Finite Group Databasebeta

← #80204

Subgroups — #80204

Order 384.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#67C211yesmaximal, characteristic, complemented
2C22×#215C211yesmaximal, characteristic, complemented
3C2×#918C211yesmaximal, characteristic, complemented
4#5630C211yesmaximal, characteristic
5C3×#451C211yesmaximal, characteristic, complemented
6#5190C211yesmaximal, characteristic, complemented
7#5187C211yesmaximal, characteristic, complemented
8#5185C211yesmaximal, characteristic, complemented
9#5051C211yesmaximal, characteristic, complemented
10#4818C211yesmaximal, characteristic, complemented
11#4817C211yesmaximal, characteristic, complemented
12#4812C211yesmaximal, characteristic, complemented
13#4806C211yesmaximal, characteristic, complemented
14#4731C211yesmaximal, characteristic, complemented
15#4730C211yesmaximal, characteristic, complemented
16#3266—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×D3×Q8C2211yescharacteristic, complemented
20C22×D12C2211yescharacteristic
21C2×C3×Q16C2211yescharacteristic, complemented
22C2×C3×Q16C2211yescharacteristic, complemented
23C3×C22⋊Q8C2211yescharacteristic
24#947C2211yescharacteristic
25#946—21no—
26#946C2211yescharacteristic, complemented
27Q8×Dic3C2211yescharacteristic, complemented
28#944—21no—
29#944C2211yescharacteristic, complemented
30C2×#216C2211yescharacteristic, complemented
31C2×#215—24no—
32C2×#215C2211yescharacteristic
33C2×#215C2211yescharacteristic
34C2×#215C2211yescharacteristic
35C2×#215C2211yescharacteristic
36C4×#67—21no—
37C4×#67C2211yescharacteristic
38#924—21no—
39#924C2211yescharacteristic
40C22×C3⋊C8C2211yescharacteristic
41#918—24no—
42C2×#204C2211yescharacteristic
43C2×#204C2211yescharacteristic
44C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
45#895—21no—
46#895C2211yescharacteristic
47#894—21no—
48#893C2211yescharacteristic
49#882—21no—
50D3×C22⋊C4—21no—
51C3×#103C2211yescharacteristic, complemented
52C3×#103C2211yescharacteristic, complemented
53C3×C22⋊C8C2211yescharacteristic
54#835C2211yescharacteristic, complemented
55#821C2211yescharacteristic
56#821C2211yescharacteristic
57(C2×D3)⋊C8C2211yescharacteristic, complemented
58#815C2211yescharacteristic, complemented
59Dic3⋊C8C2211yescharacteristic, complemented
60#810C2211yescharacteristic, complemented
61#807C2211yescharacteristic, complemented
62C2×#137—31no—
63C22×SD16—31no—
64D4×Q8—31no—
65#546—31no—
66#497—31no—
67#468—31no—
68#467—31no—
69#464—31no—
70#461—31no—
71#452—31no—
72#451D311yescharacteristic, complemented
73#450—31no—
74#448—31no—
75#440—31no—
76#435—31no—
77C23×D3—21no—
78C2×C3×Q8—21no—
79C2×C3×Q8—22no—
80C2×C3×Q8—41no—
81C2×C3×Q8C2311yescharacteristic
82C2×C3×Q8C2311yescharacteristic
83C2×C3×Q8C2311yescharacteristic
84C22×C3×C4—21no—
85C22×C3×C4C2311yescharacteristic
86C2×#67D411yescharacteristic
87#239—22no—
88#239—41no—
89D3×Q8—22no—
90D3×Q8—41no—
91C2×D12—22no—
92C2×D12—21no—
93C2×D12—21no—
94C2×D12—21no—
95C2×D12—41no—
96C2×D12C2311yescharacteristic
97C2×D12C2311yescharacteristic
98C2×C4×D3—21no—
99C2×C4×D3—21no—
100C2×C4×D3—21no—
101C2×C4×D3C2311yescharacteristic
102C2×Dic6—21no—
103C2×Dic6C2311yescharacteristic
104C3×Q16—22no—
105C3×Q16—41no—
106C2×C3×C8C2311yescharacteristic
107C2×C3×C8C2311yescharacteristic
108C3×C4⋊C4—21no—
109C3×C4⋊C4C2311yescharacteristic
110C3×C22⋊C4—21no—
111#217—21no—
112#217D411yescharacteristic
113#216—22no—
114#215—22no—
115#215—22no—
116#215—22no—
117#215D414yescomplemented
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4D411yescharacteristic
124C4⋊Dic3—21no—
125C4⋊Dic3C2311yescharacteristic
126Dic3⋊C4—21no—
127Dic3⋊C4—21no—
128Dic3⋊C4—21no—
129Dic3⋊C4—21no—
130Dic3⋊C4D411yescharacteristic
131C4×Dic3—21no—
132C4×Dic3C2311yescharacteristic
133C2×C3⋊C8—22no—
134C2×C3⋊C8C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136#204—22no—
137#204—41no—
138ΓL(1,𝔽25)—22no—
139C2×#41—31no—
140C22×Q8—31no—
141C22×Q8C2×D311yescharacteristic
142C22×D4—31no—
143#137—64no—
144C2×Q16—31no—
145C2×Q16C2×D311yescharacteristic, complemented
146C2×Q16C2×D311yescharacteristic, complemented
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—31no—
152C2×SD16—31no—
153C2×SD16—64no—
154C2×M16—31no—
155C22×C8—31no—
156C4⋊Q8—31no—
157C4⋊Q8—61no—
158#124—31no—
159#123—61no—
160C22⋊Q8—31no—
161C22⋊Q8—31no—
162C22⋊Q8—61no—
163C22⋊Q8—61no—
164C22⋊Q8C2×D311yescharacteristic
165#121—31no—
166#121—61no—
167#120—61no—
168C4×Q8—31no—
169C4×D4—31no—
170C4×D4—31no—
171C4×D4—61no—
172C2×C22⋊C4—61no—
173#107—31no—
174C4⋊C8—31no—
175#103—31no—
176#103—31no—
177#103C2×D311yescharacteristic, complemented
178#103C2×D311yescharacteristic, complemented
179#102—31no—
180#102—31no—
181C22⋊C8—31no—
182C22⋊C8C2×D311yescharacteristic
183#97—31no—
184C23×C3C2×D411yescharacteristic
185C22×D3—21no—
186C22×D3—21no—
187C22×D3—42no—
188C22×D3—42no—
189C22×D3—81no—
190C22×D3C2×D411yescharacteristic
191C3×Q8—21no—
192C3×Q8—21no—
193C3×Q8—21no—
194C3×Q8—21no—
195C3×Q8—21no—
196C3×Q8—42no—
197C3×Q8C2×D412yes—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—42no—
205C2×C3×C4C2×D411yescharacteristic
206C2×C3×C4C2411yescharacteristic
207#67—22no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3C2×D411yescharacteristic
211C2×Dic3C2×D411yescharacteristic
212C2×Dic3C2×D411yescharacteristic
213D12—21no—
214D12—21no—
215D12—21no—
216D12—42no—
217D12—42no—
218D12—81no—
219D12C2×D412yes—
220C4×D3—21no—
221C4×D3—21no—
222C4×D3—42no—
223C4×D3—42no—
224C4×D3—81no—
225Dic6—21no—
226Dic6—21no—
227Dic6—42no—
228C3×C8—21no—
229C3×C8—21no—
230C3⋊C8—21no—
231C3⋊C8C2×D412yes—
232C24—61no—
233#41—62no—
234#41—121no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—31no—
238C2×Q8—41no—
239C2×Q8—61no—
240C2×Q8—62no—
241C2×Q8—121no—
242C2×Q8C22×D311yescharacteristic
243C2×Q8C22×D311yescharacteristic
244C2×Q8C22×D311yescharacteristic
245C2×D4—31no—
246C2×D4—31no—
247C2×D4—31no—
248C2×D4—62no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—121no—
253C22×C4—21no—
254C22×C4—31no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4C22×D311yescharacteristic
259Q16—22no—
260Q16—41no—
261Q16—62no—
262SD16—34no—
263SD16—62no—
264SD16—62no—
265SD16—62no—
266SD16—62no—
267SD16—121no—
268M16—62no—
269C2×C8—31no—
270C2×C8—31no—
271C2×C8—62no—
272C2×C8C22×D311yescharacteristic
273C2×C8C22×D311yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—31no—
276C4⋊C4—31no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4C22×D311yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—31no—
285C22⋊C4—31no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C42—31no—
293C42—61no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3#13712yescomplemented
297C22×C3C22×D411yescharacteristic
298C2×D3—21no—
299C2×D3—22no—
300C2×D3—21no—
301C2×D3—21no—
302C2×D3—41no—
303C2×D3—41no—
304C2×D3—41no—
305C2×D3—41no—
306C2×D3—41no—
307C2×D3—84no—
308C3×C4—22no—
309C3×C4—21no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4C22×D411yescharacteristic
314C3×C4H(5,𝔽2)11yescharacteristic
315Dic3—21no—
316Dic3—21no—
317Dic3—21no—
318Dic3—41no—
319Dic3—41no—
320C23—31no—
321C23—61no—
322C23—61no—
323C23—122no—
324C23—122no—
325C23—241no—
326C23D3×D411yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—61no—
334Q8—61no—
335Q8—122no—
336Q8D3×D412yes—
337D4—32no—
338D4—62no—
339D4—61no—
340D4—61no—
341D4—61no—
342D4—122no—
343D4—122no—
344D4—241no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—31no—
352C2×C4—31no—
353C2×C4—31no—
354C2×C4—42no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—241no—
362C2×C4D3×D411yescharacteristic
363C2×C4C23×D311yescharacteristic
364C8—21no—
365C8—21no—
366C8—32no—
367C8—61no—
368C2×C3—22no—
369C2×C3D4211yescharacteristic
370C2×C3C2×#13711yescharacteristic
371C2×C3#57711yescharacteristic
372D3—41no—
373D3—42no—
374D3—81no—
375C22—21no—
376C22—21no—
377C22—61no—
378C22—61no—
379C22—61no—
380C22—62no—
381C22—121no—
382C22—121no—
383C22—121no—
384C22—121no—
385C22—121no—
386C22—244no—
387C22#91812yescomplemented
388C22C2×D3×D411yescharacteristic
389C4—21no—
390C4—22no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—61no—
395C4—61no—
396C4—61no—
397C4—121no—
398C4—121no—
399C4C2×D3×D411yescharacteristic
400C4#100411yescharacteristic
401C3#326611yescharacteristic, complemented, Sylow
402C2—22no—
403C2—121no—
404C2—122no—
405C2—241no—
406C2#559011yescharacteristic
407C2C2×#91811yescharacteristic
408C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#451C2
socleC22×C3C22×D4
prime radical#5630C2
hypercenter#451D3
hypocenterC3#3266
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. Q8×#67C2semidirect
2. C22×#215C2semidirect
3. C2×#918C2semidirect
5. C3×#451C2semidirect
6. #5190C2semidirect
7. #5187C2semidirect
8. #5185C2semidirect
9. #5051C2semidirect
10. #4818C2semidirect
11. #4817C2semidirect
12. #4812C2semidirect
13. #4806C2semidirect
14. #4731C2semidirect
15. #4730C2semidirect
19. C2×D3×Q8C22semidirect
21. C2×C3×Q16C22semidirect
22. C2×C3×Q16C22semidirect
26. #946C22semidirect
27. Q8×Dic3C22semidirect
29. #944C22semidirect
30. C2×#216C22semidirect
44. C2×ΓL(1,𝔽25)C22semidirect
51. C3×#103C22semidirect
52. C3×#103C22semidirect
54. #835C22semidirect
57. (C2×D3)⋊C8C22semidirect
58. #815C22semidirect
59. Dic3⋊C8C22semidirect
60. #810C22semidirect
61. #807C22semidirect
72. #451D3semidirect
117. #215D4semidirect
145. C2×Q16C2×D3semidirect
146. C2×Q16C2×D3semidirect
177. #103C2×D3semidirect
178. #103C2×D3semidirect
296. C22×C3#137semidirect
387. C22#918semidirect
401. C3#3266semidirect

Maximal subgroups

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

subgroupindex
Q8×#672
C22×#2152
C2×#9182
#56302
C3×#4512
#51902
#51872
#51852
#50512
#48182
#48172
#48122
#48062
#47312
#47302
#32663