Finite Group Databasebeta

← #80249

Subgroups — #80249

Order 384.

396 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5834C211yesmaximal, characteristic
2#5641C211yesmaximal, characteristic, complemented
3C3×#452C211yesmaximal, characteristic, complemented
4#5233C211yesmaximal, characteristic, complemented
5#5187C211yesmaximal, characteristic, complemented
6Dic3×Q16C211yesmaximal, characteristic, complemented
7#5171C211yesmaximal, characteristic, complemented
8#5163C211yesmaximal, characteristic, complemented
9#5048C211yesmaximal, characteristic, complemented
10#4804C211yesmaximal, characteristic, complemented
11#4791C211yesmaximal, characteristic, complemented
12#4784C211yesmaximal, characteristic, complemented
13#4760C211yesmaximal, characteristic, complemented
14#4737C211yesmaximal, characteristic, complemented
15#67⋊C8C211yesmaximal, characteristic, complemented
16#3304—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C3×C22⋊Q8C2211yescharacteristic
22#946C2211yescharacteristic
23Q8×Dic3C2211yescharacteristic
24Q8×Dic3C2211yescharacteristic
25C2×#215C2211yescharacteristic
26#940—21no—
27#940C2211yescharacteristic
28#938—21no—
29#938C2211yescharacteristic
30#930—21no—
31#930C2211yescharacteristic
32C4×#67—21no—
33C4×#67C2211yescharacteristic
34#926C2211yescharacteristic
35C22×C3⋊C8C2211yescharacteristic
36C2×D3×C8C2211yescharacteristic
37#898—21no—
38#892C2211yescharacteristic
39#888C2211yescharacteristic
40#887—21no—
41#885—21no—
42#879—21no—
43C3×#103C2211yescharacteristic
44C3×#102C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51#821C2211yescharacteristic
52#821C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54#818C2211yescharacteristic
55#817C2211yescharacteristic
56Dic3⋊C8C2211yescharacteristic
57C8×Dic3C2211yescharacteristic
58#809C2211yescharacteristic
59#808C2211yescharacteristic
60#556—31no—
61#550—31no—
62#486—31no—
63#482—31no—
64#482—31no—
65#475—31no—
66#465—31no—
67#465—31no—
68#460—31no—
69#452—31no—
70#452D311yescharacteristic, complemented
71C4×Q16—31no—
72C8×D4—31no—
73#416—31no—
74#416—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×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67D411yescharacteristic
88#239—22no—
89#239—41no—
90C2×D12—21no—
91C2×D12C2311yescharacteristic
92C2×C4×D3—21no—
93C2×C4×D3C2311yescharacteristic
94C2×Dic6—21no—
95C3×Q16—22no—
96C3×SD16—41no—
97C2×C3×C8C2311yescharacteristic
98C2×C3×C8C2311yescharacteristic
99C3×C4⋊C4—21no—
100C3×C4⋊C4C2311yescharacteristic
101C3×C22⋊C4—21no—
102#217—21no—
103#217—21no—
104#217D411yescharacteristic
105#215—41no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4D411yescharacteristic
110C4⋊Dic3—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114C4⋊Dic3C2311yescharacteristic
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4D411yescharacteristic
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3—21no—
121C4×Dic3—21no—
122C4×Dic3C2311yescharacteristic
123C2×C3⋊C8—41no—
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126D3×C8—41no—
127C2×#41—31no—
128C2×#41C2×D311yescharacteristic
129C2×Q16C2×D311yescharacteristic
130C2×SD16—31no—
131C2×SD16C2×D311yescharacteristic
132C22×C8—31no—
133C22×C8—31no—
134C4⋊Q8—31no—
135#127—31no—
136#127—61no—
137#126—61no—
138#126—61no—
139#124—61no—
140C22⋊Q8—31no—
141C22⋊Q8C2×D311yescharacteristic
142#121—31no—
143#121—31no—
144#121—61no—
145#121—61no—
146C4×Q8—31no—
147C4×Q8—31no—
148C4×Q8—61no—
149C4×D4—31no—
150C4×D4—61no—
151#117—31no—
152#117—31no—
153#107—31no—
154#106—31no—
155#106—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103C2×D311yescharacteristic
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8C2×D311yescharacteristic
169C4×C8—31no—
170C23×C3—21no—
171C23×C3C2×D411yescharacteristic
172C22×D3—21no—
173C22×D3C2×D411yescharacteristic
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×Q8#4112yes—
178C3×D4—21no—
179C3×D4—21no—
180C3×D4—42no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—42no—
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2411yescharacteristic
189#67—22no—
190#67—42no—
191#67—81no—
192#67—81no—
193#67—81no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3C2×D411yescharacteristic
198C2×Dic3C2×D411yescharacteristic
199C2×Dic3C2×D411yescharacteristic
200D12—21no—
201D12—21no—
202D12—42no—
203C4×D3—21no—
204C4×D3—21no—
205C4×D3—42no—
206Dic6—42no—
207C3×C8—21no—
208C3×C8—21no—
209C3⋊C8—21no—
210C3⋊C8—21no—
211#41—22no—
212#41—41no—
213#41—62no—
214#41—121no—
215C2×Q8—61no—
216C2×Q8C22×D311yescharacteristic
217C2×Q8C22×D311yescharacteristic
218C2×D4—21no—
219C2×D4—31no—
220C2×D4—31no—
221C2×D4—61no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4C22×D311yescharacteristic
227C22×C4—21no—
228C22×C4—31no—
229C22×C4—61no—
230C22×C4C22×D311yescharacteristic
231Q16—22no—
232SD16—41no—
233SD16—121no—
234C2×C8—31no—
235C2×C8—31no—
236C2×C8—121no—
237C2×C8—121no—
238C2×C8C22×D311yescharacteristic
239C2×C8C22×D311yescharacteristic
240C4⋊C4—21no—
241C4⋊C4—31no—
242C4⋊C4—31no—
243C4⋊C4—31no—
244C4⋊C4—31no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4C22×D311yescharacteristic
250C22⋊C4—21no—
251C22⋊C4—31no—
252C22⋊C4—31no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C42—31no—
259C42—61no—
260C42—61no—
261C42—61no—
262C42—61no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3C22×D411yescharacteristic
270C2×D3—21no—
271C2×D3—21no—
272C2×D3—21no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—41no—
276C3×C4—22no—
277C3×C4—21no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4C2×#4111yescharacteristic
281C3×C4H(5,𝔽2)11yescharacteristic
282Dic3—21no—
283Dic3—21no—
284Dic3—41no—
285Dic3—41no—
286Dic3—41no—
287Dic3#13512yes—
288C23—21no—
289C23—31no—
290C23—61no—
291C23D3×D411yescharacteristic
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—122no—
296Q8#23712yes—
297D4—21no—
298D4—21no—
299D4—42no—
300D4—62no—
301D4—61no—
302D4—61no—
303D4—122no—
304D4—122no—
305D4—241no—
306D4—241no—
307D4—241no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—31no—
316C2×C4—42no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—122no—
323C2×C4D3×D411yescharacteristic
324C2×C4C23×D311yescharacteristic
325C8—21no—
326C8—21no—
327C8—61no—
328C8—61no—
329C2×C3—41no—
330C2×C3—81no—
331C2×C3#54611yescharacteristic
332C2×C3C2×#13511yescharacteristic
333C2×C3#57711yescharacteristic
334D3—41no—
335D3—81no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—61no—
343C22—61no—
344C22—61no—
345C22—121no—
346C22—121no—
347C22—121no—
348C22C2×D3×D411yescharacteristic
349C4—21no—
350C4—22no—
351C4—32no—
352C4—41no—
353C4—41no—
354C4—61no—
355C4—61no—
356C4—121no—
357C4—121no—
358C4—121no—
359C4C2×#23711yescharacteristic
360C4#100411yescharacteristic
361C3#330411yescharacteristic, complemented, Sylow
362C2—41no—
363C2—81no—
364C2—121no—
365C2—241no—
366C2#559111yescharacteristic
367C2D3×#13511yescharacteristic
368C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#452C2
socleC22×C3C22×D4
prime radical#5834C2
hypercenter#452D3
hypocenterC3#3304
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. #5641C2semidirect
3. C3×#452C2semidirect
4. #5233C2semidirect
5. #5187C2semidirect
6. Dic3×Q16C2semidirect
7. #5171C2semidirect
8. #5163C2semidirect
9. #5048C2semidirect
10. #4804C2semidirect
11. #4791C2semidirect
12. #4784C2semidirect
13. #4760C2semidirect
14. #4737C2semidirect
15. #67⋊C8C2semidirect
70. #452D3semidirect
361. C3#3304semidirect

Maximal subgroups

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

subgroupindex
#58342
#56412
C3×#4522
#52332
#51872
Dic3×Q162
#51712
#51632
#50482
#48042
#47912
#47842
#47602
#47372
#67⋊C82
#33043