Finite Group Databasebeta

← #86586

Subgroups — #86586

Order 384.

1060 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#544C211yesmaximal, characteristic, complemented
2#5729C212yesmaximal, complemented
3#5727C212yesmaximal, complemented
4D3×C4⋊Q8C212yesmaximal, complemented
5#5722C212yesmaximal, complemented
6#5690C212yesmaximal, complemented
7#5689C212yesmaximal, complemented
8#5688C212yesmaximal, complemented
9#5685C212yesmaximal, complemented
10#5636C214yesmaximal, complemented
11#5632C214yesmaximal, complemented
12#5523C211yesmaximal, characteristic
13#5522C211yesmaximal, characteristic
14#5519C211yesmaximal, characteristic, complemented
15#5517C211yesmaximal, characteristic, complemented
16#5504C211yesmaximal, characteristic
17C2×#888C211yesmaximal, characteristic, complemented
18#3503—31nomaximal, Sylow
19C2×D3×Q8C2214yes—
20C2×#235C2211yescharacteristic
21C22×Dic6C2212yes—
22C3×C4⋊Q8C2212yes—
23C3×C4⋊Q8C2212yes—
24C3×#125C2212yes—
25C3×#125C2212yes—
26C3×C22⋊Q8C2214yes—
27C3×#117C2211yescharacteristic
28C3×#117C2211yescharacteristic
29C2×C3×C4⋊C4C2211yescharacteristic
30#946C2214yes—
31#944C2214yes—
32#944C2214yes—
33C4×#67C2214yes—
34#926C2211yescharacteristic
35C2×C4⋊Dic3C2211yescharacteristic
36C2×Dic3⋊C4C2212yes—
37C2×C4×Dic3C2211yescharacteristic
38#897C2214yes—
39#896C2214yes—
40#896C2214yes—
41#896C2214yes—
42#896C2214yes—
43#893C2211yescharacteristic
44#893C2212yes—
45#893C2211yescharacteristic
46#892C2212yes—
47#892C2212yes—
48D3×C4⋊C4C2214yes—
49D3×C4⋊C4C2212yes—
50D3×C4⋊C4C2212yes—
51#890C2212yes—
52#890C2211yescharacteristic
53#890C2211yescharacteristic
54#889C2214yes—
55#889C2214yes—
56#888C2214yes—
57#888C2214yes—
58#888C2212yes—
59#888C2211yescharacteristic
60#888C2212yes—
61#888C2212yes—
62#888C2212yes—
63#888C2212yes—
64#888C2211yescharacteristic
65#887C2214yes—
66#887C2214yes—
67#887C2211yescharacteristic
68#887C2212yes—
69#887C2211yescharacteristic
70#67⋊C4C2212yes—
71#67⋊C4C2214yes—
72#67⋊C4C2212yes—
73#878C2212yes—
74#878C2214yes—
75#878C2212yes—
76C4×D12C2212yes—
77#872C2212yes—
78C42×D3C2212yes—
79#870C2212yes—
80#869C2212yes—
81C4×Dic6C2212yes—
82C4×Dic6C2212yes—
83C4×Dic6C2212yes—
84Q82—32no—
85#557—32no—
86#554—32no—
87#554—34no—
88D4×Q8—32no—
89D4×Q8—34no—
90#544—32no—
91#544—32no—
92#544—31no—
93#544D311yescharacteristic, complemented
94#533—32no—
95#533—31no—
96C2×C4⋊Q8—32no—
97C2×C4⋊Q8—31no—
98#520—31no—
99#520—31no—
100C4×#41—31no—
101C2×C3×Q8C2314yes—
102C22×C3×C4C2312yes—
103C22×C3×C4C2311yescharacteristic
104C2×#67C2312yes—
105C22×Dic3C2311yescharacteristic
106C22×Dic3C2311yescharacteristic
107C22×Dic3C2312yes—
108D3×Q8—48no—
109#235Q818yescomplemented
110C2×D12C2311yescharacteristic
111C2×C4×D3C2312yes—
112C2×C4×D3C2312yes—
113C2×C4×D3C2314yes—
114C2×C4×D3C2314yes—
115C2×C4×D3C2312yes—
116C2×Dic6—44no—
117C2×Dic6C2312yes—
118C2×Dic6C2312yes—
119C2×Dic6C2314yes—
120C2×Dic6C2314yes—
121C2×Dic6C2314yes—
122C2×Dic6C2311yescharacteristic
123C3×C4⋊C4C2312yes—
124C3×C4⋊C4C2311yescharacteristic
125C3×C4⋊C4C2314yes—
126C3×C4⋊C4C2314yes—
127C3×C4⋊C4C2311yescharacteristic
128C3×C4⋊C4C2312yes—
129C3×C4⋊C4C2314yes—
130C3×C4⋊C4C2311yescharacteristic
131C3×C4⋊C4C2311yescharacteristic
132C3×C22⋊C4C2312yes—
133C3×C22⋊C4C2312yes—
134C3×C42C2312yes—
135C3×C42C2312yes—
136#217C2312yes—
137(C2×D3)⋊C4C2312yes—
138(C2×D3)⋊C4C2312yes—
139(C2×D3)⋊C4C2314yes—
140(C2×D3)⋊C4C2314yes—
141C4⋊Dic3C2312yes—
142C4⋊Dic3C2312yes—
143C4⋊Dic3C2312yes—
144C4⋊Dic3C2311yescharacteristic
145C4⋊Dic3C2312yes—
146C4⋊Dic3C2314yes—
147C4⋊Dic3C2311yescharacteristic
148Dic3⋊C4C2312yes—
149Dic3⋊C4C2312yes—
150Dic3⋊C4C2312yes—
151Dic3⋊C4C2312yes—
152Dic3⋊C4C2314yes—
153Dic3⋊C4C2314yes—
154Dic3⋊C4C2314yes—
155Dic3⋊C4C2312yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2314yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2314yes—
160C4×Dic3C2311yescharacteristic
161C4×Dic3C2311yescharacteristic
162C4×Dic3C2312yes—
163C4×Dic3C2314yes—
164C4×Dic3C2311yescharacteristic
165C4×Dic3C2312yes—
166C4×Dic3C2312yes—
167C4×Dic3C2311yescharacteristic
168C4×Dic3C2314yes—
169C2×#41—31no—
170C22×Q8—34no—
171C22×Q8—32no—
172C4⋊Q8—34no—
173C4⋊Q8—34no—
174C4⋊Q8—31no—
175C4⋊Q8—32no—
176C4⋊Q8—32no—
177C4⋊Q8—34no—
178C4⋊Q8—32no—
179C4⋊Q8—32no—
180C4⋊Q8—31no—
181C4⋊Q8—34no—
182C4⋊Q8—32no—
183C4⋊Q8—32no—
184C4⋊Q8C2×D312yes—
185C4⋊Q8C2×D312yes—
186#125—31no—
187#125—31no—
188#125—34no—
189#125—34no—
190#125—32no—
191#125—32no—
192#125C2×D312yes—
193#125C2×D312yes—
194C22⋊Q8—34no—
195C22⋊Q8—34no—
196C22⋊Q8—34no—
197C22⋊Q8—34no—
198C22⋊Q8—34no—
199C22⋊Q8—34no—
200C22⋊Q8—32no—
201C22⋊Q8—32no—
202C22⋊Q8—34no—
203C22⋊Q8C2×D314yes—
204C4×Q8—31no—
205C4×Q8—31no—
206C4×Q8—32no—
207C4×Q8—34no—
208C4×Q8—34no—
209C4×Q8—32no—
210C4×Q8—32no—
211C4×Q8—32no—
212C4×D4—34no—
213C4×D4—31no—
214C4×D4—31no—
215C4×D4—32no—
216C4×D4—32no—
217C4×D4—32no—
218C4×D4—34no—
219C4×D4—32no—
220#117—31no—
221#117—32no—
222#117—32no—
223#117—32no—
224#117C2×D311yescharacteristic
225#117C2×D311yescharacteristic
226C2×C4⋊C4—31no—
227C2×C4⋊C4—32no—
228C2×C4⋊C4—32no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—34no—
231C2×C4⋊C4C2×D311yescharacteristic
232C2×C42—31no—
233C2×C42—32no—
234C23×C3C2411yescharacteristic
235C22×D3C2412yes—
236C3×Q8—44no—
237C2×C3×C4—42no—
238C2×C3×C4C2×Q814yescomplemented
239C2×C3×C4C2412yes—
240C2×C3×C4C2412yes—
241C2×C3×C4C2412yes—
242C2×C3×C4C2414yes—
243C2×C3×C4C2412yes—
244C2×C3×C4C2411yescharacteristic
245C2×C3×C4C2411yescharacteristic
246#67C2×Q818yescomplemented
247C2×Dic3—22no—
248C2×Dic3—22no—
249C2×Dic3—42no—
250C2×Dic3C2412yes—
251C2×Dic3C2411yescharacteristic
252C2×Dic3C2411yescharacteristic
253C2×Dic3C2414yes—
254C2×Dic3C2412yes—
255C2×Dic3C2412yes—
256C2×Dic3C2411yescharacteristic
257C2×Dic3C2411yescharacteristic
258D12C2×Q814yescomplemented
259C4×D3—24no—
260C4×D3—24no—
261C4×D3—44no—
262C4×D3—44no—
263C4×D3C2×Q818yescomplemented
264Dic6—28no—
265Dic6—24no—
266Dic6—24no—
267Dic6—44no—
268Dic6—44no—
269Dic6C2×Q814yescomplemented
270#41—38no—
271C2×Q8—31no—
272C2×Q8—32no—
273C2×Q8—34no—
274C2×Q8—32no—
275C2×Q8—34no—
276C2×Q8—34no—
277C2×Q8—124no—
278C2×Q8—128no—
279C2×Q8C22×D314yes—
280C2×D4—32no—
281C2×D4—31no—
282C22×C4—32no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—32no—
286C22×C4—32no—
287C22×C4—32no—
288C22×C4—34no—
289C22×C4—34no—
290C22×C4C22×D312yes—
291C22×C4C22×D311yescharacteristic
292C4⋊C4—32no—
293C4⋊C4—31no—
294C4⋊C4—34no—
295C4⋊C4—32no—
296C4⋊C4—34no—
297C4⋊C4—34no—
298C4⋊C4—32no—
299C4⋊C4—34no—
300C4⋊C4—34no—
301C4⋊C4—31no—
302C4⋊C4—32no—
303C4⋊C4—32no—
304C4⋊C4—32no—
305C4⋊C4—34no—
306C4⋊C4—32no—
307C4⋊C4—32no—
308C4⋊C4—32no—
309C4⋊C4—34no—
310C4⋊C4—32no—
311C4⋊C4C22×D311yescharacteristic
312C4⋊C4C22×D312yes—
313C4⋊C4C22×D314yes—
314C4⋊C4C22×D312yes—
315C4⋊C4C22×D311yescharacteristic
316C4⋊C4C22×D311yescharacteristic
317C4⋊C4C22×D311yescharacteristic
318C4⋊C4C22×D314yes—
319C4⋊C4C22×D314yes—
320C22⋊C4—32no—
321C22⋊C4—34no—
322C22⋊C4—32no—
323C22⋊C4—32no—
324C22⋊C4—34no—
325C22⋊C4C22×D312yes—
326C22⋊C4C22×D312yes—
327C42—31no—
328C42—31no—
329C42—32no—
330C42—32no—
331C42—31no—
332C42—31no—
333C42—34no—
334C42—34no—
335C42—32no—
336C42C22×D312yes—
337C42C22×D312yes—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3C22×Q812yes—
341C22×C3C2511yescharacteristic
342C2×D3—22no—
343C2×D3—22no—
344C2×D3C22×Q814yes—
345C3×C4—24no—
346C3×C4—22no—
347C3×C4—22no—
348C3×C4—22no—
349C3×C4—22no—
350C3×C4C22×Q812yes—
351C3×C4C22×Q812yes—
352Dic3—22no—
353Dic3—24no—
354Dic3—21no—
355Dic3—22no—
356Dic3—21no—
357Dic3C22×Q814yes—
358Dic3H(5,𝔽2)12yes—
359Dic32−1+412yes—
360C23—32no—
361C23C23×D311yescharacteristic
362Q8—34no—
363Q8—44no—
364Q8—68no—
365Q8—64no—
366Q8—64no—
367Q8—124no—
368Q8—124no—
369D4—38no—
370D4—34no—
371C2×C4—32no—
372C2×C4—31no—
373C2×C4—31no—
374C2×C4—38no—
375C2×C4—32no—
376C2×C4—31no—
377C2×C4—31no—
378C2×C4—32no—
379C2×C4—34no—
380C2×C4—42no—
381C2×C4—64no—
382C2×C4—64no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—124no—
386C2×C4—124no—
387C2×C4—122no—
388C2×C4D3×Q814yescomplemented
389C2×C4C23×D312yes—
390C2×C4C23×D312yes—
391C2×C4C23×D312yes—
392C2×C4C23×D314yes—
393C2×C4C23×D312yes—
394C2×C4C23×D311yescharacteristic
395C2×C4C23×D311yescharacteristic
396C2×C3—22no—
397C2×C3C23×Q811yescharacteristic
398C2×C3C2×H(5,𝔽2)11yescharacteristic
399C2×C3C2×2−1+411yescharacteristic
400D3—24no—
401C22—21no—
402C22—21no—
403C22—34no—
404C22—62no—
405C22—62no—
406C22C2×D3×Q812yes—
407C22C24×D311yescharacteristic
408C4—24no—
409C4—22no—
410C4—22no—
411C4—22no—
412C4—22no—
413C4—34no—
414C4—32no—
415C4—32no—
416C4—61no—
417C4—62no—
418C4—64no—
419C4—61no—
420C4—62no—
421C4C2×D3×Q812yes—
422C4C2×D3×Q812yes—
423C3#350311yescharacteristic, complemented, Sylow
424C2—22no—
425C2—64no—
426C2C22×D3×Q811yescharacteristic
427C2D3×H(5,𝔽2)11yescharacteristic
428C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#544C2
prime radicalC2×#235C22
hypercenter#544D3
hypocenterC3#3503
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #544

Complements

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

NH ≅ G/Ntype
1. C3×#544C2semidirect
2. #5729C2semidirect
3. #5727C2semidirect
4. D3×C4⋊Q8C2semidirect
5. #5722C2semidirect
6. #5690C2semidirect
7. #5689C2semidirect
8. #5688C2semidirect
9. #5685C2semidirect
10. #5636C2semidirect
11. #5632C2semidirect
14. #5519C2semidirect
15. #5517C2semidirect
17. C2×#888C2semidirect
93. #544D3semidirect
109. #235Q8semidirect
238. C2×C3×C4C2×Q8semidirect
246. #67C2×Q8semidirect
258. D12C2×Q8semidirect
263. C4×D3C2×Q8semidirect
269. Dic6C2×Q8semidirect
388. C2×C4D3×Q8semidirect
423. C3#3503semidirect

Maximal subgroups

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

subgroupindex
C3×#5442
#57292
#57272
D3×C4⋊Q82
#57222
#56902
#56892
#56882
#56852
#56362
#56322
#55232
#55222
#55192
#55172
#55042
C2×#8882
#35033