Finite Group Databasebeta

← #86265

Subgroups — #86265

Order 384.

1010 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#533C211yesmaximal, characteristic, complemented
2#5734C211yesmaximal, characteristic, complemented
3#5732C211yesmaximal, characteristic, complemented
4#5727C211yesmaximal, characteristic, complemented
5#5722C211yesmaximal, characteristic, complemented
6#5691C212yesmaximal, complemented
7#5688C212yesmaximal, complemented
8#5641C212yesmaximal, complemented
9#5640C214yesmaximal, complemented
10#5633C212yesmaximal, complemented
11#5582C212yesmaximal, complemented
12#5572C212yesmaximal, complemented
13#5568C214yesmaximal, complemented
14#5533C212yesmaximal, complemented
15#5532C212yesmaximal, complemented
16#5479C211yesmaximal, characteristic, complemented
17C4×#235C211yesmaximal, characteristic, complemented
18#3518—31nomaximal, Sylow
19C2×D3×Q8C2212yes—
20C2×#235C2211yescharacteristic
21C3×C4⋊Q8C2211yescharacteristic
22C3×C4⋊Q8C2211yescharacteristic
23C3×#125C2212yes—
24C3×C22⋊Q8C2212yes—
25C3×C22⋊Q8C2212yes—
26C3×C4×Q8C2212yes—
27C3×C4×Q8C2212yes—
28C3×#117C2212yes—
29C2×C3×C42C2211yescharacteristic
30#947C2212yes—
31#947C2214yes—
32#946C2212yes—
33Q8×Dic3C2212yes—
34#944C2214yes—
35#930C2211yescharacteristic
36#929C2212yes—
37C4×#67C2212yes—
38C4×#67C2212yes—
39C4×#67C2212yes—
40#926C2211yescharacteristic
41#924C2211yescharacteristic
42#898C2214yes—
43#898C2214yes—
44#897C2212yes—
45#896C2214yes—
46#896C2214yes—
47#896C2214yes—
48#895C2212yes—
49#895C2212yes—
50#894C2214yes—
51#893C2214yes—
52#893C2212yes—
53#892C2214yes—
54#892C2212yes—
55#892C2212yes—
56#892C2212yes—
57#888C2212yes—
58#888C2212yes—
59#888C2212yes—
60#887C2212yes—
61#887C2214yes—
62#887C2214yes—
63#887C2212yes—
64#885C2214yes—
65#885C2214yes—
66#884C2214yes—
67#879C2214yes—
68#876C2212yes—
69#876C2214yes—
70#875C2211yescharacteristic
71#875C2214yes—
72C4×D12C2211yescharacteristic
73C4×D12C2211yescharacteristic
74C4×D12C2211yescharacteristic
75#872C2212yes—
76#872C2212yes—
77#872C2214yes—
78C42×D3C2212yes—
79#870C2211yescharacteristic
80C4×Dic6C2211yescharacteristic
81C4×Dic6C2211yescharacteristic
82C4×Dic6C2211yescharacteristic
83C4×Dic6C2214yes—
84Q82—31no—
85#556—31no—
86#556—32no—
87#556—34no—
88#556—32no—
89#555—31no—
90#555—32no—
91D4×Q8—31no—
92#548—32no—
93#542—34no—
94#542—32no—
95#533—32no—
96#533D311yescharacteristic, complemented
97#529—32no—
98#529—31no—
99#519—32no—
100C4×#41—31no—
101C2×C3×Q8C2312yes—
102C2×C3×Q8C2312yes—
103C22×C3×C4C2311yescharacteristic
104C22×C3×C4C2311yescharacteristic
105C22×C3×C4C2311yescharacteristic
106C2×#67C2312yes—
107C2×#67C2312yes—
108D3×Q8—28no—
109#235—24no—
110C2×D12C2311yescharacteristic
111C2×D12C2311yescharacteristic
112C2×D12C2314yes—
113C2×C4×D3C2312yes—
114C2×C4×D3C2312yes—
115C2×C4×D3C2312yes—
116C2×C4×D3C2314yes—
117C2×C4×D3C2312yes—
118C2×Dic6C2311yescharacteristic
119C2×Dic6C2314yes—
120C2×Dic6C2314yes—
121C2×Dic6C2311yescharacteristic
122C3×C4⋊C4C2312yes—
123C3×C4⋊C4C2312yes—
124C3×C4⋊C4C2314yes—
125C3×C4⋊C4C2314yes—
126C3×C4⋊C4C2312yes—
127C3×C4⋊C4C2312yes—
128C3×C22⋊C4C2314yes—
129C3×C42C2311yescharacteristic
130C3×C42C2311yescharacteristic
131C3×C42C2314yes—
132C3×C42C2311yescharacteristic
133C3×C42C2311yescharacteristic
134#217C2312yes—
135#217C2312yes—
136(C2×D3)⋊C4C2312yes—
137(C2×D3)⋊C4C2312yes—
138(C2×D3)⋊C4C2312yes—
139(C2×D3)⋊C4C2312yes—
140(C2×D3)⋊C4C2314yes—
141(C2×D3)⋊C4C2312yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2314yes—
144(C2×D3)⋊C4C2314yes—
145(C2×D3)⋊C4C2314yes—
146(C2×D3)⋊C4C2314yes—
147C4⋊Dic3C2311yescharacteristic
148C4⋊Dic3C2311yescharacteristic
149C4⋊Dic3C2311yescharacteristic
150C4⋊Dic3C2314yes—
151C4⋊Dic3C2311yescharacteristic
152Dic3⋊C4C2312yes—
153Dic3⋊C4C2312yes—
154Dic3⋊C4C2312yes—
155Dic3⋊C4C2312yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2314yes—
160Dic3⋊C4C2314yes—
161C4×Dic3C2312yes—
162C4×Dic3C2312yes—
163C4×Dic3C2312yes—
164C4×Dic3C2314yes—
165C4×Dic3C2314yes—
166C4×Dic3C2312yes—
167C4×Dic3C2314yes—
168C2×#41—31no—
169C22×Q8—32no—
170C4⋊Q8—34no—
171C4⋊Q8—32no—
172C4⋊Q8—32no—
173C4⋊Q8—32no—
174C4⋊Q8C2×D311yescharacteristic
175C4⋊Q8C2×D311yescharacteristic
176#126—34no—
177#126—34no—
178#126—34no—
179#126—32no—
180#126—34no—
181#125—31no—
182#125C2×D312yes—
183#124—32no—
184#124—34no—
185#124—34no—
186#124—34no—
187#124—31no—
188#124—34no—
189#123—32no—
190#123—34no—
191C22⋊Q8—32no—
192C22⋊Q8—31no—
193C22⋊Q8—32no—
194C22⋊Q8—34no—
195C22⋊Q8—34no—
196C22⋊Q8—34no—
197C22⋊Q8C2×D312yes—
198C22⋊Q8C2×D312yes—
199#121—31no—
200#121—32no—
201#121—32no—
202#121—34no—
203C4×Q8—32no—
204C4×Q8—34no—
205C4×Q8—32no—
206C4×Q8—32no—
207C4×Q8—34no—
208C4×Q8—31no—
209C4×Q8—31no—
210C4×Q8—34no—
211C4×Q8—31no—
212C4×Q8C2×D312yes—
213C4×Q8C2×D312yes—
214C4×D4—32no—
215C4×D4—32no—
216C4×D4—32no—
217C4×D4—34no—
218C4×D4—32no—
219C4×D4—31no—
220C4×D4—31no—
221C4×D4—31no—
222#117—31no—
223#117—34no—
224#117—32no—
225#117—32no—
226#117—32no—
227#117—32no—
228#117—32no—
229#117—34no—
230#117C2×D312yes—
231C2×C42—32no—
232C2×C42C2×D311yescharacteristic
233C23×C3C2411yescharacteristic
234C22×D3C2412yes—
235C22×D3C2412yes—
236C3×Q8—24no—
237C3×Q8—42no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—41no—
241C2×C3×C4C2411yescharacteristic
242C2×C3×C4C2411yescharacteristic
243C2×C3×C4C2414yes—
244C2×C3×C4C2414yes—
245C2×C3×C4C2411yescharacteristic
246C2×C3×C4C2411yescharacteristic
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2411yescharacteristic
249#67—24no—
250#67—42no—
251C2×Dic3C2412yes—
252C2×Dic3C2412yes—
253C2×Dic3C2412yes—
254C2×Dic3C2412yes—
255C2×Dic3C2412yes—
256C2×Dic3C2412yes—
257D12—22no—
258D12—22no—
259D12—44no—
260C4×D3—24no—
261C4×D3—24no—
262C4×D3—24no—
263C4×D3—28no—
264C4×D3—42no—
265Dic6—22no—
266Dic6—28no—
267Dic6—44no—
268Dic6#4114yes—
269#41—64no—
270C2×Q8—31no—
271C2×Q8—31no—
272C2×Q8—34no—
273C2×Q8—34no—
274C2×Q8—68no—
275C2×Q8C22×D312yes—
276C2×Q8C22×D312yes—
277C2×D4—32no—
278C2×D4—32no—
279C2×D4—31no—
280C2×D4—31no—
281C2×D4—34no—
282C22×C4—32no—
283C22×C4—32no—
284C22×C4—34no—
285C22×C4—32no—
286C22×C4—32no—
287C22×C4C22×D311yescharacteristic
288C22×C4C22×D311yescharacteristic
289C22×C4C22×D311yescharacteristic
290C4⋊C4—32no—
291C4⋊C4—31no—
292C4⋊C4—31no—
293C4⋊C4—31no—
294C4⋊C4—32no—
295C4⋊C4—31no—
296C4⋊C4—34no—
297C4⋊C4—32no—
298C4⋊C4—32no—
299C4⋊C4—34no—
300C4⋊C4—32no—
301C4⋊C4—34no—
302C4⋊C4—32no—
303C4⋊C4—34no—
304C4⋊C4C22×D312yes—
305C4⋊C4C22×D312yes—
306C4⋊C4C22×D312yes—
307C4⋊C4C22×D314yes—
308C4⋊C4C22×D314yes—
309C4⋊C4C22×D312yes—
310C22⋊C4—32no—
311C22⋊C4—32no—
312C22⋊C4—34no—
313C22⋊C4—32no—
314C22⋊C4—32no—
315C22⋊C4—32no—
316C22⋊C4—32no—
317C22⋊C4—34no—
318C22⋊C4—34no—
319C22⋊C4—32no—
320C22⋊C4—34no—
321C22⋊C4—32no—
322C22⋊C4—34no—
323C22⋊C4C22×D314yes—
324C42—32no—
325C42—32no—
326C42—32no—
327C42—34no—
328C42—34no—
329C42—34no—
330C42—32no—
331C42C22×D311yescharacteristic
332C42C22×D311yescharacteristic
333C42C22×D311yescharacteristic
334C42C22×D311yescharacteristic
335C42C22×D314yes—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3C2511yescharacteristic
340C2×D3—22no—
341C2×D3—22no—
342C2×D3—22no—
343C2×D3—22no—
344C2×D3—22no—
345C2×D3—22no—
346C3×C4—24no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—24no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4C2×#4112yes—
353C3×C42−1+412yes—
354Dic3—22no—
355Dic3—22no—
356Dic3—22no—
357Dic3—22no—
358Dic3—22no—
359Dic3C2×#4114yes—
360C23—32no—
361C23—32no—
362C23C23×D311yescharacteristic
363Q8—24no—
364Q8—34no—
365Q8—42no—
366Q8—68no—
367Q8—62no—
368Q8—124no—
369D4—64no—
370D4—62no—
371D4—62no—
372D4—124no—
373D4—122no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—32no—
377C2×C4—32no—
378C2×C4—32no—
379C2×C4—32no—
380C2×C4—32no—
381C2×C4—32no—
382C2×C4—41no—
383C2×C4—68no—
384C2×C4—64no—
385C2×C4—64no—
386C2×C4—64no—
387C2×C4—122no—
388C2×C4C23×D311yescharacteristic
389C2×C4C23×D311yescharacteristic
390C2×C4C23×D314yes—
391C2×C4C23×D314yes—
392C2×C4C23×D311yescharacteristic
393C2×C4C23×D311yescharacteristic
394C2×C4C23×D311yescharacteristic
395C2×C4C23×D311yescharacteristic
396C2×C3—41no—
397C2×C3C22×#4111yescharacteristic
398C2×C3C2×2−1+411yescharacteristic
399C2×C3#58511yescharacteristic
400D3—42no—
401D3—42no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—62no—
406C22—62no—
407C22—62no—
408C22—62no—
409C22—62no—
410C22—62no—
411C22C24×D311yescharacteristic
412C4—21no—
413C4—21no—
414C4—24no—
415C4—21no—
416C4—21no—
417C4—24no—
418C4—34no—
419C4—62no—
420C4—62no—
421C4—62no—
422C4—62no—
423C4—62no—
424C4#100712yes—
425C4D3×#4112yes—
426C3#351811yescharacteristic, complemented, Sylow
427C2—41no—
428C2—122no—
429C2—122no—
430C2C2×#100711yescharacteristic
431C2C2×D3×#4111yescharacteristic
432C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#533C2
hypercenter#533D3
hypocenterC3#3518
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #533

Complements

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

NH ≅ G/Ntype
1. C3×#533C2semidirect
2. #5734C2semidirect
3. #5732C2semidirect
4. #5727C2semidirect
5. #5722C2semidirect
6. #5691C2semidirect
7. #5688C2semidirect
8. #5641C2semidirect
9. #5640C2semidirect
10. #5633C2semidirect
11. #5582C2semidirect
12. #5572C2semidirect
13. #5568C2semidirect
14. #5533C2semidirect
15. #5532C2semidirect
16. #5479C2semidirect
17. C4×#235C2semidirect
96. #533D3semidirect
426. C3#3518semidirect

Maximal subgroups

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

subgroupindex
C3×#5332
#57342
#57322
#57272
#57222
#56912
#56882
#56412
#56402
#56332
#55822
#55722
#55682
#55332
#55322
#54792
C4×#2352
#35183