Finite Group Databasebeta

← #80688

Subgroups — #80688

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×#915C211yesmaximal, characteristic, complemented
2C2×#904C211yesmaximal, characteristic, complemented
3#5633C211yesmaximal, characteristic, complemented
4#5599C211yesmaximal, characteristic, complemented
5C3×#465C211yesmaximal, characteristic, complemented
6#5176C211yesmaximal, characteristic, complemented
7#5111C211yesmaximal, characteristic, complemented
8#5047C211yesmaximal, characteristic, complemented
9#5037C211yesmaximal, characteristic, complemented
10#4865C211yesmaximal, characteristic, complemented
11#4853C211yesmaximal, characteristic, complemented
12#4806C211yesmaximal, characteristic, complemented
13#4798C211yesmaximal, characteristic, complemented
14#4781C211yesmaximal, characteristic, complemented
15#4762C211yesmaximal, characteristic, complemented
16#3271—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C2×#235C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C22×C3×C8C2211yescharacteristic
23C3×C22⋊Q8C2211yescharacteristic
24C3×#121C2211yescharacteristic
25#947C2211yescharacteristic
26C2×#216C2211yescharacteristic
27C2×#215C2211yescharacteristic
28#940—21no—
29#940C2211yescharacteristic
30#939—21no—
31C2×#214C2211yescharacteristic
32C2×#213C2211yescharacteristic
33C4×#67C2211yescharacteristic
34#915—24no—
35C2×Dic12C2211yescharacteristic
36#904—24no—
37C2×D24C2211yescharacteristic
38C2×#204C2211yescharacteristic
39C2×#204C2211yescharacteristic
40C2×ΓL(1,𝔽25)C2211yescharacteristic
41#896—21no—
42#895C2211yescharacteristic
43#895C2211yescharacteristic
44#893—21no—
45#887C2211yescharacteristic
46#887C2211yescharacteristic
47#885—21no—
48#884—21no—
49#884—21no—
50#67⋊C4—21no—
51C3×#106C2211yescharacteristic
52C3×#103C2211yescharacteristic
53C3×#102C2211yescharacteristic
54#830C2211yescharacteristic
55(C2×D3)⋊C8C2211yescharacteristic
56#815C2211yescharacteristic
57Dic3⋊C8C2211yescharacteristic
58#810C2211yescharacteristic
59#810C2211yescharacteristic
60#809C2211yescharacteristic
61#809C2211yescharacteristic
62C2×#137—31no—
63C2×#135—31no—
64#550—31no—
65#548—31no—
66#496—31no—
67#469—31no—
68#465D311yescharacteristic, complemented
69#464—31no—
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×D4—21no—
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67D411yescharacteristic
87C22×Dic3—21no—
88D3×Q8—22no—
89D3×Q8—41no—
90#237—22no—
91#237—41no—
92#235—22no—
93#235—22no—
94#235—41no—
95#235—41no—
96C2×D12—21no—
97C2×D12—21no—
98C2×D12C2311yescharacteristic
99C2×D12C2311yescharacteristic
100C2×C4×D3—21no—
101C2×C4×D3—21no—
102C2×C4×D3—21no—
103C2×C4×D3C2311yescharacteristic
104C2×Dic6—21no—
105C2×Dic6C2311yescharacteristic
106C2×Dic6C2311yescharacteristic
107C3×SD16—22no—
108C2×C3×C8—22no—
109C2×C3×C8C2311yescharacteristic
110C2×C3×C8C2311yescharacteristic
111C3×C4⋊C4—21no—
112C3×C4⋊C4C2311yescharacteristic
113C3×C4⋊C4C2311yescharacteristic
114C3×C22⋊C4—21no—
115C3×C22⋊C4—21no—
116#217D411yescharacteristic
117#216—22no—
118#215—41no—
119#214—22no—
120#213—41no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4D411yescharacteristic
126Dic3⋊C4—21no—
127Dic3⋊C4—21no—
128Dic3⋊C4D411yescharacteristic
129C4×Dic3—21no—
130C4×Dic3—21no—
131C4×Dic3C2311yescharacteristic
132C2×C3⋊C8C2311yescharacteristic
133C2×C3⋊C8C2311yescharacteristic
134Dic12D414yescomplemented
135D24—22no—
136#204—22no—
137#204—22no—
138ΓL(1,𝔽25)—22no—
139C2×#41—31no—
140C2×#41—31no—
141C2×#41—31no—
142C22×Q8—31no—
143#137—64no—
144#135—64no—
145C2×Q16—31no—
146C2×Q16—31no—
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×C8C2×D311yescharacteristic
156#127—31no—
157#127—61no—
158#124—31no—
159#124—61no—
160C22⋊Q8—61no—
161C22⋊Q8C2×D311yescharacteristic
162#121—31no—
163#121—31no—
164#121—61no—
165#121—61no—
166#121—61no—
167#121C2×D311yescharacteristic
168C4×Q8—31no—
169C4×Q8—31no—
170C4×D4—31no—
171C4×D4—61no—
172C4×D4—61no—
173#106C2×D311yescharacteristic
174C4⋊C8—31no—
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—
185C23×C3C2×D411yescharacteristic
186C22×D3—21no—
187C22×D3—21no—
188C22×D3C2×D411yescharacteristic
189C3×Q8—21no—
190C3×Q8—21no—
191C3×D4—21no—
192C3×D4—21no—
193C3×D4—81no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2411yescharacteristic
201#67—41no—
202#67—42no—
203#67—42no—
204#67—42no—
205#67—42no—
206#67—81no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—42no—
210C2×Dic3C2×D411yescharacteristic
211C2×Dic3C2×D411yescharacteristic
212C2×Dic3C2×D411yescharacteristic
213D12—21no—
214D12—21no—
215D12—21no—
216D12—21no—
217D12—42no—
218D12—81no—
219C4×D3—21no—
220C4×D3—21no—
221C4×D3—42no—
222C4×D3—42no—
223C4×D3—42no—
224Dic6—21no—
225Dic6—21no—
226Dic6—42no—
227Dic6C2×D412yes—
228Dic6C2×D412yes—
229C3×C8—21no—
230C3×C8C2×D412yes—
231C3⋊C8—21no—
232C3⋊C8—21no—
233#41—62no—
234#41—62no—
235#41—62no—
236#41—121no—
237#41—121no—
238#41—121no—
239C2×Q8—31no—
240C2×Q8—31no—
241C2×Q8—61no—
242C2×Q8—62no—
243C2×Q8—121no—
244C2×Q8C22×D311yescharacteristic
245C2×D4—21no—
246C2×D4—31no—
247C2×D4—31no—
248C2×D4—31no—
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—61no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4C22×D311yescharacteristic
263Q16—34no—
264Q16—62no—
265SD16—22no—
266SD16—62no—
267SD16—62no—
268SD16—62no—
269SD16—121no—
270D8—62no—
271D8—121no—
272M16—62no—
273C2×C8—22no—
274C2×C8—31no—
275C2×C8—31no—
276C2×C8C22×D311yescharacteristic
277C2×C8C22×D311yescharacteristic
278C4⋊C4—21no—
279C4⋊C4—31no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4C22×D311yescharacteristic
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—31no—
293C42—61no—
294C42—61no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3C22×D411yescharacteristic
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—21no—
312C3×C4—41no—
313C3×C4—41no—
314C3×C4—41no—
315C3×C4C22×D411yescharacteristic
316C3×C4H(5,𝔽2)11yescharacteristic
317Dic3—22no—
318Dic3—22no—
319Dic3—21no—
320Dic3—21no—
321Dic3—21no—
322C23—21no—
323C23—31no—
324C23—61no—
325C23—61no—
326C23D3×D411yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—32no—
330Q8—32no—
331Q8—61no—
332Q8—61no—
333Q8—122no—
334D4—21no—
335D4—21no—
336D4—61no—
337D4—61no—
338D4—61no—
339D4—61no—
340D4—81no—
341D4—122no—
342D4—121no—
343D4—122no—
344D4—122no—
345D4—122no—
346D4—122no—
347D4—241no—
348D4—241no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—31no—
355C2×C4—31no—
356C2×C4—31no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4D3×D411yescharacteristic
366C2×C4C23×D311yescharacteristic
367C8—21no—
368C8—61no—
369C8—61no—
370C8D3×D412yes—
371C2×C3—41no—
372C2×C3—81no—
373C2×C3D4211yescharacteristic
374C2×C3#57511yescharacteristic
375C2×C3#57711yescharacteristic
376D3—41no—
377D3—81no—
378D3—81no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—61no—
386C22—61no—
387C22—61no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22C2×D3×D411yescharacteristic
395C4—21no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—61no—
400C4—61no—
401C4—61no—
402C4—62no—
403C4—62no—
404C4C2×D3×D411yescharacteristic
405C4#100911yescharacteristic
406C3#327111yescharacteristic, complemented, Sylow
407C2—41no—
408C2—81no—
409C2—121no—
410C2—241no—
411C2—241no—
412C2#560911yescharacteristic
413C2#576211yescharacteristic
414C2#576811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#465C2
socleC22×C3C22×D4
hypercenter#465D3
hypocenterC3#3271
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#915C2semidirect
2. C2×#904C2semidirect
3. #5633C2semidirect
4. #5599C2semidirect
5. C3×#465C2semidirect
6. #5176C2semidirect
7. #5111C2semidirect
8. #5047C2semidirect
9. #5037C2semidirect
10. #4865C2semidirect
11. #4853C2semidirect
12. #4806C2semidirect
13. #4798C2semidirect
14. #4781C2semidirect
15. #4762C2semidirect
68. #465D3semidirect
134. Dic12D4semidirect
406. C3#3271semidirect

Maximal subgroups

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

subgroupindex
C2×#9152
C2×#9042
#56332
#55992
C3×#4652
#51762
#51112
#50472
#50372
#48652
#48532
#48062
#47982
#47812
#47622
#32713