Finite Group Databasebeta

← #79799

Subgroups — #79799

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#5577C211yesmaximal, characteristic, complemented
2#5558C211yesmaximal, characteristic, complemented
3C3×#440C211yesmaximal, characteristic, complemented
4#5172C211yesmaximal, characteristic, complemented
5#5172C211yesmaximal, characteristic, complemented
6#5022C211yesmaximal, characteristic, complemented
7#5015C211yesmaximal, characteristic, complemented
8#4885C211yesmaximal, characteristic, complemented
9#4882C211yesmaximal, characteristic, complemented
10#4804C211yesmaximal, characteristic, complemented
11#4803C211yesmaximal, characteristic, complemented
12#4782C211yesmaximal, characteristic, complemented
13#4775C211yesmaximal, characteristic, complemented
14#4713C211yesmaximal, characteristic, complemented
15#4706C211yesmaximal, characteristic, complemented
16#3309—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C3×C4×Q8C2211yescharacteristic
21C3×C4×D4C2211yescharacteristic
22#938C2211yescharacteristic
23#938C2211yescharacteristic
24C2×#213C2211yescharacteristic
25C2×#213C2211yescharacteristic
26#929—21no—
27C4×#67—21no—
28C2×ΓL(1,𝔽25)C2211yescharacteristic
29C2×D3×C8C2211yescharacteristic
30#895—21no—
31#895—21no—
32#895C2211yescharacteristic
33#895C2211yescharacteristic
34#892C2211yescharacteristic
35D3×C4⋊C4C2211yescharacteristic
36#884—21no—
37#879—21no—
38#874—21no—
39#874C2211yescharacteristic
40C4×D12—21no—
41C4×D12C2211yescharacteristic
42#870C2211yescharacteristic
43C3×#107C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#102C2211yescharacteristic
46C3×#97C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#821C2211yescharacteristic
52#821C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54(C2×D3)⋊C8C2211yescharacteristic
55#809C2211yescharacteristic
56#809C2211yescharacteristic
57#808C2211yescharacteristic
58#807C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
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#449—31no—
71#440D311yescharacteristic, complemented
72#435—31no—
73#417—31no—
74#416—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4—21no—
78C2×#67—21no—
79C2×#67—21no—
80#239—22no—
81#239—22no—
82#239—41no—
83#239—41no—
84C2×D12—21no—
85C2×D12—21no—
86C2×D12—21no—
87C2×D12—21no—
88C2×D12—21no—
89C2×D12—21no—
90C2×D12D411yescharacteristic
91C2×D12C2311yescharacteristic
92C2×D12C2311yescharacteristic
93C2×C4×D3—21no—
94C2×C4×D3—21no—
95C2×C4×D3—21no—
96C2×C4×D3C2311yescharacteristic
97C2×C4×D3C2311yescharacteristic
98C3×SD16—22no—
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C3×C4⋊C4—21no—
102C3×C4⋊C4C2311yescharacteristic
103C3×C4⋊C4C2311yescharacteristic
104C3×C22⋊C4—21no—
105C3×C42—21no—
106C3×C42C2311yescharacteristic
107#217—21no—
108#217—21no—
109#213—41no—
110#213—41no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4D411yescharacteristic
115(C2×D3)⋊C4D411yescharacteristic
116C4⋊Dic3D411yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122C4×Dic3—21no—
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125ΓL(1,𝔽25)—41no—
126D3×C8—41no—
127C2×#41—31no—
128C2×#41—31no—
129C2×SD16C2×D311yescharacteristic
130C2×D8—31no—
131C2×D8—31no—
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—31no—
143#121—61no—
144#121—61no—
145#121—61no—
146C4×Q8C2×D311yescharacteristic
147C4×D4—31no—
148C4×D4—61no—
149C4×D4—61no—
150C4×D4C2×D311yescharacteristic
151#117—31no—
152C2×C4⋊C4—31no—
153#107—31no—
154#107C2×D311yescharacteristic
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⋊C8—31no—
169#97C2×D311yescharacteristic
170C23×C3—21no—
171C22×D3—21no—
172C22×D3—21no—
173C22×D3C2×D411yescharacteristic
174C22×D3C2×D411yescharacteristic
175C3×Q8—21no—
176C3×Q8#4112yes—
177C3×D4—21no—
178C3×D4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—81no—
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2411yescharacteristic
186#67—42no—
187#67—81no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192D12—21no—
193D12—21no—
194D12—21no—
195D12—21no—
196D12—41no—
197D12—42no—
198D12—42no—
199D12—42no—
200D12—81no—
201D12—81no—
202D12—81no—
203C4×D3—21no—
204C4×D3—21no—
205C4×D3—21no—
206C4×D3—21no—
207C4×D3—42no—
208C4×D3—42no—
209C4×D3—81no—
210C3×C8—21no—
211C3×C8—21no—
212C3⋊C8—21no—
213C3⋊C8—21no—
214#41—62no—
215#41—62no—
216#41—121no—
217#41—121no—
218C2×Q8C22×D311yescharacteristic
219C2×D4—31no—
220C2×D4—31no—
221C2×D4—31no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4C22×D311yescharacteristic
231C22×C4—21no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—61no—
237SD16—22no—
238D8—121no—
239D8—121no—
240M16—121no—
241C2×C8—31no—
242C2×C8—31no—
243C2×C8—121no—
244C2×C8C22×D311yescharacteristic
245C2×C8C22×D311yescharacteristic
246C4⋊C4—21no—
247C4⋊C4—31no—
248C4⋊C4—31no—
249C4⋊C4—31no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4C22×D311yescharacteristic
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—21no—
264C42—61no—
265C42C22×D311yescharacteristic
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3C22×D411yescharacteristic
270C2×D3—21no—
271C2×D3—21no—
272C2×D3—21no—
273C2×D3—21no—
274C2×D3—21no—
275C2×D3—21no—
276C2×D3—41no—
277C2×D3—41no—
278C2×D3—41no—
279C2×D3—41no—
280C2×D3—41no—
281C2×D3—41no—
282C3×C4—22no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—41no—
286C3×C4—41no—
287C3×C4C2×#4111yescharacteristic
288C3×C4H(5,𝔽2)11yescharacteristic
289Dic3—21no—
290Dic3—21no—
291Dic3—41no—
292Dic3—41no—
293C23—21no—
294C23—31no—
295C23—31no—
296C23—61no—
297C23—61no—
298Q8—21no—
299Q8#23512yes—
300D4—21no—
301D4—21no—
302D4—61no—
303D4—61no—
304D4—61no—
305D4—61no—
306D4—121no—
307D4—122no—
308D4—122no—
309D4—122no—
310D4—122no—
311D4—241no—
312D4—241no—
313D4—241no—
314D4—241no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—31no—
319C2×C4—31no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—81no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4—241no—
330C2×C4D3×D411yescharacteristic
331C2×C4D3×D411yescharacteristic
332C2×C4C23×D311yescharacteristic
333C8—21no—
334C8—21no—
335C8—61no—
336C8—61no—
337C2×C3—81no—
338C2×C3#54611yescharacteristic
339C2×C3#57511yescharacteristic
340C2×C3#57611yescharacteristic
341D3—41no—
342D3—41no—
343D3—81no—
344D3—81no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—61no—
349C22—61no—
350C22—61no—
351C22—61no—
352C22—61no—
353C22—61no—
354C22—121no—
355C22—121no—
356C22—121no—
357C22—121no—
358C22—121no—
359C22—121no—
360C22C2×D3×D411yescharacteristic
361C4—21no—
362C4—21no—
363C4—22no—
364C4—41no—
365C4—41no—
366C4—61no—
367C4—61no—
368C4—121no—
369C4—121no—
370C4C2×#23511yescharacteristic
371C4#100911yescharacteristic
372C3#330911yescharacteristic, complemented, Sylow
373C2—81no—
374C2—121no—
375C2—121no—
376C2—241no—
377C2—241no—
378C2#555011yescharacteristic
379C2#577411yescharacteristic
380C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#440C2
socleC22×C3C22×D4
hypercenter#440D3
hypocenterC3#3309
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #5577C2semidirect
2. #5558C2semidirect
3. C3×#440C2semidirect
4. #5172C2semidirect
5. #5172C2semidirect
6. #5022C2semidirect
7. #5015C2semidirect
8. #4885C2semidirect
9. #4882C2semidirect
10. #4804C2semidirect
11. #4803C2semidirect
12. #4782C2semidirect
13. #4775C2semidirect
14. #4713C2semidirect
15. #4706C2semidirect
71. #440D3semidirect
372. C3#3309semidirect

Maximal subgroups

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

subgroupindex
#55772
#55582
C3×#4402
#51722
#50222
#50152
#48852
#48822
#48042
#48032
#47822
#47752
#47132
#47062
#33093