Finite Group Databasebeta

← #80255

Subgroups — #80255

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#5831C211yesmaximal, characteristic
2#5635C211yesmaximal, characteristic, complemented
3C3×#452C211yesmaximal, characteristic, complemented
4#5232C211yesmaximal, characteristic, complemented
5#5186C211yesmaximal, characteristic, complemented
6#5184C211yesmaximal, characteristic, complemented
7#5171C211yesmaximal, characteristic, complemented
8#5166C211yesmaximal, characteristic, complemented
9#5052C211yesmaximal, characteristic, complemented
10D3×#103C211yesmaximal, characteristic, complemented
11#4791C211yesmaximal, characteristic, complemented
12#4777C211yesmaximal, characteristic, complemented
13#4765C211yesmaximal, characteristic, complemented
14#4728C211yesmaximal, characteristic, complemented
15#4721C211yesmaximal, characteristic, complemented
16#3316—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C3×C22⋊Q8C2211yescharacteristic
22#946C2211yescharacteristic
23Q8×Dic3C2211yescharacteristic
24Q8×Dic3C2211yescharacteristic
25C2×#216C2211yescharacteristic
26#938—21no—
27#938C2211yescharacteristic
28#936—21no—
29#936C2211yescharacteristic
30C4×#67—21no—
31C4×#67C2211yescharacteristic
32C2×C4⋊Dic3C2211yescharacteristic
33#924—21no—
34#924C2211yescharacteristic
35C2×#208C2211yescharacteristic
36C2×D3×C8C2211yescharacteristic
37#897—21no—
38D3×C4⋊C4C2211yescharacteristic
39#889—21no—
40#888C2211yescharacteristic
41#67⋊C4—21no—
42#878—21no—
43C3×#103C2211yescharacteristic
44C3×#102C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#817C2211yescharacteristic
53#817C2211yescharacteristic
54#816C2211yescharacteristic
55#816C2211yescharacteristic
56#815C2211yescharacteristic
57Dic3⋊C8C2211yescharacteristic
58#810C2211yescharacteristic
59#808C2211yescharacteristic
60#554—31no—
61#548—31no—
62#489—31no—
63#483—31no—
64#481—31no—
65#475—31no—
66#470—31no—
67#465—31no—
68#464—31no—
69#452D311yescharacteristic, complemented
70#451—31no—
71#441—31no—
72#435—31no—
73#417—31no—
74C2×#103—31no—
75C3×#41—22no—
76C3×#41—41no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4C2311yescharacteristic
83C2×#67—21no—
84C2×#67D411yescharacteristic
85C22×Dic3—21no—
86D3×Q8—22no—
87D3×Q8—41no—
88C2×C4×D3—21no—
89C2×C4×D3—21no—
90C2×C4×D3C2311yescharacteristic
91C2×Dic6—21no—
92C2×Dic6—21no—
93C2×Dic6C2311yescharacteristic
94C3×Q16—22no—
95C3×SD16—41no—
96C2×C3×C8C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C3×C4⋊C4—21no—
99C3×C4⋊C4C2311yescharacteristic
100C3×C22⋊C4—21no—
101#217—21no—
102#217—21no—
103#217—21no—
104#217—21no—
105#217D411yescharacteristic
106#216—41no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4D411yescharacteristic
109C4⋊Dic3—21no—
110C4⋊Dic3—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114C4⋊Dic3C2311yescharacteristic
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4D411yescharacteristic
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3C2311yescharacteristic
123#208—41no—
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126D3×C8—41no—
127C2×#41C2×D311yescharacteristic
128C22×Q8—31no—
129C2×Q16—31no—
130C2×Q16C2×D311yescharacteristic
131C2×SD16C2×D311yescharacteristic
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8—31no—
135#125—61no—
136#124—31no—
137#124—61no—
138C22⋊Q8—31no—
139C22⋊Q8—31no—
140C22⋊Q8—61no—
141C22⋊Q8—61no—
142C22⋊Q8—61no—
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#106—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103—31no—
163#103C2×D311yescharacteristic
164#102—31no—
165#102—31no—
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8C2×D311yescharacteristic
169#97—31no—
170C23×C3—21no—
171C23×C3C2×D411yescharacteristic
172C22×D3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8#4112yes—
177C3×D4—21no—
178C3×D4—21no—
179C3×D4—42no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—42no—
186C2×C3×C4C2×D411yescharacteristic
187C2×C3×C4C2411yescharacteristic
188#67—22no—
189#67—42no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—81no—
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197C2×Dic3C2×D411yescharacteristic
198C4×D3—21no—
199C4×D3—21no—
200C4×D3—42no—
201C4×D3—81no—
202Dic6—21no—
203Dic6—21no—
204Dic6—42no—
205Dic6—81no—
206C3×C8—21no—
207C3×C8—21no—
208C3⋊C8—21no—
209C3⋊C8—21no—
210#41—22no—
211#41—41no—
212C2×Q8—31no—
213C2×Q8—61no—
214C2×Q8—61no—
215C2×Q8—62no—
216C2×Q8—121no—
217C2×Q8C22×D311yescharacteristic
218C2×Q8C22×D311yescharacteristic
219C2×D4—21no—
220C2×D4—31no—
221C2×D4—61no—
222C2×D4C22×D311yescharacteristic
223C22×C4—21no—
224C22×C4—31no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4C22×D311yescharacteristic
229Q16—22no—
230Q16—121no—
231SD16—41no—
232M16—121no—
233C2×C8—31no—
234C2×C8—31no—
235C2×C8—121no—
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⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4C22×D311yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—31no—
253C22⋊C4—31no—
254C22⋊C4—61no—
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×C3—41no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3C22×D411yescharacteristic
269C2×D3—21no—
270C2×D3—21no—
271C2×D3#13712yes—
272C3×C4—22no—
273C3×C4—21no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4C2×#4111yescharacteristic
277C3×C4H(5,𝔽2)11yescharacteristic
278Dic3—21no—
279Dic3—21no—
280Dic3—21no—
281Dic3—41no—
282Dic3—41no—
283Dic3—41no—
284Dic3—41no—
285C23—21no—
286C23—31no—
287C23D3×D411yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—61no—
292Q8—61no—
293Q8—122no—
294Q8—241no—
295Q8#23712yes—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—62no—
300D4—122no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—42no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
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—81no—
327C2×C3#54611yescharacteristic
328C2×C3C2×#13711yescharacteristic
329C2×C3#57711yescharacteristic
330D3—22no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—32no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—61no—
339C22—61no—
340C22C2×D3×D411yescharacteristic
341C4—21no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—61no—
346C4—61no—
347C4—61no—
348C4—121no—
349C4—121no—
350C4—121no—
351C4—121no—
352C4C2×#23711yescharacteristic
353C4#100411yescharacteristic
354C3#331611yescharacteristic, complemented, Sylow
355C2—41no—
356C2—62no—
357C2—81no—
358C2#559111yescharacteristic
359C2#577011yescharacteristic
360C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#452C2
socleC22×C3C22×D4
prime radical#5831C2
hypercenter#452D3
hypocenterC3#3316
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. #5635C2semidirect
3. C3×#452C2semidirect
4. #5232C2semidirect
5. #5186C2semidirect
6. #5184C2semidirect
7. #5171C2semidirect
8. #5166C2semidirect
9. #5052C2semidirect
10. D3×#103C2semidirect
11. #4791C2semidirect
12. #4777C2semidirect
13. #4765C2semidirect
14. #4728C2semidirect
15. #4721C2semidirect
69. #452D3semidirect
354. C3#3316semidirect

Maximal subgroups

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

subgroupindex
#58312
#56352
C3×#4522
#52322
#51862
#51842
#51712
#51662
#50522
D3×#1032
#47912
#47772
#47652
#47282
#47212
#33163