Finite Group Databasebeta

← #80612

Subgroups — #80612

Order 384.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#918C211yesmaximal, characteristic, complemented
2C2×#916C211yesmaximal, characteristic, complemented
3#5677C211yesmaximal, characteristic, complemented
4#5577C211yesmaximal, characteristic, complemented
5C3×#464C211yesmaximal, characteristic, complemented
6#5187C211yesmaximal, characteristic, complemented
7#5172C211yesmaximal, characteristic, complemented
8#5058C211yesmaximal, characteristic, complemented
9#5030C211yesmaximal, characteristic, complemented
10#4848C211yesmaximal, characteristic, complemented
11#4836C211yesmaximal, characteristic, complemented
12#4807C211yesmaximal, characteristic, complemented
13#4806C211yesmaximal, characteristic, complemented
14#4782C211yesmaximal, characteristic, complemented
15#4781C211yesmaximal, characteristic, complemented
16#3271—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×D3×Q8C2211yescharacteristic
20C2×#237C2211yescharacteristic
21C2×C3×Q16C2211yescharacteristic
22C2×C3×SD16C2211yescharacteristic
23C3×#124C2211yescharacteristic
24C3×C4×Q8C2211yescharacteristic
25#946C2211yescharacteristic
26C2×#216C2211yescharacteristic
27C2×#215C2211yescharacteristic
28C2×#215C2211yescharacteristic
29#938C2211yescharacteristic
30C2×#213C2211yescharacteristic
31C2×#213C2211yescharacteristic
32#918—24no—
33#916—24no—
34C2×#204C2211yescharacteristic
35C2×#204C2211yescharacteristic
36C2×ΓL(1,𝔽25)C2211yescharacteristic
37C2×D3×C8C2211yescharacteristic
38#895—21no—
39#895—21no—
40#895C2211yescharacteristic
41#895C2211yescharacteristic
42#885—21no—
43#884—21no—
44#67⋊C4—21no—
45#878—21no—
46#874—21no—
47#874C2211yescharacteristic
48C4×D12—21no—
49C4×D12C2211yescharacteristic
50C4×Dic6C2211yescharacteristic
51C3×C4⋊C8C2211yescharacteristic
52C3×#103C2211yescharacteristic
53C3×#102C2211yescharacteristic
54#835C2211yescharacteristic
55#821C2211yescharacteristic
56#821C2211yescharacteristic
57(C2×D3)⋊C8C2211yescharacteristic
58(C2×D3)⋊C8C2211yescharacteristic
59#810C2211yescharacteristic
60#808C2211yescharacteristic
61#803C2211yescharacteristic
62C2×#137—31no—
63C2×#135—31no—
64#550—31no—
65#548—31no—
66#496—31no—
67#469—31no—
68#465—31no—
69#464D311yescharacteristic, complemented
70#460—31no—
71#452—31no—
72#449—31no—
73#449—31no—
74#448—31no—
75#441—31no—
76#435—31no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4C2311yescharacteristic
80C2×#67—21no—
81C2×#67—21no—
82C22×Dic3—21no—
83#239—22no—
84#239—22no—
85#239—41no—
86#239—41no—
87D3×Q8—22no—
88D3×Q8—41no—
89#237—22no—
90#237—41no—
91C2×D12—21no—
92C2×D12—21no—
93C2×D12—21no—
94C2×D12—21no—
95C2×D12—21no—
96C2×D12—21no—
97C2×D12D411yescharacteristic
98C2×D12C2311yescharacteristic
99C2×D12C2311yescharacteristic
100C2×C4×D3—21no—
101C2×C4×D3—21no—
102C2×C4×D3—21no—
103C2×C4×D3C2311yescharacteristic
104C2×C4×D3C2311yescharacteristic
105C2×Dic6—21no—
106C2×Dic6C2311yescharacteristic
107C3×Q16—22no—
108C3×SD16—22no—
109C2×C3×C8C2311yescharacteristic
110C2×C3×C8C2311yescharacteristic
111C3×C4⋊C4—21no—
112C3×C4⋊C4C2311yescharacteristic
113C3×C22⋊C4—21no—
114C3×C22⋊C4—21no—
115C3×C42—21no—
116C3×C42C2311yescharacteristic
117#217—21no—
118#216D414yescomplemented
119#215—22no—
120#215—41no—
121#213—22no—
122#213—41no—
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4D411yescharacteristic
127(C2×D3)⋊C4D411yescharacteristic
128C4⋊Dic3D411yescharacteristic
129C4⋊Dic3C2311yescharacteristic
130Dic3⋊C4—21no—
131Dic3⋊C4—21no—
132C4×Dic3—21no—
133C2×C3⋊C8C2311yescharacteristic
134C2×C3⋊C8C2311yescharacteristic
135#204—22no—
136#204—22no—
137ΓL(1,𝔽25)—22no—
138D3×C8—22no—
139C2×#41—31no—
140C2×#41—31no—
141C2×#41—31no—
142C22×Q8—31no—
143#137—64no—
144#135—64no—
145C2×Q16—31no—
146C2×Q16C2×D311yescharacteristic
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16C2×D311yescharacteristic
152C2×D8—31no—
153C2×D8—31no—
154C2×M16—31no—
155C22×C8—31no—
156#127—31no—
157#127—61no—
158#124—61no—
159#124C2×D311yescharacteristic
160C22⋊Q8—31no—
161C22⋊Q8—61no—
162#121—31no—
163#121—31no—
164#121—31no—
165#121—61no—
166#121—61no—
167#121—61no—
168C4×Q8—31no—
169C4×Q8C2×D311yescharacteristic
170C4×D4—31no—
171C4×D4—61no—
172C4×D4—61no—
173#106—31no—
174C4⋊C8C2×D311yescharacteristic
175#103—31no—
176#103—31no—
177#103C2×D311yescharacteristic
178#102—31no—
179#102—31no—
180#102C2×D311yescharacteristic
181C22⋊C8—31no—
182C22⋊C8—31no—
183#97—31no—
184C23×C3—21no—
185C22×D3—21no—
186C22×D3—21no—
187C22×D3C2×D411yescharacteristic
188C22×D3C2×D411yescharacteristic
189C3×Q8—21no—
190C3×Q8—21no—
191C3×Q8—21no—
192C3×Q8C2×D412yes—
193C3×D4—21no—
194C3×D4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2411yescharacteristic
201#67—42no—
202#67—42no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—42no—
206C2×Dic3C2×D411yescharacteristic
207C2×Dic3C2×D411yescharacteristic
208D12—21no—
209D12—21no—
210D12—21no—
211D12—21no—
212D12—41no—
213D12—42no—
214D12—42no—
215D12—42no—
216D12—81no—
217D12—81no—
218D12—81no—
219C4×D3—21no—
220C4×D3—21no—
221C4×D3—21no—
222C4×D3—21no—
223C4×D3—42no—
224C4×D3—42no—
225C4×D3—42no—
226Dic6—21no—
227Dic6—42no—
228Dic6C2×D412yes—
229C3×C8—21no—
230C3×C8—21no—
231C3⋊C8—21no—
232C3⋊C8C2×D412yes—
233#41—62no—
234#41—62no—
235#41—62no—
236#41—121no—
237#41—121no—
238#41—121no—
239C2×Q8—31no—
240C2×Q8—61no—
241C2×Q8—62no—
242C2×Q8—121no—
243C2×Q8C22×D311yescharacteristic
244C2×Q8C22×D311yescharacteristic
245C2×D4—31no—
246C2×D4—31no—
247C2×D4—31no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4C22×D311yescharacteristic
257C22×C4—31no—
258C22×C4—31no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—61no—
263Q16—22no—
264Q16—34no—
265SD16—22no—
266SD16—62no—
267SD16—62no—
268SD16—62no—
269SD16—121no—
270D8—62no—
271D8—121no—
272M16—62no—
273C2×C8—31no—
274C2×C8—31no—
275C2×C8—62no—
276C2×C8C22×D311yescharacteristic
277C2×C8C22×D311yescharacteristic
278C4⋊C4—21no—
279C4⋊C4—31no—
280C4⋊C4—31no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4C22×D311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—31no—
287C22⋊C4—31no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C42—21no—
293C42—61no—
294C42C22×D311yescharacteristic
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3C22×D411yescharacteristic
299C2×D3—21no—
300C2×D3—21no—
301C2×D3—21no—
302C2×D3—21no—
303C2×D3—21no—
304C2×D3—21no—
305C2×D3—41no—
306C2×D3—41no—
307C2×D3—41no—
308C2×D3—41no—
309C2×D3—41no—
310C2×D3—41no—
311C3×C4—22no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—41no—
315C3×C4—41no—
316C3×C4C22×D411yescharacteristic
317C3×C4H(5,𝔽2)11yescharacteristic
318Dic3—22no—
319Dic3—21no—
320Dic3—21no—
321Dic3—41no—
322C23—21no—
323C23—31no—
324C23—31no—
325C23—61no—
326C23—61no—
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—32no—
331Q8—61no—
332Q8—122no—
333Q8D3×D412yes—
334D4—21no—
335D4—21no—
336D4—61no—
337D4—61no—
338D4—61no—
339D4—61no—
340D4—121no—
341D4—122no—
342D4—122no—
343D4—122no—
344D4—122no—
345D4—122no—
346D4—241no—
347D4—241no—
348D4—241no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—31no—
353C2×C4—31no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4D3×D411yescharacteristic
365C2×C4D3×D411yescharacteristic
366C2×C4C23×D311yescharacteristic
367C8—21no—
368C8—21no—
369C8—32no—
370C8—61no—
371C2×C3—81no—
372C2×C3D4211yescharacteristic
373C2×C3#57511yescharacteristic
374C2×C3#57711yescharacteristic
375D3—41no—
376D3—41no—
377D3—81no—
378D3—81no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—61no—
383C22—61no—
384C22—61no—
385C22—61no—
386C22—61no—
387C22—61no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22C2×D3×D411yescharacteristic
395C4—21no—
396C4—21no—
397C4—22no—
398C4—41no—
399C4—41no—
400C4—61no—
401C4—61no—
402C4—62no—
403C4—121no—
404C4C2×D3×D411yescharacteristic
405C4#100911yescharacteristic
406C3#327111yescharacteristic, complemented, Sylow
407C2—81no—
408C2—121no—
409C2—121no—
410C2—241no—
411C2—241no—
412C2#560911yescharacteristic
413C2#577011yescharacteristic
414C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#464C2
socleC22×C3C22×D4
hypercenter#464D3
hypocenterC3#3271
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#918C2semidirect
2. C2×#916C2semidirect
3. #5677C2semidirect
4. #5577C2semidirect
5. C3×#464C2semidirect
6. #5187C2semidirect
7. #5172C2semidirect
8. #5058C2semidirect
9. #5030C2semidirect
10. #4848C2semidirect
11. #4836C2semidirect
12. #4807C2semidirect
13. #4806C2semidirect
14. #4782C2semidirect
15. #4781C2semidirect
69. #464D3semidirect
118. #216D4semidirect
406. C3#3271semidirect

Maximal subgroups

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

subgroupindex
C2×#9182
C2×#9162
#56772
#55772
C3×#4642
#51872
#51722
#50582
#50302
#48482
#48362
#48072
#48062
#47822
#47812
#32713