126C2×SD16—31no—
127C2×SD16—31no—
128C2×D8—31no—
129C2×D8C2×D311yescharacteristic
130C22×C8—31no—
131C22×C8—31no—
132C4⋊Q8—31no—
133#127—31no—
134#126—61no—
135#125—31no—
136#124—31no—
137#124—61no—
138#123—61no—
139C22⋊Q8—31no—
140#121—31no—
141#121—31no—
142#121—31no—
143#121—61no—
144C4×Q8—31no—
145C4×D4—31no—
146C4×D4—61no—
147C4×D4—61no—
148C4×D4C2×D311yescharacteristic
149C4×D4C2×D311yescharacteristic
150#117—61no—
151C2×C42—31no—
152#107—31no—
153#107—31no—
154#106—31no—
155#106C2×D311yescharacteristic
156#103—31no—
157#103—31no—
158#102—31no—
159#102—31no—
160#102—31no—
161#102—31no—
162#102C2×D311yescharacteristic
163#102C2×D311yescharacteristic
164#97—31no—
165#97—31no—
166#97—31no—
167#97C2×D311yescharacteristic
168C23×C3—21no—
169C23×C3—21no—
170C22×D3—21no—
171C22×D3C2×D411yescharacteristic
172C3×D4—21no—
173C3×D4—21no—
174C3×D4—21no—
175C3×D4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4C2×D411yescharacteristic
181C2×C3×C4C2×D411yescharacteristic
182C2×C3×C4C2411yescharacteristic
183#67—42no—
184#67—42no—
185#67—42no—
186#67—42no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193D12—21no—
194D12—21no—
195D12—42no—
196C4×D3—22no—
197C4×D3—21no—
198C4×D3—21no—
199C4×D3—41no—
200C4×D3—42no—
201Dic6—21no—
202Dic6—21no—
203Dic6—42no—
204C3×C8—21no—
205C3×C8#4112yescomplemented
206C3⋊C8—21no—
207C3⋊C8#4112yescomplemented
208#41—124no—
209C2×Q8—31no—
210C2×Q8—61no—
211C2×D4—31no—
212C2×D4—61no—
213C2×D4—61no—
214C2×D4—61no—
215C2×D4—61no—
216C2×D4—61no—
217C2×D4C22×D311yescharacteristic
218C2×D4C22×D311yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—61no—
225SD16—62no—
226SD16—62no—
227D8—22no—
228D8—62no—
229C2×C8—31no—
230C2×C8—31no—
231C2×C8—62no—
232C2×C8—62no—
233C2×C8C22×D311yescharacteristic
234C2×C8C22×D311yescharacteristic
235C4⋊C4—31no—
236C4⋊C4—31no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4C22×D311yescharacteristic
243C4⋊C4C22×D311yescharacteristic
244C22⋊C4—21no—
245C22⋊C4—21no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C42—31no—
254C42—31no—
255C42—31no—
256C42—61no—
257C42C22×D311yescharacteristic
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3C22×D411yescharacteristic
265C2×D3—21no—
266C2×D3—21no—
267C2×D3—41no—
268C2×D3—41no—
269C2×D3—41no—
270C2×D3AGL(1,ℤ/8)12yes—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4C2×#4111yescharacteristic
276C3×C4C2×#4111yescharacteristic
277Dic3—21no—
278Dic3—21no—
279Dic3—21no—
280Dic3—41no—
281Dic3—41no—
282Dic3—41no—
283C23—21no—
284C23—21no—
285C23—31no—
286C23—61no—
287Q8—61no—
288Q8—61no—
289Q8—122no—
290D4—21no—
291D4—21no—
292D4—21no—
293D4—21no—
294D4—61no—
295D4—61no—
296D4—122no—
297D4—122no—
298D4—122no—
299D4—122no—
300D4—122no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—62no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—121no—
315C2×C4—122no—
316C2×C4D3×D411yescharacteristic
317C2×C4D3×D411yescharacteristic
318C2×C4C23×D311yescharacteristic
319C8—21no—
320C8—32no—
321C8—61no—
322C8#23512yescomplemented
323C2×C3—81no—
324C2×C3—81no—
325C2×C3#53211yescharacteristic
326C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
327C2×C3#57511yescharacteristic
328D3—22no—
329D3—81no—
330C22—32no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—61no—
338C22—61no—
339C22—121no—
340C22—121no—
341C22—121no—
342C22C2×D3×D411yescharacteristic
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—61no—
348C4—61no—
349C4—61no—
350C4—121no—
351C4—121no—
352C4—121no—
353C4C2×#23511yescharacteristic
354C4D3×#4111yescharacteristic
355C3#318011yescharacteristic, complemented, Sylow
356C2—62no—
357C2—81no—
358C2—81no—
359C2—241no—
360C2#554811yescharacteristic
361C2D3×AGL(1,ℤ/8)11yescharacteristic
362C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#442C2
socleC22×C3C22×D4
hypercenter#442D3
hypocenterC3#3180
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #5557C2semidirect
2. #5548C2semidirect
3. C3×#442C2semidirect
4. #5157C2semidirect
5. #5154C2semidirect
6. #5018C2semidirect
7. #5014C2semidirect
8. #4864C2semidirect
9. #4856C2semidirect
10. #4778C2semidirect
11. #4775C2semidirect
12. #4765C2semidirect
13. #4764C2semidirect
14. #4710C2semidirect
15. D3×#97C2semidirect
71. #442D3semidirect
205. C3×C8#41semidirect
207. C3⋊C8#41semidirect
322. C8#235semidirect
355. C3#3180semidirect

Maximal subgroups

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

subgroupindex
#55572
#55482
C3×#4422
#51572
#51542
#50182
#50142
#48642
#48562
#47782
#47752
#47652
#47642
#47102
D3×#972
#31803