Finite Group Databasebeta

← #86737

Subgroups — #86737

Order 384.

1492 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1010C211yesmaximal, characteristic, complemented
2C2×#1007C211yesmaximal, characteristic, complemented
3C3×#548C211yesmaximal, characteristic, complemented
4#5834C211yesmaximal, characteristic, complemented
5Q8×#67C211yesmaximal, characteristic, complemented
6#5677C213yesmaximal, complemented
7#5668C213yesmaximal, complemented
8#5632C213yesmaximal, complemented
9#5629C213yesmaximal, complemented
10#5608C213yesmaximal, complemented
11#5599C213yesmaximal, complemented
12#5576C211yesmaximal, characteristic, complemented
13C4×D3×Q8C211yesmaximal, characteristic, complemented
14#5552C213yesmaximal, complemented
15#5548C213yesmaximal, complemented
16#3497—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic
19#1010—28no—
20#1007—28no—
21C2×#239C2213yes—
22C2×#239C2211yescharacteristic
23C2×D3×Q8C2213yes—
24C2×D3×Q8C2211yescharacteristic
25C2×D3×Q8C2211yescharacteristic
26C2×#237C2213yes—
27C2×#237C2213yes—
28C2×#235C2213yes—
29C2×#235C2213yes—
30C2×#235C2213yes—
31C22×Dic6C2213yes—
32C3×#124C2213yes—
33C3×C22⋊Q8C2213yes—
34C3×#121C2213yes—
35C3×C4×Q8C2211yescharacteristic
36C3×C4×D4C2213yes—
37#946C2213yes—
38Q8×Dic3C2211yescharacteristic
39#944C2213yes—
40#940C2213yes—
41#940C2213yes—
42#939C2216yes—
43#938C2213yes—
44#938C2213yes—
45#930C2213yes—
46C4×#67C2213yes—
47C4×#67C2213yes—
48#924C2213yes—
49#897C2213yes—
50#896C2216yes—
51#895C2213yes—
52#895C2213yes—
53D3×C4⋊C4C2213yes—
54#888C2213yes—
55#887C2213yes—
56#885C2213yes—
57#885C2213yes—
58#884C2213yes—
59#884C2216yes—
60#884C2216yes—
61#884C2211yescharacteristic
62#67⋊C4C2216yes—
63#67⋊C4C2216yes—
64#878C2216yes—
65#875C2213yes—
66C4×D12C2213yes—
67C42×D3C2213yes—
68C4×Dic6C2213yes—
69C2×2−1+4—31no—
70C2×2−1+4—31no—
71#550—31no—
72#550—33no—
73D4×Q8—31no—
74D4×Q8—33no—
75#548—33no—
76#548—31no—
77#548—33no—
78#548D311yescharacteristic, complemented
79#537—33no—
80#537—33no—
81#532—33no—
82#532—33no—
83C2×C4×Q8—31no—
84C3×#41—24no—
85C2×C3×Q8—24no—
86C2×C3×Q8C2313yes—
87C2×C3×Q8C2311yescharacteristic
88C2×C3×D4C2313yes—
89C2×C3×D4C2313yes—
90C22×C3×C4C2313yes—
91C22×C3×C4C2313yes—
92C2×#67C2313yes—
93C2×#67C2313yes—
94C2×#67C2313yes—
95C2×#67C2311yescharacteristic
96C2×#67C2311yescharacteristic
97C2×#67C2313yes—
98C2×#67C2316yes—
99C22×Dic3C2316yes—
100#239—24no—
101#239—212no—
102D3×Q8—24no—
103D3×Q8—212no—
104D3×Q8D418yescomplemented
105#237—212no—
106#237—212no—
107#235—212no—
108#235—212no—
109#235—212no—
110C2×D12C2313yes—
111C2×D12C2313yes—
112C2×D12C2311yescharacteristic
113C2×D12C2313yes—
114C2×C4×D3C2311yescharacteristic
115C2×C4×D3C2313yes—
116C2×C4×D3C2313yes—
117C2×C4×D3C2316yes—
118C2×C4×D3C2313yes—
119C2×C4×D3C2313yes—
120C2×Dic6—212no—
121C2×Dic6C2313yes—
122C2×Dic6C2313yes—
123C2×Dic6C2313yes—
124C2×Dic6C2313yes—
125C3×C4⋊C4C2313yes—
126C3×C4⋊C4C2313yes—
127C3×C22⋊C4C2316yes—
128C3×C22⋊C4C2313yes—
129C3×C22⋊C4C2311yescharacteristic
130C3×C42C2313yes—
131#217C2313yes—
132#217C2313yes—
133(C2×D3)⋊C4C2311yescharacteristic
134(C2×D3)⋊C4C2313yes—
135(C2×D3)⋊C4C2316yes—
136(C2×D3)⋊C4C2313yes—
137(C2×D3)⋊C4C2313yes—
138C4⋊Dic3C2313yes—
139C4⋊Dic3C2313yes—
140Dic3⋊C4C2316yes—
141Dic3⋊C4C2316yes—
142Dic3⋊C4C2313yes—
143Dic3⋊C4C2311yescharacteristic
144C4×Dic3C2313yes—
145C4×Dic3C2313yes—
146C4×Dic3C2313yes—
1472−1+4—68no—
1482−1+4—68no—
149C2×#41—33no—
150C2×#41—31no—
151C2×#41—33no—
152C2×#41—33no—
153C2×#41—33no—
154C2×#41—33no—
155C2×#41—33no—
156C2×#41C2×D311yescharacteristic
157C22×Q8—31no—
158C22×Q8—33no—
159C22×Q8—31no—
160C22×Q8—33no—
161C22×Q8C2×D311yescharacteristic
162C4⋊Q8—33no—
163C4⋊Q8—33no—
164#127—33no—
165#127—33no—
166#124—33no—
167#124—33no—
168#124—33no—
169#124C2×D313yes—
170C22⋊Q8—33no—
171C22⋊Q8—33no—
172C22⋊Q8—36no—
173C22⋊Q8—33no—
174C22⋊Q8—36no—
175C22⋊Q8C2×D313yes—
176#121—33no—
177#121—33no—
178#121—36no—
179#121—33no—
180#121—33no—
181#121—33no—
182#121—33no—
183#121—31no—
184#121—36no—
185#121—36no—
186#121C2×D313yes—
187C4×Q8—31no—
188C4×Q8—33no—
189C4×Q8—33no—
190C4×Q8C2×D311yescharacteristic
191C4×D4—33no—
192C4×D4—33no—
193C4×D4—36no—
194C4×D4—36no—
195C4×D4—33no—
196C4×D4C2×D313yes—
197C2×C4⋊C4—33no—
198C2×C42—33no—
199C23×C3C2411yescharacteristic
200C23×C3C2413yes—
201C22×D3C2413yes—
202C22×D3C2411yescharacteristic
203C22×D3C2411yescharacteristic
204C3×Q8—26no—
205C3×Q8C2×D414yes—
206C3×D4—26no—
207C3×D4—43no—
208C2×C3×C4—26no—
209C2×C3×C4—26no—
210C2×C3×C4C2411yescharacteristic
211C2×C3×C4C2413yes—
212C2×C3×C4C2413yes—
213C2×C3×C4C2411yescharacteristic
214C2×C3×C4C2413yes—
215#67—22no—
216#67—26no—
217#67—26no—
218#67—26no—
219#67—26no—
220#67—22no—
221#67—46no—
222C2×Dic3—212no—
223C2×Dic3C2413yes—
224C2×Dic3C2413yes—
225C2×Dic3C2411yescharacteristic
226C2×Dic3C2413yes—
227C2×Dic3C2411yescharacteristic
228D12—22no—
229D12—26no—
230D12—26no—
231D12—43no—
232C4×D3—26no—
233C4×D3—26no—
234C4×D3—212no—
235C4×D3—22no—
236C4×D3—43no—
237C4×D3C2×D4112yes—
238Dic6—26no—
239Dic6—26no—
240Dic6—26no—
241Dic6C2×D4112yes—
242#41—24no—
243#41—612no—
244#41—612no—
245#41—612no—
246#41—612no—
247#41—612no—
248#41—612no—
249#41—64no—
250C2×Q8—24no—
251C2×Q8—38no—
252C2×Q8—33no—
253C2×Q8—33no—
254C2×Q8—33no—
255C2×Q8—33no—
256C2×Q8—612no—
257C2×Q8—612no—
258C2×Q8—64no—
259C2×Q8C22×D313yes—
260C2×Q8C22×D311yescharacteristic
261C2×D4—31no—
262C2×D4—33no—
263C2×D4—31no—
264C2×D4—33no—
265C2×D4—33no—
266C2×D4—33no—
267C2×D4—36no—
268C2×D4—31no—
269C2×D4—33no—
270C2×D4—33no—
271C2×D4—33no—
272C2×D4C22×D313yes—
273C2×D4C22×D313yes—
274C22×C4—33no—
275C22×C4—36no—
276C22×C4—31no—
277C22×C4—36no—
278C22×C4—33no—
279C22×C4—33no—
280C22×C4—33no—
281C22×C4C22×D313yes—
282C22×C4C22×D313yes—
283C4⋊C4—36no—
284C4⋊C4—33no—
285C4⋊C4—31no—
286C4⋊C4—33no—
287C4⋊C4—36no—
288C4⋊C4—33no—
289C4⋊C4C22×D313yes—
290C4⋊C4C22×D313yes—
291C22⋊C4—33no—
292C22⋊C4—33no—
293C22⋊C4—33no—
294C22⋊C4—31no—
295C22⋊C4—33no—
296C22⋊C4—33no—
297C22⋊C4—36no—
298C22⋊C4C22×D313yes—
299C22⋊C4C22×D316yes—
300C22⋊C4C22×D311yescharacteristic
301C42—33no—
302C42—33no—
303C42—33no—
304C42C22×D313yes—
305C22×C3—23no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—23no—
309C22×C3—23no—
310C22×C3—21no—
311C22×C3C2511yescharacteristic
312C2×D3—21no—
313C2×D3—21no—
314C2×D3—23no—
315C2×D3—21no—
316C2×D3—23no—
317C2×D3—21no—
318C2×D3—21no—
319C2×D3—23no—
320C2×D3C22×D412yes—
321C3×C4—21no—
322C3×C4—23no—
323C3×C4—21no—
324C3×C4—23no—
325C3×C4C22×D416yes—
326Dic3—21no—
327Dic3—23no—
328Dic3—23no—
329Dic3C22×D416yes—
330Dic32−1+412yescomplemented
331C23—31no—
332C23—31no—
333C23—33no—
334C23C23×D311yescharacteristic
335C23C23×D313yes—
336Q8—26no—
337Q8—312no—
338Q8—66no—
339Q8—66no—
340Q8—66no—
341Q8D3×D414yes—
342D4—26no—
343D4—43no—
344D4—62no—
345D4—66no—
346D4—66no—
347D4—66no—
348D4—62no—
349D4—66no—
350D4—62no—
351D4—66no—
352D4—66no—
353D4—123no—
354D4—126no—
355C2×C4—26no—
356C2×C4—26no—
357C2×C4—31no—
358C2×C4—33no—
359C2×C4—312no—
360C2×C4—33no—
361C2×C4—31no—
362C2×C4—33no—
363C2×C4—612no—
364C2×C4—62no—
365C2×C4—612no—
366C2×C4—66no—
367C2×C4—66no—
368C2×C4—123no—
369C2×C4C23×D311yescharacteristic
370C2×C4C23×D313yes—
371C2×C4C23×D311yescharacteristic
372C2×C4C23×D313yes—
373C2×C4C23×D313yes—
374C2×C3—43no—
375C2×C3—41no—
376C2×C3C23×D411yescharacteristic
377C2×C3C2×2−1+411yescharacteristic
378C2×C3#58511yescharacteristic
379D3—22no—
380D3—43no—
381D3—41no—
382C22—21no—
383C22—23no—
384C22—21no—
385C22—23no—
386C22—21no—
387C22—23no—
388C22—32no—
389C22—61no—
390C22—61no—
391C22—63no—
392C22—61no—
393C22—61no—
394C22—63no—
395C22—61no—
396C22—63no—
397C22C24×D311yescharacteristic
398C4—21no—
399C4—23no—
400C4—23no—
401C4—21no—
402C4—36no—
403C4—32no—
404C4—63no—
405C4—61no—
406C4—63no—
407C4C2×D3×D416yes—
408C3#349711yescharacteristic, complemented, Sylow
409C2—41no—
410C2—43no—
411C2—62no—
412C2—123no—
413C2—121no—
414C2C22×D3×D411yescharacteristic
415C2#596411yescharacteristic
416C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#548C2
hypercenter#548D3
hypocenterC3#3497
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#1010C2semidirect
2. C2×#1007C2semidirect
3. C3×#548C2semidirect
4. #5834C2semidirect
5. Q8×#67C2semidirect
6. #5677C2semidirect
7. #5668C2semidirect
8. #5632C2semidirect
9. #5629C2semidirect
10. #5608C2semidirect
11. #5599C2semidirect
12. #5576C2semidirect
13. C4×D3×Q8C2semidirect
14. #5552C2semidirect
15. #5548C2semidirect
78. #548D3semidirect
104. D3×Q8D4semidirect
330. Dic32−1+4semidirect
408. C3#3497semidirect

Maximal subgroups

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

subgroupindex
C2×#10102
C2×#10072
C3×#5482
#58342
Q8×#672
#56772
#56682
#56322
#56292
#56082
#55992
#55762
C4×D3×Q82
#55522
#55482
#34973