Finite Group Databasebeta

← #86133

Subgroups — #86133

Order 384.

1004 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×#529C211yesmaximal, characteristic, complemented
2#5712C214yesmaximal, complemented
3#5689C212yesmaximal, complemented
4#5682C212yesmaximal, complemented
5#5648C214yesmaximal, complemented
6#5642C212yesmaximal, complemented
7#5611C212yesmaximal, complemented
8#5579C211yesmaximal, characteristic, complemented
9Q8×Dic6C211yesmaximal, characteristic, complemented
10#5563C211yesmaximal, characteristic, complemented
11#5555C212yesmaximal, complemented
12#5543C211yesmaximal, characteristic, complemented
13#5541C212yesmaximal, complemented
14#5534C212yesmaximal, complemented
15#5519C212yesmaximal, complemented
16#5479C211yesmaximal, characteristic, complemented
17C2×#869C211yesmaximal, characteristic, complemented
18#3544—31nomaximal, Sylow
19C2×#237C2214yes—
20C22×Dic6C2212yes—
21C3×#126C2212yes—
22C3×#125C2211yescharacteristic
23C3×#124C2211yescharacteristic
24C3×#123C2212yes—
25C3×C22⋊Q8C2211yescharacteristic
26C3×#121C2211yescharacteristic
27C3×C4×Q8C2211yescharacteristic
28C3×C4×D4C2212yes—
29C3×C4×D4C2211yescharacteristic
30C3×#117C2212yes—
31C2×C3×C42C2211yescharacteristic
32#946C2212yes—
33Q8×Dic3C2212yes—
34#938C2212yes—
35#935C2214yes—
36D4×Dic3C2212yes—
37#930C2211yescharacteristic
38#930C2212yes—
39#929C2212yes—
40C4×#67C2212yes—
41#926C2212yes—
42C2×C4⋊Dic3C2212yes—
43C2×C4⋊Dic3C2212yes—
44C2×C4⋊Dic3C2212yes—
45#924C2211yescharacteristic
46#924C2212yes—
47#898C2214yes—
48#897C2214yes—
49#896C2214yes—
50#896C2214yes—
51#892C2212yes—
52D3×C4⋊C4C2212yes—
53#890C2212yes—
54#890C2214yes—
55#889C2214yes—
56#888C2214yes—
57#888C2214yes—
58#888C2212yes—
59#886C2214yes—
60#886C2214yes—
61#885C2214yes—
62#885C2214yes—
63#883C2214yes—
64#879C2214yes—
65#878C2214yes—
66#878C2214yes—
67#878C2214yes—
68#877C2214yes—
69#876C2212yes—
70#875C2211yescharacteristic
71C4×D12C2211yescharacteristic
72C4×D12C2212yes—
73#872C2212yes—
74#870C2211yescharacteristic
75#870C2212yes—
76#869C2212yes—
77#869C2212yes—
78#869C2212yes—
79#869C2211yescharacteristic
80#869C2211yescharacteristic
81#869C2212yes—
82C4×Dic6C2211yescharacteristic
83C4×Dic6C2212yes—
84#563—34no—
85Q82—31no—
86#555—31no—
87#554—31no—
88#554—31no—
89#552—32no—
90#552—32no—
91#547—32no—
92#547—32no—
93#544—32no—
94#544—32no—
95#542—32no—
96#542—32no—
97#536—34no—
98C2×C4⋊Q8—31no—
99#529—31no—
100#529D311yescharacteristic, complemented
101C2×C3×Q8C2311yescharacteristic
102C2×C3×D4C2311yescharacteristic
103C2×C3×D4C2312yes—
104C22×C3×C4C2311yescharacteristic
105C22×C3×C4C2312yes—
106C22×C3×C4C2311yescharacteristic
107C22×C3×C4C2311yescharacteristic
108C2×#67C2312yes—
109C2×#67C2314yes—
110C22×Dic3C2314yes—
111C22×Dic3C2314yes—
112#237—48no—
113C2×D12C2311yescharacteristic
114C2×C4×D3C2312yes—
115C2×C4×D3C2314yes—
116C2×Dic6—44no—
117C2×Dic6C2311yescharacteristic
118C2×Dic6C2312yes—
119C2×Dic6C2314yes—
120C2×Dic6C2312yes—
121C2×Dic6C2314yes—
122C3×C4⋊C4C2311yescharacteristic
123C3×C4⋊C4C2312yes—
124C3×C4⋊C4C2312yes—
125C3×C4⋊C4C2312yes—
126C3×C4⋊C4C2311yescharacteristic
127C3×C4⋊C4C2312yes—
128C3×C22⋊C4C2312yes—
129C3×C22⋊C4C2312yes—
130C3×C22⋊C4C2312yes—
131C3×C22⋊C4C2312yes—
132C3×C22⋊C4C2312yes—
133C3×C42C2311yescharacteristic
134C3×C42C2312yes—
135C3×C42C2311yescharacteristic
136C3×C42C2311yescharacteristic
137C3×C42C2311yescharacteristic
138#217C2312yes—
139#217C2314yes—
140#217C2314yes—
141(C2×D3)⋊C4C2312yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2314yes—
145(C2×D3)⋊C4C2312yes—
146(C2×D3)⋊C4C2314yes—
147C4⋊Dic3C2312yes—
148C4⋊Dic3C2311yescharacteristic
149C4⋊Dic3C2312yes—
150C4⋊Dic3C2314yes—
151C4⋊Dic3C2314yes—
152C4⋊Dic3C2312yes—
153C4⋊Dic3C2312yes—
154C4⋊Dic3C2314yes—
155C4⋊Dic3C2311yescharacteristic
156C4⋊Dic3C2314yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2312yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2314yes—
162Dic3⋊C4C2314yes—
163Dic3⋊C4C2312yes—
164Dic3⋊C4C2314yes—
165C4×Dic3C2312yes—
166C4×Dic3C2314yes—
167C4×Dic3C2314yes—
168C2×#41—34no—
169C22×Q8—32no—
170C4⋊Q8—34no—
171C4⋊Q8—34no—
172C4⋊Q8—32no—
173C4⋊Q8—32no—
174C4⋊Q8—32no—
175C4⋊Q8—31no—
176C4⋊Q8—31no—
177C4⋊Q8—32no—
178C4⋊Q8—32no—
179#126—34no—
180#126—34no—
181#126—32no—
182#126C2×D312yes—
183#125—32no—
184#125—34no—
185#125—34no—
186#125—31no—
187#125—32no—
188#125C2×D311yescharacteristic
189#124—34no—
190#124—34no—
191#124—31no—
192#124C2×D311yescharacteristic
193#123—34no—
194#123—32no—
195#123—34no—
196#123—34no—
197#123—34no—
198#123C2×D312yes—
199C22⋊Q8—32no—
200C22⋊Q8—31no—
201C22⋊Q8—32no—
202C22⋊Q8—34no—
203C22⋊Q8—34no—
204C22⋊Q8—34no—
205C22⋊Q8—34no—
206C22⋊Q8—34no—
207C22⋊Q8—34no—
208C22⋊Q8C2×D311yescharacteristic
209#121—32no—
210#121—31no—
211#121—32no—
212#121C2×D311yescharacteristic
213C4×Q8—32no—
214C4×Q8—31no—
215C4×Q8—32no—
216C4×Q8C2×D311yescharacteristic
217C4×D4—32no—
218C4×D4—32no—
219C4×D4—31no—
220C4×D4—32no—
221C4×D4C2×D312yes—
222C4×D4C2×D311yescharacteristic
223#117—32no—
224#117—32no—
225#117—34no—
226#117—32no—
227#117C2×D312yes—
228C2×C4⋊C4—32no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—32no—
232C2×C42C2×D311yescharacteristic
233C23×C3C2412yes—
234C23×C3C2411yescharacteristic
235C22×D3C2412yes—
236C3×Q8—22no—
237C3×D4—24no—
238C3×D4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—24no—
242C2×C3×C4—22no—
243C2×C3×C4C2412yes—
244C2×C3×C4C2412yes—
245C2×C3×C4C2412yes—
246C2×C3×C4C2411yescharacteristic
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2411yescharacteristic
249C2×C3×C4C2411yescharacteristic
250C2×C3×C4C2411yescharacteristic
251C2×C3×C4C2411yescharacteristic
252#67—24no—
253#67—44no—
254C2×Dic3—28no—
255C2×Dic3—44no—
256C2×Dic3C2412yes—
257C2×Dic3C2412yes—
258C2×Dic3C2414yes—
259C2×Dic3C2412yes—
260C2×Dic3C2414yes—
261D12—22no—
262C4×D3—24no—
263C4×D3—44no—
264Dic6—22no—
265Dic6—42no—
266Dic6—44no—
267Dic6—42no—
268Dic6—44no—
269#41—128no—
270C2×Q8—31no—
271C2×Q8—32no—
272C2×Q8—34no—
273C2×Q8—34no—
274C2×Q8—32no—
275C2×Q8—124no—
276C2×Q8C22×D311yescharacteristic
277C2×D4—32no—
278C2×D4—34no—
279C2×D4—31no—
280C2×D4C22×D311yescharacteristic
281C2×D4C22×D312yes—
282C22×C4—34no—
283C22×C4—34no—
284C22×C4—32no—
285C22×C4—34no—
286C22×C4C22×D311yescharacteristic
287C22×C4C22×D311yescharacteristic
288C22×C4C22×D311yescharacteristic
289C22×C4C22×D312yes—
290C4⋊C4—32no—
291C4⋊C4—32no—
292C4⋊C4—31no—
293C4⋊C4—32no—
294C4⋊C4—32no—
295C4⋊C4—34no—
296C4⋊C4—34no—
297C4⋊C4—34no—
298C4⋊C4—34no—
299C4⋊C4—31no—
300C4⋊C4—34no—
301C4⋊C4—34no—
302C4⋊C4—32no—
303C4⋊C4—32no—
304C4⋊C4—32no—
305C4⋊C4—32no—
306C4⋊C4—34no—
307C4⋊C4—34no—
308C4⋊C4C22×D311yescharacteristic
309C4⋊C4C22×D311yescharacteristic
310C4⋊C4C22×D312yes—
311C4⋊C4C22×D312yes—
312C4⋊C4C22×D312yes—
313C4⋊C4C22×D312yes—
314C22⋊C4—32no—
315C22⋊C4—34no—
316C22⋊C4—34no—
317C22⋊C4—32no—
318C22⋊C4—34no—
319C22⋊C4—32no—
320C22⋊C4—34no—
321C22⋊C4—32no—
322C22⋊C4—32no—
323C22⋊C4C22×D312yes—
324C22⋊C4C22×D312yes—
325C22⋊C4C22×D312yes—
326C22⋊C4C22×D312yes—
327C22⋊C4C22×D312yes—
328C42—34no—
329C42—32no—
330C42—34no—
331C42C22×D311yescharacteristic
332C42C22×D311yescharacteristic
333C42C22×D311yescharacteristic
334C42C22×D311yescharacteristic
335C42C22×D312yes—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3H(5,𝔽2)12yes—
342C22×C3C2511yescharacteristic
343C2×D3—22no—
344C2×D3—22no—
345C2×D3—22no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—22no—
353C3×C42−1+412yes—
354C3×C42−1+412yes—
355Dic3—24no—
356Dic3—24no—
357Dic3—22no—
358Dic3—22no—
359Dic3—22no—
360C23—32no—
361C23C23×D312yes—
362C23C23×D311yescharacteristic
363Q8—22no—
364Q8—62no—
365Q8—122no—
366Q8—124no—
367Q8—122no—
368Q8—124no—
369D4—24no—
370D4—22no—
371D4—64no—
372D4—62no—
373D4—124no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—32no—
379C2×C4—32no—
380C2×C4—34no—
381C2×C4—32no—
382C2×C4—34no—
383C2×C4—68no—
384C2×C4—64no—
385C2×C4—124no—
386C2×C4—124no—
387C2×C4C23×D312yes—
388C2×C4C23×D312yes—
389C2×C4C23×D311yescharacteristic
390C2×C4C23×D311yescharacteristic
391C2×C4C23×D311yescharacteristic
392C2×C4C23×D311yescharacteristic
393C2×C4C23×D312yes—
394C2×C4C23×D311yescharacteristic
395C2×C4C23×D311yescharacteristic
396C2×C3—22no—
397C2×C3—42no—
398C2×C3C2×H(5,𝔽2)11yescharacteristic
399C2×C3C2×2−1+411yescharacteristic
400C2×C3C2×2−1+411yescharacteristic
401D3—42no—
402C22—21no—
403C22—21no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—62no—
408C22—62no—
409C22—62no—
410C22#100912yes—
411C22C24×D311yescharacteristic
412C4—21no—
413C4—21no—
414C4—22no—
415C4—22no—
416C4—21no—
417C4—21no—
418C4—22no—
419C4—64no—
420C4—64no—
421C4—62no—
422C4—62no—
423C4—62no—
424C4#101012yes—
425C4#101012yes—
426C3#354411yescharacteristic, complemented, Sylow
427C2—22no—
428C2—42no—
429C2—122no—
430C2C2×#100911yescharacteristic
431C2C2×#101011yescharacteristic
432C2C2×#101011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#529C2
hypercenter#529D3
hypocenterC3#3544
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C3×#529C2semidirect
2. #5712C2semidirect
3. #5689C2semidirect
4. #5682C2semidirect
5. #5648C2semidirect
6. #5642C2semidirect
7. #5611C2semidirect
8. #5579C2semidirect
9. Q8×Dic6C2semidirect
10. #5563C2semidirect
11. #5555C2semidirect
12. #5543C2semidirect
13. #5541C2semidirect
14. #5534C2semidirect
15. #5519C2semidirect
16. #5479C2semidirect
17. C2×#869C2semidirect
100. #529D3semidirect
426. C3#3544semidirect

Maximal subgroups

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

subgroupindex
C3×#5292
#57122
#56892
#56822
#56482
#56422
#56112
#55792
Q8×Dic62
#55632
#55552
#55432
#55412
#55342
#55192
#54792
C2×#8692
#35443