Finite Group Databasebeta

← #80111

Subgroups — #80111

Order 384.

404 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5834C211yesmaximal, characteristic, complemented
2#5618C211yesmaximal, characteristic, complemented
3C3×#449C211yesmaximal, characteristic, complemented
4#5232C211yesmaximal, characteristic, complemented
5#5172C211yesmaximal, characteristic, complemented
6#5164C211yesmaximal, characteristic, complemented
7#5157C211yesmaximal, characteristic, complemented
8#5150C211yesmaximal, characteristic, complemented
9#5039C211yesmaximal, characteristic, complemented
10#4804C211yesmaximal, characteristic, complemented
11#4796C211yesmaximal, characteristic, complemented
12#4784C211yesmaximal, characteristic, complemented
13#4764C211yesmaximal, characteristic, complemented
14#4736C211yesmaximal, characteristic, complemented
15#4728C211yesmaximal, characteristic, complemented
16#3309—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C2×C3×D8C2211yescharacteristic
21C3×#121C2211yescharacteristic
22Q8×Dic3C2211yescharacteristic
23#940—21no—
24#940C2211yescharacteristic
25#939—21no—
26#938—21no—
27#938C2211yescharacteristic
28#938C2211yescharacteristic
29D4×Dic3—21no—
30D4×Dic3C2211yescharacteristic
31C2×#213C2211yescharacteristic
32#930—21no—
33#930C2211yescharacteristic
34C4×#67—21no—
35C4×#67C2211yescharacteristic
36C2×C4⋊Dic3C2211yescharacteristic
37C2×#208C2211yescharacteristic
38C2×D3×C8C2211yescharacteristic
39#892C2211yescharacteristic
40#890C2211yescharacteristic
41#886—21no—
42#879—21no—
43C3×#103C2211yescharacteristic
44C3×#102C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic
46#835C2211yescharacteristic
47#835C2211yescharacteristic
48#832C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51#821C2211yescharacteristic
52#821C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54#818C2211yescharacteristic
55#818C2211yescharacteristic
56#815C2211yescharacteristic
57Dic3⋊C8C2211yescharacteristic
58#809C2211yescharacteristic
59#808C2211yescharacteristic
60#553—31no—
61#550—31no—
62#490—31no—
63#482—31no—
64#480—31no—
65#479—31no—
66#469—31no—
67#466—31no—
68#459—31no—
69#449—31no—
70#449D311yescharacteristic, complemented
71#440—31no—
72#435—31no—
73#417—31no—
74#416—31no—
75C3×#41—22no—
76C3×#41—41no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4—21no—
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
82C22×C3×C4—21no—
83C22×C3×C4C2311yescharacteristic
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C2×#67D411yescharacteristic
90C22×Dic3—21no—
91C22×Dic3—21no—
92#239—22no—
93#239—41no—
94C2×D12—21no—
95C2×D12C2311yescharacteristic
96C2×C4×D3—21no—
97C2×C4×D3C2311yescharacteristic
98C3×SD16—22no—
99C3×D8—41no—
100C2×C3×C8C2311yescharacteristic
101C2×C3×C8C2311yescharacteristic
102C3×C4⋊C4C2311yescharacteristic
103C3×C22⋊C4—21no—
104#217—21no—
105#217—21no—
106#217—21no—
107#217D411yescharacteristic
108#213—41no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4D411yescharacteristic
112C4⋊Dic3—21no—
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3C2311yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117C4⋊Dic3C2311yescharacteristic
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×#41—31no—
128C2×#41C2×D311yescharacteristic
129C2×SD16C2×D311yescharacteristic
130C2×D8—31no—
131C2×D8C2×D311yescharacteristic
132C2×M16—31no—
133C22×C8—31no—
134#127—31no—
135#127—61no—
136#126—61no—
137#125—31no—
138#123—61no—
139#121—31no—
140#121—31no—
141#121—31no—
142#121—61no—
143#121—61no—
144#121—61no—
145#121C2×D311yescharacteristic
146C4×Q8—31no—
147C4×D4—31no—
148C4×D4—31no—
149C4×D4—61no—
150C4×D4—61no—
151#117—31no—
152C2×C4⋊C4—31no—
153#107—31no—
154#107—31no—
155#106—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103C2×D311yescharacteristic
160#102—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8C2×D311yescharacteristic
169#97—31no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3C2×D411yescharacteristic
173C22×D3—21no—
174C22×D3C2×D411yescharacteristic
175C3×Q8—21no—
176C3×Q8#4112yes—
177C3×D4—21no—
178C3×D4—21no—
179C3×D4—21no—
180C3×D4—21no—
181C3×D4—42no—
182C3×D4—42no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—42no—
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2411yescharacteristic
190#67—41no—
191#67—42no—
192#67—81no—
193#67—81no—
194#67—81no—
195#67—81no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—81no—
200C2×Dic3—81no—
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203C2×Dic3C2×D411yescharacteristic
204D12—21no—
205D12—21no—
206D12—42no—
207C4×D3—21no—
208C4×D3—21no—
209C4×D3—42no—
210C3×C8—21no—
211C3×C8—21no—
212C3⋊C8—21no—
213C3⋊C8—21no—
214#41—22no—
215#41—41no—
216#41—62no—
217#41—121no—
218C2×Q8C22×D311yescharacteristic
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—31no—
222C2×D4—31no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4C22×D311yescharacteristic
230C2×D4C22×D311yescharacteristic
231C22×C4—21no—
232C22×C4—31no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4C22×D311yescharacteristic
237SD16—22no—
238D8—41no—
239D8—121no—
240M16—121no—
241C2×C8—31no—
242C2×C8—31no—
243C2×C8—121no—
244C2×C8C22×D311yescharacteristic
245C2×C8C22×D311yescharacteristic
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—31no—
249C4⋊C4—31no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4C22×D311yescharacteristic
255C22⋊C4—21no—
256C22⋊C4—31no—
257C22⋊C4—31no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C42—31no—
264C42—61no—
265C42—61no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3C22×D411yescharacteristic
276C2×D3—21no—
277C2×D3—21no—
278C2×D3—21no—
279C2×D3—41no—
280C2×D3—41no—
281C2×D3—41no—
282C3×C4—22no—
283C3×C4—21no—
284C3×C4—41no—
285C3×C4C2×#4111yescharacteristic
286C3×C4H(5,𝔽2)11yescharacteristic
287Dic3—21no—
288Dic3—21no—
289Dic3—21no—
290Dic3—41no—
291Dic3—41no—
292Dic3—41no—
293C23—21no—
294C23—21no—
295C23—31no—
296C23—61no—
297C23D3×D411yescharacteristic
298Q8—21no—
299Q8#23712yes—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—42no—
305D4—42no—
306D4—61no—
307D4—61no—
308D4—122no—
309D4—121no—
310D4—122no—
311D4—241no—
312D4—241no—
313D4—241no—
314D4—241no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—31no—
322C2×C4—42no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—122no—
329C2×C4—241no—
330C2×C4—241no—
331C2×C4D3×D411yescharacteristic
332C2×C4C23×D311yescharacteristic
333C8—21no—
334C8—21no—
335C8—61no—
336C8—61no—
337C2×C3—41no—
338C2×C3—81no—
339C2×C3—81no—
340C2×C3#54611yescharacteristic
341C2×C3#57511yescharacteristic
342C2×C3#57611yescharacteristic
343D3—41no—
344D3—81no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—61no—
355C22—61no—
356C22—61no—
357C22—121no—
358C22—121no—
359C22—121no—
360C22C2×D3×D411yescharacteristic
361C4—21no—
362C4—22no—
363C4—41no—
364C4—61no—
365C4—61no—
366C4—61no—
367C4—121no—
368C4—121no—
369C4—121no—
370C4C2×#23711yescharacteristic
371C4#100411yescharacteristic
372C3#330911yescharacteristic, complemented, Sylow
373C2—41no—
374C2—81no—
375C2—81no—
376C2—121no—
377C2—241no—
378C2#559111yescharacteristic
379C2#576911yescharacteristic
380C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#449C2
socleC22×C3C22×D4
hypercenter#449D3
hypocenterC3#3309
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #5834C2semidirect
2. #5618C2semidirect
3. C3×#449C2semidirect
4. #5232C2semidirect
5. #5172C2semidirect
6. #5164C2semidirect
7. #5157C2semidirect
8. #5150C2semidirect
9. #5039C2semidirect
10. #4804C2semidirect
11. #4796C2semidirect
12. #4784C2semidirect
13. #4764C2semidirect
14. #4736C2semidirect
15. #4728C2semidirect
70. #449D3semidirect
372. C3#3309semidirect

Maximal subgroups

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

subgroupindex
#58342
#56182
C3×#4492
#52322
#51722
#51642
#51572
#51502
#50392
#48042
#47962
#47842
#47642
#47362
#47282
#33093