Finite Group Databasebeta

← #87027

Subgroups — #87027

Order 384.

890 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#557C211yesmaximal, characteristic, complemented
2#5731C212yesmaximal
3#5728C211yesmaximal, characteristic
4#5721C211yesmaximal, characteristic, complemented
5#5721C212yesmaximal, complemented
6#5698C212yesmaximal
7#5696C214yesmaximal
8C3⋊#564C214yesmaximal, complemented
9#5685C212yesmaximal, complemented
10#5583C212yesmaximal
11#5583C212yesmaximal
12#5578C211yesmaximal, characteristic
13#5578C211yesmaximal, characteristic
14#5570C211yesmaximal, characteristic, complemented
15#5569C212yesmaximal, complemented
16#5567C212yesmaximal, complemented
17#5567C211yesmaximal, characteristic, complemented
18#3556—31nomaximal, Sylow
19C3×C4⋊Q8C2211yescharacteristic
20C3×C4⋊Q8C2212yes—
21C3×#125C2212yes—
22C3×#125C2214yes—
23C3×C4×Q8C2211yescharacteristic
24C3×C4×Q8C2212yes—
25C3×C4×Q8C2212yes—
26C3×C4×Q8C2211yescharacteristic
27#946C2212yes—
28#946C2211yescharacteristic
29#946C2211yescharacteristic
30Q8×Dic3C2212yes—
31Q8×Dic3C2212yes—
32Q8×Dic3C2212yes—
33#944C2211yescharacteristic
34#944C2212yes—
35#944C2211yescharacteristic
36#944C2212yes—
37#898C2214yes—
38#898C2214yes—
39#898C2214yes—
40#897C2211yescharacteristic
41#897C2212yes—
42#897C2211yescharacteristic
43#896C2212yes—
44#896C2212yes—
45#892C2212yes—
46#890C2212yes—
47#890C2212yes—
48#890C2214yes—
49#890C2214yes—
50#890C2214yes—
51#890C2212yes—
52#889C2214yes—
53#889C2214yes—
54#889C2212yes—
55#889C2212yes—
56#889C2214yes—
57#889C2214yes—
58#889C2214yes—
59#889C2214yes—
60#889C2214yes—
61#888C2211yescharacteristic
62#888C2214yes—
63#888C2211yescharacteristic
64#887C2214yes—
65#887C2212yes—
66#887C2212yes—
67#887C2212yes—
68#876C2214yes—
69#872C2212yes—
70#872C2212yes—
71C42×D3C2211yescharacteristic
72#870C2214yes—
73#870C2214yes—
74#870C2212yes—
75#869C2212yes—
76#869C2211yescharacteristic
77C4×Dic6C2212yes—
78C4×Dic6C2212yes—
79C4×Dic6C2214yes—
80C4×Dic6C2211yescharacteristic
81C4×Dic6C2212yes—
82C4×Dic6C2212yes—
83C4×Dic6C2211yescharacteristic
84#564—34no—
85#557—32no—
86#557—31no—
87#557—32no—
88#557—32no—
89#557—31no—
90#557—32no—
91#557—31no—
92#557D311yescharacteristic, complemented
93#541—32no—
94#541—34no—
95#541—32no—
96#541—32no—
97#541—32no—
98#533—31no—
99#533—31no—
100#533—31no—
101C2×C3×Q8C2311yescharacteristic
102C2×C3×Q8C2311yescharacteristic
103C2×C3×Q8C2312yes—
104C2×C4×D3C2311yescharacteristic
105C2×C4×D3C2311yescharacteristic
106C2×C4×D3C2311yescharacteristic
107C2×Dic6C2312yes—
108C2×Dic6C2311yescharacteristic
109C2×Dic6C2312yes—
110C2×Dic6C2311yescharacteristic
111C2×Dic6C2312yes—
112C3×C4⋊C4C2311yescharacteristic
113C3×C4⋊C4C2312yes—
114C3×C4⋊C4C2314yes—
115C3×C4⋊C4C2314yes—
116C3×C4⋊C4C2311yescharacteristic
117C3×C4⋊C4C2312yes—
118C3×C4⋊C4C2314yes—
119C3×C4⋊C4C2312yes—
120C3×C4⋊C4C2312yes—
121C3×C42C2311yescharacteristic
122C3×C42C2312yes—
123C3×C42C2312yes—
124C3×C42C2314yes—
125(C2×D3)⋊C4C2312yes—
126(C2×D3)⋊C4C2312yes—
127(C2×D3)⋊C4C2312yes—
128(C2×D3)⋊C4C2312yes—
129(C2×D3)⋊C4C2314yes—
130C4⋊Dic3C2311yescharacteristic
131C4⋊Dic3C2314yes—
132C4⋊Dic3C2312yes—
133C4⋊Dic3C2311yescharacteristic
134C4⋊Dic3C2312yes—
135C4⋊Dic3C2312yes—
136C4⋊Dic3C2312yes—
137C4⋊Dic3C2314yes—
138C4⋊Dic3C2314yes—
139C4⋊Dic3C2314yes—
140Dic3⋊C4C2312yes—
141Dic3⋊C4C2312yes—
142Dic3⋊C4C2314yes—
143Dic3⋊C4C2314yes—
144Dic3⋊C4C2314yes—
145Dic3⋊C4C2312yes—
146Dic3⋊C4C2312yes—
147Dic3⋊C4C2312yes—
148Dic3⋊C4C2312yes—
149Dic3⋊C4C2312yes—
150Dic3⋊C4C2312yes—
151Dic3⋊C4C2314yes—
152Dic3⋊C4C2314yes—
153Dic3⋊C4C2314yes—
154Dic3⋊C4C2312yes—
155Dic3⋊C4C2312yes—
156Dic3⋊C4C2314yes—
157C4×Dic3C2311yescharacteristic
158C4×Dic3C2311yescharacteristic
159C4×Dic3C2312yes—
160C4×Dic3C2312yes—
161C4×Dic3C2314yes—
162C4×Dic3C2314yes—
163C4×Dic3C2314yes—
164C4×Dic3C2314yes—
165C4×Dic3C2311yescharacteristic
166C4⋊Q8—31no—
167C4⋊Q8—31no—
168C4⋊Q8—32no—
169C4⋊Q8—32no—
170C4⋊Q8—31no—
171C4⋊Q8—31no—
172C4⋊Q8—34no—
173C4⋊Q8—31no—
174C4⋊Q8—32no—
175C4⋊Q8C2×D311yescharacteristic
176C4⋊Q8C2×D312yes—
177#126—34no—
178#126—34no—
179#126—34no—
180#126—34no—
181#125—32no—
182#125—32no—
183#125—32no—
184#125—34no—
185#125—32no—
186#125—32no—
187#125—34no—
188#125—34no—
189#125—34no—
190#125—34no—
191#125—34no—
192#125—34no—
193#125—34no—
194#125—34no—
195#125—34no—
196#125—32no—
197#125—34no—
198#125—34no—
199#125C2×D312yes—
200#125C2×D314yes—
201C22⋊Q8—32no—
202C22⋊Q8—31no—
203C22⋊Q8—31no—
204C22⋊Q8—32no—
205C22⋊Q8—32no—
206C22⋊Q8—32no—
207C22⋊Q8—31no—
208C22⋊Q8—31no—
209C4×Q8—32no—
210C4×Q8—32no—
211C4×Q8—32no—
212C4×Q8—32no—
213C4×Q8—34no—
214C4×Q8—32no—
215C4×Q8—32no—
216C4×Q8—32no—
217C4×Q8—32no—
218C4×Q8—31no—
219C4×Q8—31no—
220C4×Q8—34no—
221C4×Q8—32no—
222C4×Q8—32no—
223C4×Q8C2×D311yescharacteristic
224C4×Q8C2×D311yescharacteristic
225C4×Q8C2×D312yes—
226C4×Q8C2×D312yes—
227#117—32no—
228#117—32no—
229#117—32no—
230C2×C42—31no—
231C22×D3C2411yescharacteristic
232C3×Q8—22no—
233C3×Q8—22no—
234C3×Q8—24no—
235C2×C3×C4C2414yes—
236C2×C3×C4C2412yes—
237C2×C3×C4C2412yes—
238C2×C3×C4C2411yescharacteristic
239C2×C3×C4C2412yes—
240C2×C3×C4C2412yes—
241C2×C3×C4C2411yescharacteristic
242C2×C3×C4C2411yescharacteristic
243C2×Dic3C2412yes—
244C2×Dic3C2412yes—
245C2×Dic3C2412yes—
246C2×Dic3C2412yes—
247C2×Dic3C2414yes—
248C2×Dic3C2411yescharacteristic
249C2×Dic3C2411yescharacteristic
250C2×Dic3C2411yescharacteristic
251C4×D3—22no—
252C4×D3—22no—
253C4×D3—22no—
254Dic6—24no—
255Dic6—22no—
256Dic6—22no—
257Dic6—24no—
258Dic6—24no—
259C2×Q8—32no—
260C2×Q8—32no—
261C2×Q8—32no—
262C2×Q8—31no—
263C2×Q8—31no—
264C2×Q8C22×D312yes—
265C2×Q8C22×D311yescharacteristic
266C2×Q8C22×D311yescharacteristic
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—31no—
270C4⋊C4—32no—
271C4⋊C4—32no—
272C4⋊C4—32no—
273C4⋊C4—32no—
274C4⋊C4—32no—
275C4⋊C4—32no—
276C4⋊C4—32no—
277C4⋊C4—31no—
278C4⋊C4—32no—
279C4⋊C4—31no—
280C4⋊C4—32no—
281C4⋊C4—32no—
282C4⋊C4—34no—
283C4⋊C4—32no—
284C4⋊C4—32no—
285C4⋊C4—34no—
286C4⋊C4—32no—
287C4⋊C4—32no—
288C4⋊C4—34no—
289C4⋊C4—34no—
290C4⋊C4—34no—
291C4⋊C4—34no—
292C4⋊C4—34no—
293C4⋊C4—34no—
294C4⋊C4—34no—
295C4⋊C4—34no—
296C4⋊C4—34no—
297C4⋊C4C22×D312yes—
298C4⋊C4C22×D312yes—
299C4⋊C4C22×D311yescharacteristic
300C4⋊C4C22×D314yes—
301C4⋊C4C22×D311yescharacteristic
302C4⋊C4C22×D312yes—
303C4⋊C4C22×D312yes—
304C4⋊C4C22×D314yes—
305C4⋊C4C22×D314yes—
306C22⋊C4—34no—
307C22⋊C4—32no—
308C22⋊C4—32no—
309C22⋊C4—32no—
310C22⋊C4—32no—
311C42—31no—
312C42—31no—
313C42—31no—
314C42—34no—
315C42—32no—
316C42—32no—
317C42—34no—
318C42—34no—
319C42—34no—
320C42C22×D311yescharacteristic
321C42C22×D312yes—
322C42C22×D312yes—
323C42C22×D314yes—
324C22×C3C2511yescharacteristic
325C2×D3—21no—
326C2×D3—21no—
327C2×D3—21no—
328C3×C4—22no—
329C3×C4—22no—
330C3×C4—21no—
331C3×C4—22no—
332C3×C4—24no—
333C3×C4—22no—
334C3×C42−1+412yes—
335C3×C42−1+412yes—
336Dic3—22no—
337Dic3—24no—
338Dic3—22no—
339Dic3—22no—
340Dic3—22no—
341Dic3—21no—
342Dic3—21no—
343Dic32−1+412yes—
344C23—31no—
345Q8—22no—
346Q8—24no—
347Q8—22no—
348Q8—64no—
349Q8—64no—
350Q8—64no—
351Q8—62no—
352Q8—62no—
353C2×C4—31no—
354C2×C4—32no—
355C2×C4—34no—
356C2×C4—32no—
357C2×C4—32no—
358C2×C4—32no—
359C2×C4—31no—
360C2×C4—31no—
361C2×C4—62no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4C23×D314yes—
365C2×C4C23×D312yes—
366C2×C4C23×D312yes—
367C2×C4C23×D311yescharacteristic
368C2×C4C23×D312yes—
369C2×C4C23×D312yes—
370C2×C4C23×D311yescharacteristic
371C2×C4C23×D311yescharacteristic
372C2×C3C2×2−1+411yescharacteristic
373C2×C3C2×2−1+411yescharacteristic
374C2×C3C2×2−1+411yescharacteristic
375D3—41no—
376C22—61no—
377C22—61no—
378C22—61no—
379C22C24×D311yescharacteristic
380C4—22no—
381C4—24no—
382C4—22no—
383C4—21no—
384C4—22no—
385C4—22no—
386C4—32no—
387C4—61no—
388C4—62no—
389C4—64no—
390C4—61no—
391C4—62no—
392C4—62no—
393C4—62no—
394C4#100712yes—
395C4#101012yes—
396C3#355611yescharacteristic, complemented, Sylow
397C2—121no—
398C2C2×#100711yescharacteristic
399C2C2×#101011yescharacteristic
400C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#557C2
prime radicalC22×D3C24
hypercenter#557D3
hypocenterC3#3556
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. C3×#557C2semidirect
4. #5721C2semidirect
5. #5721C2semidirect
8. C3⋊#564C2semidirect
9. #5685C2semidirect
14. #5570C2semidirect
15. #5569C2semidirect
16. #5567C2semidirect
17. #5567C2semidirect
92. #557D3semidirect
396. C3#3556semidirect

Maximal subgroups

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

subgroupindex
C3×#5572
#57312
#57282
#57212
#56982
#56962
C3⋊#5642
#56852
#55832
#55782
#55702
#55692
#55672
#35563