Finite Group Databasebeta

← #80321

Subgroups — #80321

Order 384.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5718C211yesmaximal, characteristic, complemented
2#5539C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5152C211yesmaximal, characteristic, complemented
5#5150C211yesmaximal, characteristic, complemented
6#5150C211yesmaximal, characteristic, complemented
7Dic3×D8C211yesmaximal, characteristic, complemented
8#5071C211yesmaximal, characteristic, complemented
9#5014C211yesmaximal, characteristic, complemented
10#4847C211yesmaximal, characteristic, complemented
11#4828C211yesmaximal, characteristic, complemented
12#4766C211yesmaximal, characteristic, complemented
13#4764C211yesmaximal, characteristic, complemented
14#4761C211yesmaximal, characteristic, complemented
15#4760C211yesmaximal, characteristic, complemented
16#3366—31nomaximal, Sylow
17C2×#237—21no—
18C2×C3×D8C2211yescharacteristic
19C2×C3×D8C2211yescharacteristic
20C3×#127C2211yescharacteristic
21C3×C4×D4C2211yescharacteristic
22#940C2211yescharacteristic
23#940C2211yescharacteristic
24#939—21no—
25#939—21no—
26#938—21no—
27D4×Dic3C2211yescharacteristic
28D4×Dic3C2211yescharacteristic
29D4×Dic3C2211yescharacteristic
30D4×Dic3C2211yescharacteristic
31C2×#213C2211yescharacteristic
32#924—21no—
33C2×Dic3⋊C4—21no—
34#890C2211yescharacteristic
35#888C2211yescharacteristic
36#878—21no—
37#878—21no—
38C4×D12C2211yescharacteristic
39#870C2211yescharacteristic
40C4×Dic6C2211yescharacteristic
41C3×C4⋊C8C2211yescharacteristic
42C3×#102C2211yescharacteristic
43C3×#102C2211yescharacteristic
44#832C2211yescharacteristic
45#832C2211yescharacteristic
46#832C2211yescharacteristic
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#818C2211yescharacteristic
52#817C2211yescharacteristic
53#815C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56C8×Dic3C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic
59#803C2211yescharacteristic
60#554—31no—
61#550—31no—
62#490—31no—
63#490—31no—
64#486—31no—
65#478—31no—
66#478—31no—
67#476—31no—
68#459—31no—
69#459—31no—
70#459D311yescharacteristic, complemented
71#446—31no—
72#442—31no—
73#442—31no—
74C4×D8—31no—
75C2×C3×D4—21no—
76C2×C3×D4—21no—
77C2×C3×D4C2311yescharacteristic
78C2×C3×D4C2311yescharacteristic
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4—21no—
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C22×Dic3—21no—
86C22×Dic3—21no—
87C22×Dic3—21no—
88C22×Dic3—21no—
89#237—44no—
90C2×D12C2311yescharacteristic
91C2×C4×D3—21no—
92C2×Dic6—21no—
93C2×Dic6D411yescharacteristic
94C3×D8—22no—
95C3×D8—22no—
96C2×C3×C8C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C22⋊C4—21no—
100C3×C42C2311yescharacteristic
101#217—21no—
102#217—21no—
103#217—21no—
104#217—21no—
105#213—22no—
106(C2×D3)⋊C4—21no—
107C4⋊Dic3—21no—
108C4⋊Dic3D411yescharacteristic
109C4⋊Dic3C2311yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111C4⋊Dic3C2311yescharacteristic
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4D411yescharacteristic
117Dic3⋊C4D411yescharacteristic
118C4×Dic3C2311yescharacteristic
119C4×Dic3C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122C2×#41—61no—
123C2×D8—31no—
124C2×D8C2×D311yescharacteristic
125C2×D8C2×D311yescharacteristic
126C4⋊Q8—31no—
127#127—31no—
128#127—31no—
129#127C2×D311yescharacteristic
130#125—31no—
131#125—31no—
132C22⋊Q8—61no—
133C22⋊Q8—61no—
134C22⋊Q8—61no—
135#121—61no—
136#121—61no—
137#121—61no—
138C4×Q8—31no—
139C4×D4—31no—
140C4×D4—31no—
141C4×D4—31no—
142C4×D4—31no—
143C4×D4—31no—
144C4×D4C2×D311yescharacteristic
145C2×C4⋊C4—61no—
146#107—31no—
147#106—31no—
148#106—31no—
149C4⋊C8—31no—
150C4⋊C8—31no—
151C4⋊C8C2×D311yescharacteristic
152#102—31no—
153#102—31no—
154#102—31no—
155#102—31no—
156#102—31no—
157#102—31no—
158#102—31no—
159#102—31no—
160#102C2×D311yescharacteristic
161#102C2×D311yescharacteristic
162#97—31no—
163#97—31no—
164C4×C8—31no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C22×D3—21no—
169C3×D4—21no—
170C3×D4—21no—
171C3×D4—21no—
172C3×D4—21no—
173C3×D4—21no—
174C3×D4—42no—
175C3×D4—81no—
176C3×D4#4112yescomplemented
177C2×C3×C4—21no—
178C2×C3×C4—81no—
179C2×C3×C4C2×D411yescharacteristic
180C2×C3×C4C2×D411yescharacteristic
181C2×C3×C4C2411yescharacteristic
182#67—42no—
183#67—42no—
184#67—81no—
185#67—81no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—42no—
190C2×Dic3—42no—
191C2×Dic3—42no—
192C2×Dic3—81no—
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197D12—21no—
198D12—21no—
199C4×D3—42no—
200Dic6—22no—
201Dic6—81no—
202C3×C8—21no—
203C3×C8—21no—
204C3⋊C8—21no—
205C3⋊C8—21no—
206#41—124no—
207C2×Q8—31no—
208C2×Q8—61no—
209C2×D4—21no—
210C2×D4—21no—
211C2×D4—31no—
212C2×D4—61no—
213C2×D4—61no—
214C2×D4—61no—
215C2×D4—61no—
216C2×D4C22×D311yescharacteristic
217C2×D4C22×D311yescharacteristic
218C2×D4C22×D311yescharacteristic
219C22×C4—21no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—61no—
225D8—22no—
226D8—22no—
227D8—62no—
228C2×C8—31no—
229C2×C8—31no—
230C2×C8C22×D311yescharacteristic
231C2×C8C22×D311yescharacteristic
232C4⋊C4—31no—
233C4⋊C4—31no—
234C4⋊C4—31no—
235C4⋊C4—31no—
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4C22×D311yescharacteristic
244C22⋊C4—21no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C42—31no—
251C42—31no—
252C42C22×D311yescharacteristic
253C22×C3—22no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3C22×D411yescharacteristic
263C2×D3—41no—
264C2×D3—41no—
265C2×D3—41no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—41no—
269C3×C4C2×#4111yescharacteristic
270C3×C42−1+411yescharacteristic
271Dic3—21no—
272Dic3—21no—
273Dic3—21no—
274Dic3—41no—
275Dic3—41no—
276Dic3—41no—
277Dic3AGL(1,ℤ/8)12yes—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—61no—
282Q8—62no—
283Q8—241no—
284D4—21no—
285D4—21no—
286D4—21no—
287D4—21no—
288D4—21no—
289D4—42no—
290D4—61no—
291D4—61no—
292D4—81no—
293D4—122no—
294D4—122no—
295D4—241no—
296D4—241no—
297D4#23712yescomplemented
298C2×C4—21no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—61no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—81no—
307C2×C4—122no—
308C2×C4—122no—
309C2×C4—122no—
310C2×C4—122no—
311C2×C4—241no—
312C2×C4D3×D411yescharacteristic
313C2×C4D3×D411yescharacteristic
314C2×C4C23×D311yescharacteristic
315C8—21no—
316C8—21no—
317C8—61no—
318C8—61no—
319C2×C3—42no—
320C2×C3—81no—
321C2×C3—81no—
322C2×C3#54811yescharacteristic
323C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
324C2×C3#57611yescharacteristic
325D3—81no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—121no—
336C22—121no—
337C22—121no—
338C22C2×D3×D411yescharacteristic
339C4—21no—
340C4—21no—
341C4—32no—
342C4—41no—
343C4—61no—
344C4—61no—
345C4—61no—
346C4—121no—
347C4—121no—
348C4—121no—
349C4C2×#23711yescharacteristic
350C4#101011yescharacteristic
351C3#336611yescharacteristic, complemented, Sylow
352C2—42no—
353C2—81no—
354C2—81no—
355C2—241no—
356C2#559811yescharacteristic
357C2#575711yescharacteristic
358C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3366
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #5718C2semidirect
2. #5539C2semidirect
3. C3×#459C2semidirect
4. #5152C2semidirect
5. #5150C2semidirect
6. #5150C2semidirect
7. Dic3×D8C2semidirect
8. #5071C2semidirect
9. #5014C2semidirect
10. #4847C2semidirect
11. #4828C2semidirect
12. #4766C2semidirect
13. #4764C2semidirect
14. #4761C2semidirect
15. #4760C2semidirect
70. #459D3semidirect
176. C3×D4#41semidirect
297. D4#237semidirect
351. C3#3366semidirect

Maximal subgroups

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

subgroupindex
#57182
#55392
C3×#4592
#51522
#51502
Dic3×D82
#50712
#50142
#48472
#48282
#47662
#47642
#47612
#47602
#33663