Finite Group Databasebeta

← #80322

Subgroups — #80322

Order 384.

358 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5713C211yesmaximal, characteristic, complemented
2#5539C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic
4#5153C211yesmaximal, characteristic, complemented
5#5153C211yesmaximal, characteristic, complemented
6#5152C211yesmaximal, characteristic, complemented
7Dic3×D8C211yesmaximal, characteristic, complemented
8#5071C211yesmaximal, characteristic, complemented
9#5018C211yesmaximal, characteristic, complemented
10#4829C211yesmaximal, characteristic, complemented
11#4828C211yesmaximal, characteristic, complemented
12#4768C211yesmaximal, characteristic, complemented
13#4765C211yesmaximal, characteristic, complemented
14#4763C211yesmaximal, characteristic, complemented
15#4759C211yesmaximal, characteristic, complemented
16#3370—31nomaximal, Sylow
17C2×C3×D8C2211yescharacteristic
18C2×C3×D8C2211yescharacteristic
19C3×#127C2211yescharacteristic
20C3×C4×D4C2211yescharacteristic
21#936—21no—
22#936C2211yescharacteristic
23#936C2211yescharacteristic
24#935—21no—
25#935—21no—
26D4×Dic3—21no—
27D4×Dic3C2211yescharacteristic
28D4×Dic3C2211yescharacteristic
29D4×Dic3C2211yescharacteristic
30D4×Dic3C2211yescharacteristic
31C2×#214C2211yescharacteristic
32#924—21no—
33C2×Dic3⋊C4—21no—
34#890C2211yescharacteristic
35#888C2211yescharacteristic
36#878—21no—
37#878—21no—
38#870C2211yescharacteristic
39C4×Dic6C2211yescharacteristic
40C4×Dic6C2211yescharacteristic
41C3×C4⋊C8C2211yescharacteristic
42C3×#102C2211yescharacteristic
43C3×#102C2211yescharacteristic
44#832C2211yescharacteristic
45#832C2211yescharacteristic
46#832C2211yescharacteristic
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#818C2211yescharacteristic
50#817C2211yescharacteristic
51#816C2211yescharacteristic
52#816C2211yescharacteristic
53#815C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56C8×Dic3C2211yescharacteristic
57#810C2211yescharacteristic
58#807C2211yescharacteristic
59#803C2211yescharacteristic
60#559—31no—
61#554—31no—
62#490—31no—
63#489—31no—
64#488—31no—
65#479—31no—
66#478—31no—
67#474—31no—
68#463—31no—
69#463—31no—
70#459D311yescharacteristic
71#446—31no—
72#442—31no—
73#440—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—
81C22×Dic3—21no—
82C22×Dic3—21no—
83C22×Dic3—21no—
84C22×Dic3—21no—
85C2×Dic6—21no—
86C2×Dic6D411yescharacteristic
87C2×Dic6C2311yescharacteristic
88C3×D8—22no—
89C3×D8—22no—
90C2×C3×C8C2311yescharacteristic
91C2×C3×C8C2311yescharacteristic
92C3×C4⋊C4C2311yescharacteristic
93C3×C22⋊C4—21no—
94C3×C42C2311yescharacteristic
95#217—21no—
96#217—21no—
97#217—21no—
98#217—21no—
99#217—21no—
100#217—21no—
101#217—21no—
102#217—21no—
103#214—22no—
104C4⋊Dic3—21no—
105C4⋊Dic3D411yescharacteristic
106C4⋊Dic3C2311yescharacteristic
107C4⋊Dic3C2311yescharacteristic
108C4⋊Dic3C2311yescharacteristic
109Dic3⋊C4—21no—
110Dic3⋊C4—21no—
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4D411yescharacteristic
115Dic3⋊C4D411yescharacteristic
116C4×Dic3—21no—
117C4×Dic3C2311yescharacteristic
118C4×Dic3C2311yescharacteristic
119C2×C3⋊C8C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121C2×SD16—31no—
122C2×D8C2×D311yescharacteristic
123C2×D8C2×D311yescharacteristic
124C4⋊Q8—31no—
125#127C2×D311yescharacteristic
126#125—31no—
127#125—31no—
128#124—31no—
129#124—31no—
130#124—61no—
131#123—61no—
132#123—61no—
133C22⋊Q8—61no—
134C22⋊Q8—61no—
135C22⋊Q8—61no—
136C4×Q8—31no—
137C4×Q8—31no—
138C4×D4—31no—
139C4×D4—31no—
140C4×D4—31no—
141C4×D4—31no—
142C4×D4—61no—
143C4×D4C2×D311yescharacteristic
144C2×C4⋊C4—61no—
145#107—31no—
146#107—31no—
147#106—31no—
148C4⋊C8—31no—
149C4⋊C8—31no—
150C4⋊C8C2×D311yescharacteristic
151#103—31no—
152#103—31no—
153#103—31no—
154#102—31no—
155#102—31no—
156#102—31no—
157#102—31no—
158#102—31no—
159#102C2×D311yescharacteristic
160#102C2×D311yescharacteristic
161#97—31no—
162#97—31no—
163C4×C8—31no—
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C3×D4—21no—
168C3×D4—21no—
169C3×D4—21no—
170C3×D4—21no—
171C3×D4—21no—
172C3×D4—42no—
173C3×D4—81no—
174C3×D4#4112yes—
175C2×C3×C4—21no—
176C2×C3×C4—81no—
177C2×C3×C4C2×D411yescharacteristic
178C2×C3×C4C2×D411yescharacteristic
179C2×C3×C4C2411yescharacteristic
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—42no—
185C2×Dic3—81no—
186C2×Dic3—81no—
187C2×Dic3—81no—
188C2×Dic3C2×D411yescharacteristic
189C2×Dic3C2×D411yescharacteristic
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192Dic6—21no—
193Dic6—21no—
194Dic6—41no—
195Dic6—81no—
196C3×C8—21no—
197C3×C8—21no—
198C3⋊C8—21no—
199C3⋊C8—21no—
200C2×Q8—31no—
201C2×Q8—31no—
202C2×Q8—61no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4C22×D311yescharacteristic
206C2×D4C22×D311yescharacteristic
207C2×D4C22×D311yescharacteristic
208C22×C4—21no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4—61no—
213SD16—62no—
214D8—22no—
215D8—22no—
216C2×C8—31no—
217C2×C8—31no—
218C2×C8C22×D311yescharacteristic
219C2×C8C22×D311yescharacteristic
220C4⋊C4—31no—
221C4⋊C4—31no—
222C4⋊C4—31no—
223C4⋊C4—31no—
224C4⋊C4—31no—
225C4⋊C4—31no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4C22×D311yescharacteristic
233C22⋊C4—21no—
234C22⋊C4—61no—
235C22⋊C4—61no—
236C22⋊C4—61no—
237C22⋊C4—61no—
238C22⋊C4—61no—
239C22⋊C4—61no—
240C22⋊C4—61no—
241C22⋊C4—61no—
242C42—31no—
243C42—31no—
244C42—61no—
245C42C22×D311yescharacteristic
246C22×C3—22no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3C22×D411yescharacteristic
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—41no—
259C3×C4C2×#4111yescharacteristic
260C3×C42−1+411yescharacteristic
261Dic3—21no—
262Dic3—21no—
263Dic3—21no—
264Dic3—21no—
265Dic3—41no—
266Dic3—41no—
267Dic3—41no—
268Dic3—41no—
269C23—21no—
270C23—21no—
271C23—21no—
272Q8—61no—
273Q8—61no—
274Q8—121no—
275Q8—241no—
276D4—21no—
277D4—21no—
278D4—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—81no—
283D4#23712yes—
284C2×C4—21no—
285C2×C4—31no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—81no—
294C2×C4—122no—
295C2×C4—241no—
296C2×C4—241no—
297C2×C4—241no—
298C2×C4D3×D411yescharacteristic
299C2×C4D3×D411yescharacteristic
300C2×C4C23×D311yescharacteristic
301C8—21no—
302C8—21no—
303C8—61no—
304C8—61no—
305C2×C3—42no—
306C2×C3—81no—
307C2×C3—81no—
308C2×C3#54811yescharacteristic
309C2×C3#57511yescharacteristic
310C2×C3#57611yescharacteristic
311C22—22no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22C2×D3×D411yescharacteristic
321C4—21no—
322C4—21no—
323C4—41no—
324C4—61no—
325C4—61no—
326C4—61no—
327C4—61no—
328C4—121no—
329C4—121no—
330C4—121no—
331C4—121no—
332C4C2×#23711yescharacteristic
333C4#101011yescharacteristic
334C3#337011yescharacteristic, complemented, Sylow
335C2—42no—
336C2—81no—
337C2—81no—
338C2#559811yescharacteristic
339C2#575711yescharacteristic
340C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3370
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. #5713C2semidirect
2. #5539C2semidirect
4. #5153C2semidirect
5. #5153C2semidirect
6. #5152C2semidirect
7. Dic3×D8C2semidirect
8. #5071C2semidirect
9. #5018C2semidirect
10. #4829C2semidirect
11. #4828C2semidirect
12. #4768C2semidirect
13. #4765C2semidirect
14. #4763C2semidirect
15. #4759C2semidirect
334. C3#3370semidirect

Maximal subgroups

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

subgroupindex
#57132
#55392
C3×#4592
#51532
#51522
Dic3×D82
#50712
#50182
#48292
#48282
#47682
#47652
#47632
#47592
#33703