Finite Group Databasebeta

← #83923

Subgroups — #83923

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×C3×#135C211yesmaximal, characteristic, complemented
2#5834C211yesmaximal, characteristic, complemented
3#5831C211yesmaximal, characteristic, complemented
4C2×#918C211yesmaximal, characteristic, complemented
5#5239C211yesmaximal, characteristic, complemented
6#5238C211yesmaximal, characteristic, complemented
7#5184C211yesmaximal, characteristic, complemented
8#5172C211yesmaximal, characteristic, complemented
9#5170C211yesmaximal, characteristic, complemented
10#5166C211yesmaximal, characteristic, complemented
11#5163C211yesmaximal, characteristic, complemented
12#5159C211yesmaximal, characteristic, complemented
13#5154C211yesmaximal, characteristic, complemented
14#5114C211yesmaximal, characteristic, complemented
15#5111C211yesmaximal, characteristic, complemented
16#3271—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×C3×#41C2211yescharacteristic
19C2×#239C2211yescharacteristic
20C2×D3×Q8C2211yescharacteristic
21C3×#135—24no—
22C2×C3×Q16C2211yescharacteristic
23C2×C3×SD16C2211yescharacteristic
24C2×C3×SD16C2211yescharacteristic
25C2×C3×D8C2211yescharacteristic
26C22×C3×C8C2211yescharacteristic
27Q8×Dic3C2211yescharacteristic
28Q8×Dic3C2211yescharacteristic
29C2×#216C2211yescharacteristic
30C2×#215C2211yescharacteristic
31#940—21no—
32#940C2211yescharacteristic
33#938—21no—
34#938—21no—
35#938C2211yescharacteristic
36#938C2211yescharacteristic
37#936—21no—
38#936C2211yescharacteristic
39C2×#214C2211yescharacteristic
40C2×#213C2211yescharacteristic
41#930—21no—
42#930C2211yescharacteristic
43C4×#67—21no—
44C4×#67—21no—
45C4×#67C2211yescharacteristic
46#924—21no—
47#924C2211yescharacteristic
48#918—24no—
49C2×#204C2211yescharacteristic
50C2×ΓL(1,𝔽25)C2211yescharacteristic
51#835C2211yescharacteristic
52#835C2211yescharacteristic
53#832C2211yescharacteristic
54#832C2211yescharacteristic
55#830C2211yescharacteristic
56#821C2211yescharacteristic
57(C2×D3)⋊C8C2211yescharacteristic
58#817C2211yescharacteristic
59#816C2211yescharacteristic
60#815C2211yescharacteristic
61Dic3⋊C8C2211yescharacteristic
62C2×#137—31no—
63C2×#135D311yescharacteristic, complemented
64#550—31no—
65#548—31no—
66#496—31no—
67#469—31no—
68#465—31no—
69#464—31no—
70#460—31no—
71#452—31no—
72#449—31no—
73#449—31no—
74#448—31no—
75#441—31no—
76#435—31no—
77C3×#41—22no—
78C3×#41—22no—
79C3×#41—41no—
80C3×#41—41no—
81C2×C3×Q8C2311yescharacteristic
82C2×C3×Q8C2311yescharacteristic
83C2×C3×D4—21no—
84C2×C3×D4—21no—
85C2×C3×D4C2311yescharacteristic
86C2×C3×D4C2311yescharacteristic
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4C2311yescharacteristic
90C2×#67—21no—
91C2×#67—21no—
92C2×#67—21no—
93C2×#67—21no—
94C2×#67—21no—
95C2×#67D411yescharacteristic
96#239—22no—
97#239—41no—
98D3×Q8—22no—
99D3×Q8—41no—
100C2×D12—21no—
101C2×D12C2311yescharacteristic
102C2×C4×D3—21no—
103C2×C4×D3—21no—
104C2×C4×D3C2311yescharacteristic
105C2×Dic6—21no—
106C2×Dic6C2311yescharacteristic
107C3×Q16D414yescomplemented
108C3×SD16—22no—
109C3×SD16—22no—
110C3×D8—22no—
111C2×C3×C8—22no—
112C2×C3×C8C2311yescharacteristic
113C2×C3×C8C2311yescharacteristic
114#217—21no—
115#217—21no—
116#217—21no—
117#217—21no—
118#217—21no—
119#217D411yescharacteristic
120#216—22no—
121#215—22no—
122#214—41no—
123#213—41no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4D411yescharacteristic
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3C2311yescharacteristic
129C4⋊Dic3C2311yescharacteristic
130Dic3⋊C4—21no—
131Dic3⋊C4D411yescharacteristic
132C4×Dic3—21no—
133C4×Dic3—21no—
134C4×Dic3C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136C2×C3⋊C8C2311yescharacteristic
137#204—22no—
138ΓL(1,𝔽25)—22no—
139C2×#41—31no—
140C2×#41C2×D311yescharacteristic
141C2×#41C2×D311yescharacteristic
142C22×Q8—31no—
143#137—64no—
144#135—24no—
145C2×Q16—31no—
146C2×Q16C2×D311yescharacteristic
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16C2×D311yescharacteristic
151C2×SD16C2×D311yescharacteristic
152C2×D8—31no—
153C2×D8C2×D311yescharacteristic
154C2×M16—31no—
155C22×C8C2×D311yescharacteristic
156#127—31no—
157#127—61no—
158#124—31no—
159#124—61no—
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×Q8—31no—
170C4×D4—31no—
171C4×D4—61no—
172C4×D4—61no—
173#106—31no—
174C4⋊C8—31no—
175#103—31no—
176#103—31no—
177#103—31no—
178#102—31no—
179#102—31no—
180#102—31no—
181C22⋊C8—31no—
182C22⋊C8—31no—
183#97—31no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3C2×D411yescharacteristic
187C22×D3—21no—
188C22×D3C2×D411yescharacteristic
189C3×Q8—21no—
190C3×Q8—21no—
191C3×Q8C2×D412yes—
192C3×Q8C2×D412yes—
193C3×D4—21no—
194C3×D4—21no—
195C3×D4—21no—
196C3×D4—21no—
197C3×D4—42no—
198C3×D4—42no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4C2×D411yescharacteristic
206C2×C3×C4C2411yescharacteristic
207#67—41no—
208#67—42no—
209#67—42no—
210#67—81no—
211#67—81no—
212#67—81no—
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3C2×D411yescharacteristic
217C2×Dic3C2×D411yescharacteristic
218C2×Dic3C2×D411yescharacteristic
219D12—21no—
220D12—21no—
221D12—42no—
222C4×D3—21no—
223C4×D3—21no—
224C4×D3—42no—
225C4×D3—42no—
226Dic6—21no—
227Dic6—21no—
228Dic6—42no—
229C3×C8—21no—
230C3×C8C2×D412yes—
231C3⋊C8—21no—
232C3⋊C8—21no—
233#41—22no—
234#41—22no—
235#41—41no—
236#41—41no—
237#41—62no—
238#41—121no—
239C2×Q8—31no—
240C2×Q8—61no—
241C2×Q8—62no—
242C2×Q8—121no—
243C2×Q8C22×D311yescharacteristic
244C2×Q8C22×D311yescharacteristic
245C2×D4—21no—
246C2×D4—21no—
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×D4C22×D311yescharacteristic
256C2×D4C22×D311yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—31no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4C22×D311yescharacteristic
263Q16—62no—
264Q16#6714yescomplemented
265SD16—22no—
266SD16—22no—
267SD16—62no—
268SD16—62no—
269SD16—121no—
270D8—22no—
271D8—121no—
272M16—62no—
273C2×C8—22no—
274C2×C8—31no—
275C2×C8—31no—
276C2×C8C22×D311yescharacteristic
277C2×C8C22×D311yescharacteristic
278C4⋊C4—31no—
279C4⋊C4—31no—
280C4⋊C4—31no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—61no—
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—
294C42—61no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3C22×D411yescharacteristic
305C2×D3—21no—
306C2×D3—21no—
307C2×D3—21no—
308C2×D3—41no—
309C2×D3—41no—
310C2×D3—41no—
311C3×C4—22no—
312C3×C4—22no—
313C3×C4—21no—
314C3×C4C22×D411yescharacteristic
315C3×C4H(5,𝔽2)11yescharacteristic
316Dic3—21no—
317Dic3—21no—
318Dic3—21no—
319Dic3—41no—
320Dic3—41no—
321Dic3—41no—
322C23—21no—
323C23—21no—
324C23—31no—
325C23—61no—
326C23D3×D411yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—61no—
330Q8—61no—
331Q8—122no—
332Q8C2×#6712yes—
333Q8C2×#6712yes—
334D4—21no—
335D4—21no—
336D4—21no—
337D4—21no—
338D4—42no—
339D4—42no—
340D4—61no—
341D4—61no—
342D4—122no—
343D4—121no—
344D4—122no—
345D4—122no—
346D4—241no—
347D4—241no—
348D4—241no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—31no—
354C2×C4—31no—
355C2×C4—31no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4D3×D411yescharacteristic
366C2×C4C23×D311yescharacteristic
367C8—21no—
368C8—61no—
369C8—61no—
370C8C2×#6712yes—
371C2×C3—41no—
372C2×C3—81no—
373C2×C3—81no—
374C2×C3D4211yescharacteristic
375C2×C3#57511yescharacteristic
376C2×C3#57711yescharacteristic
377D3—41no—
378D3—81no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—61no—
389C22—61no—
390C22—61no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22C2×D3×D411yescharacteristic
395C4—21no—
396C4—22no—
397C4—22no—
398C4—61no—
399C4—61no—
400C4—61no—
401C4—121no—
402C4—121no—
403C4—121no—
404C4#100411yescharacteristic
405C4C22×#6711yescharacteristic
406C3#327111yescharacteristic, complemented, Sylow
407C2—41no—
408C2—81no—
409C2—81no—
410C2—121no—
411C2—241no—
412C2#576811yescharacteristic
413C2#577011yescharacteristic
414C2D4×#6711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#135C2
socleC22×C3C22×D4
hypercenterC2×#135D3
hypocenterC3#3271
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×C3×#135C2semidirect
2. #5834C2semidirect
3. #5831C2semidirect
4. C2×#918C2semidirect
5. #5239C2semidirect
6. #5238C2semidirect
7. #5184C2semidirect
8. #5172C2semidirect
9. #5170C2semidirect
10. #5166C2semidirect
11. #5163C2semidirect
12. #5159C2semidirect
13. #5154C2semidirect
14. #5114C2semidirect
15. #5111C2semidirect
63. C2×#135D3semidirect
107. C3×Q16D4semidirect
264. Q16#67semidirect
406. C3#3271semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1352
#58342
#58312
C2×#9182
#52392
#52382
#51842
#51722
#51702
#51662
#51632
#51592
#51542
#51142
#51112
#32713