Finite Group Databasebeta

← #86340

Subgroups — #86340

Order 384.

1476 conjugacy classes of subgroups.

The subgroup lattice has 396 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×#536C211yesmaximal, characteristic, complemented
4#5832C211yesmaximal, characteristic, complemented
5C2×#944C211yesmaximal, characteristic, complemented
6#5724C212yesmaximal, complemented
7D3×C4⋊Q8C212yesmaximal, complemented
8#5675C212yesmaximal, complemented
9#5668C212yesmaximal, complemented
10#5655C218yesmaximal, complemented
11#5632C214yesmaximal, complemented
12#5631C214yesmaximal, complemented
13#5526C211yesmaximal, characteristic, complemented
14#5522C211yesmaximal, characteristic, complemented
15#3499—31nomaximal, Sylow
16C2×C3×#41C2211yescharacteristic
17C22×C3×Q8C2211yescharacteristic
18#1010—28no—
19#1007—28no—
20C2×#239C2212yes—
21C2×#239C2212yes—
22C2×D3×Q8C2212yes—
23C2×D3×Q8C2212yes—
24C2×D3×Q8C2212yes—
25C2×#237C2214yes—
26C2×#237C2212yes—
27C2×#235C2214yes—
28C2×#235C2211yescharacteristic
29C2×#235C2212yes—
30C2×#235C2211yescharacteristic
31C22×Dic6C2212yes—
32C22×Dic6C2211yescharacteristic
33C3×C4⋊Q8C2212yes—
34C3×#124C2212yes—
35C3×#123C2214yes—
36C3×C22⋊Q8C2214yes—
37C3×#117C2211yescharacteristic
38#947C2211yescharacteristic
39#944C2214yes—
40#944C2212yes—
41#944C2212yes—
42#944C2211yescharacteristic
43#940C2211yescharacteristic
44#939C2214yes—
45#936C2211yescharacteristic
46C4×#67C2214yes—
47C2×Dic3⋊C4C2214yes—
48C2×C4×Dic3C2211yescharacteristic
49#897C2214yes—
50#896C2218yes—
51#894C2218yes—
52#893C2212yes—
53D3×C4⋊C4C2218yes—
54#888C2218yes—
55#887C2212yes—
56#886C2214yes—
57#884C2218yes—
58#883C2218yes—
59#67⋊C4C2218yes—
60#67⋊C4C2214yes—
61#878C2218yes—
62#875C2212yes—
63#872C2212yes—
64C42×D3C2212yes—
65#869C2212yes—
66C2×2−1+4—31no—
67C2×2−1+4—31no—
68D4×Q8—34no—
69#547—38no—
70#547—34no—
71#536—32no—
72#536—32no—
73#536—31no—
74#536D311yescharacteristic, complemented
75#532—31no—
76#532—32no—
77C2×C4⋊Q8—31no—
78C2×C4⋊Q8—32no—
79C4×#41—31no—
80C3×#41—24no—
81C2×C3×Q8—24no—
82C2×C3×Q8C2312yes—
83C2×C3×Q8C2312yes—
84C2×C3×Q8C2311yescharacteristic
85C2×C3×D4C2311yescharacteristic
86C2×C3×D4C2312yes—
87C22×C3×C4C2312yes—
88C22×C3×C4C2312yes—
89C22×C3×C4C2311yescharacteristic
90C2×#67C2314yes—
91C2×#67C2314yes—
92C2×#67C2312yes—
93C2×#67C2312yes—
94C22×Dic3C2314yes—
95C22×Dic3C2312yes—
96C22×Dic3C2312yes—
97#239—28no—
98#239—28no—
99D3×Q8—28no—
100D3×Q8—28no—
101D3×Q8—28no—
102#237—216no—
103#237—28no—
104#235—24no—
105#235—216no—
106#235—28no—
107#235D418yescomplemented
108C2×D12C2311yescharacteristic
109C2×D12C2314yes—
110C2×D12C2311yescharacteristic
111C2×C4×D3C2314yes—
112C2×C4×D3C2314yes—
113C2×C4×D3C2314yes—
114C2×C4×D3C2314yes—
115C2×C4×D3C2312yes—
116C2×C4×D3C2312yes—
117C2×Dic6—24no—
118C2×Dic6—28no—
119C2×Dic6C2311yescharacteristic
120C2×Dic6C2312yes—
121C2×Dic6C2314yes—
122C2×Dic6C2314yes—
123C2×Dic6C2312yes—
124C2×Dic6C2311yescharacteristic
125C3×C4⋊C4C2312yes—
126C3×C4⋊C4C2318yes—
127C3×C22⋊C4C2318yes—
128C3×C22⋊C4C2312yes—
129C3×C42C2312yes—
130#217C2314yes—
131(C2×D3)⋊C4C2314yes—
132(C2×D3)⋊C4C2318yes—
133(C2×D3)⋊C4C2314yes—
134C4⋊Dic3C2318yes—
135Dic3⋊C4C2314yes—
136Dic3⋊C4C2318yes—
137Dic3⋊C4C2318yes—
138Dic3⋊C4C2314yes—
139C4×Dic3C2311yescharacteristic
140C4×Dic3C2314yes—
141C4×Dic3C2311yescharacteristic
142C4×Dic3C2312yes—
1432−1+4—68no—
1442−1+4—68no—
145C2×#41—32no—
146C2×#41—32no—
147C2×#41—32no—
148C2×#41—34no—
149C2×#41—31no—
150C2×#41—31no—
151C2×#41—32no—
152C2×#41—34no—
153C2×#41C2×D311yescharacteristic
154C22×Q8—32no—
155C22×Q8—32no—
156C22×Q8—32no—
157C22×Q8—31no—
158C22×Q8—32no—
159C22×Q8C2×D311yescharacteristic
160C4⋊Q8—31no—
161C4⋊Q8—32no—
162C4⋊Q8—32no—
163C4⋊Q8—34no—
164C4⋊Q8—38no—
165C4⋊Q8—32no—
166C4⋊Q8C2×D312yes—
167#127—31no—
168#124—31no—
169#124—31no—
170#124—32no—
171#124C2×D312yes—
172#123—38no—
173#123—38no—
174#123—34no—
175#123C2×D314yes—
176C22⋊Q8—38no—
177C22⋊Q8—34no—
178C22⋊Q8—38no—
179C22⋊Q8C2×D314yes—
180#121—34no—
181#121—38no—
182C4×Q8—32no—
183C4×D4—34no—
184C4×D4—32no—
185C4×D4—34no—
186C4×D4—38no—
187#117—32no—
188#117C2×D311yescharacteristic
189C2×C4⋊C4—34no—
190C2×C4⋊C4—38no—
191C2×C42—31no—
192C2×C42—32no—
193C23×C3C2412yes—
194C23×C3C2411yescharacteristic
195C22×D3C2412yes—
196C22×D3C2412yes—
197C3×Q8—22no—
198C3×Q8—24no—
199C3×Q8—24no—
200C3×D4—24no—
201C3×D4—22no—
202C2×C3×C4—24no—
203C2×C3×C4—24no—
204C2×C3×C4C2×D414yes—
205C2×C3×C4C2414yes—
206C2×C3×C4C2414yes—
207C2×C3×C4C2412yes—
208C2×C3×C4C2411yescharacteristic
209C2×C3×C4C2411yescharacteristic
210#67—24no—
211#67—28no—
212#67—28no—
213#67C2×D418yes—
214C2×Dic3—24no—
215C2×Dic3—28no—
216C2×Dic3—24no—
217C2×Dic3C2412yes—
218C2×Dic3C2414yes—
219C2×Dic3C2412yes—
220C2×Dic3C2412yes—
221C2×Dic3C2412yes—
222D12—28no—
223D12—22no—
224D12C2×D414yes—
225C4×D3—28no—
226C4×D3—28no—
227C4×D3—24no—
228C4×D3—28no—
229C4×D3—28no—
230C4×D3C2×D418yes—
231Dic6—24no—
232Dic6—24no—
233Dic6—22no—
234Dic6—28no—
235Dic6—28no—
236Dic6C2×D414yes—
237#41—24no—
238#41—38no—
239#41—616no—
240#41—68no—
241#41—68no—
242#41—68no—
243#41—616no—
244#41—68no—
245#41—64no—
246C2×Q8—24no—
247C2×Q8—31no—
248C2×Q8—34no—
249C2×Q8—34no—
250C2×Q8—32no—
251C2×Q8—31no—
252C2×Q8—32no—
253C2×Q8—68no—
254C2×Q8—68no—
255C2×Q8—68no—
256C2×Q8—68no—
257C2×Q8—64no—
258C2×Q8C22×D312yes—
259C2×Q8C22×D311yescharacteristic
260C2×Q8C22×D312yes—
261C2×D4—32no—
262C2×D4—34no—
263C2×D4—34no—
264C2×D4—32no—
265C2×D4—31no—
266C2×D4—34no—
267C2×D4—31no—
268C2×D4C22×D312yes—
269C2×D4C22×D311yescharacteristic
270C22×C4—32no—
271C22×C4—32no—
272C22×C4—32no—
273C22×C4—34no—
274C22×C4—34no—
275C22×C4—34no—
276C22×C4—34no—
277C22×C4—34no—
278C22×C4—32no—
279C22×C4C22×D312yes—
280C22×C4C22×D312yes—
281C22×C4C22×D311yescharacteristic
282C4⋊C4—34no—
283C4⋊C4—38no—
284C4⋊C4—38no—
285C4⋊C4—38no—
286C4⋊C4—34no—
287C4⋊C4C22×D318yes—
288C4⋊C4C22×D312yes—
289C22⋊C4—34no—
290C22⋊C4—34no—
291C22⋊C4—34no—
292C22⋊C4—38no—
293C22⋊C4C22×D312yes—
294C22⋊C4C22×D318yes—
295C42—31no—
296C42—31no—
297C42—34no—
298C42—32no—
299C42C22×D312yes—
300C22×C3—22no—
301C22×C3—22no—
302C22×C3—24no—
303C22×C3C22×D412yes—
304C22×C3C2511yescharacteristic
305C2×D3—24no—
306C2×D3—22no—
307C2×D3—24no—
308C2×D3C22×D414yes—
309C3×C4—22no—
310C3×C4—24no—
311C3×C4—24no—
312C3×C4C22×D412yes—
313C3×C4C22×D412yes—
314Dic3—22no—
315Dic3—24no—
316Dic3—22no—
317Dic3C22×D414yes—
318Dic32−1+414yescomplemented
319C23—32no—
320C23—32no—
321C23C23×D312yes—
322C23C23×D311yescharacteristic
323Q8—24no—
324Q8—24no—
325Q8—22no—
326Q8—34no—
327Q8—68no—
328Q8—68no—
329Q8—64no—
330Q8—62no—
331Q8—64no—
332D4—24no—
333D4—22no—
334D4—38no—
335D4—34no—
336D4—68no—
337D4—68no—
338D4—68no—
339D4—64no—
340D4—62no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—32no—
344C2×C4—32no—
345C2×C4—38no—
346C2×C4—32no—
347C2×C4—32no—
348C2×C4—34no—
349C2×C4—68no—
350C2×C4—68no—
351C2×C4—68no—
352C2×C4—68no—
353C2×C4—68no—
354C2×C4—64no—
355C2×C4—64no—
356C2×C4—64no—
357C2×C4D3×D414yes—
358C2×C4C23×D314yes—
359C2×C4C23×D312yes—
360C2×C4C23×D314yes—
361C2×C4C23×D311yescharacteristic
362C2×C4C23×D311yescharacteristic
363C2×C3—22no—
364C2×C3—42no—
365C2×C3C23×D411yescharacteristic
366C2×C3C2×2−1+412yes—
367D3—24no—
368D3—42no—
369C22—22no—
370C22—22no—
371C22—24no—
372C22—34no—
373C22—64no—
374C22—62no—
375C22—64no—
376C22C2×D3×D412yes—
377C22C24×D311yescharacteristic
378C4—22no—
379C4—24no—
380C4—24no—
381C4—34no—
382C4—34no—
383C4—62no—
384C4—64no—
385C4—62no—
386C4C2×D3×D412yes—
387C4C2×D3×D412yes—
388C3#349911yescharacteristic, complemented, Sylow
389C2—22no—
390C2—42no—
391C2—64no—
392C2—122no—
393C2C22×D3×D411yescharacteristic
394C2D3×2−1+412yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#536C2
hypercenter#536D3
hypocenterC3#3499
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#1010C2semidirect
2. C2×#1007C2semidirect
3. C3×#536C2semidirect
4. #5832C2semidirect
5. C2×#944C2semidirect
6. #5724C2semidirect
7. D3×C4⋊Q8C2semidirect
8. #5675C2semidirect
9. #5668C2semidirect
10. #5655C2semidirect
11. #5632C2semidirect
12. #5631C2semidirect
13. #5526C2semidirect
14. #5522C2semidirect
74. #536D3semidirect
107. #235D4semidirect
318. Dic32−1+4semidirect
388. C3#3499semidirect

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×#5362
#58322
C2×#9442
#57242
D3×C4⋊Q82
#56752
#56682
#56552
#56322
#56312
#55262
#55222
#34993