Finite Group Databasebeta

← #86709

Subgroups — #86709

Order 384.

1004 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#547C211yesmaximal, characteristic, complemented
2#5831C212yesmaximal, complemented
3#5825C212yesmaximal, complemented
4D3×C4⋊Q8C211yesmaximal, characteristic, complemented
5#5721C211yesmaximal, characteristic, complemented
6#5660C214yesmaximal, complemented
7#5651C214yesmaximal, complemented
8#5632C212yesmaximal, complemented
9#5625C212yesmaximal, complemented
10#5614C212yesmaximal, complemented
11#5603C212yesmaximal, complemented
12#5557C211yesmaximal, characteristic, complemented
13#5557C211yesmaximal, characteristic, complemented
14#5542C211yesmaximal, characteristic, complemented
15D4×Dic6C211yesmaximal, characteristic, complemented
16#5511C212yesmaximal, complemented
17#5500C212yesmaximal, complemented
18#3552—31nomaximal, Sylow
19C2×C3×#41C2212yes—
20C2×D3×Q8C2212yes—
21C22×Dic6C2212yes—
22C3×C4⋊Q8C2211yescharacteristic
23C3×#123C2214yes—
24C3×C22⋊Q8C2212yes—
25C3×#121C2212yes—
26C3×C4×D4C2211yescharacteristic
27C3×C4×D4C2211yescharacteristic
28C2×C3×C4⋊C4C2212yes—
29Q8×Dic3C2212yes—
30#944C2212yes—
31#944C2212yes—
32#938C2214yes—
33#938C2211yescharacteristic
34#938C2211yescharacteristic
35#936C2214yes—
36#936C2212yes—
37#936C2211yescharacteristic
38#936C2211yescharacteristic
39#935C2214yes—
40#935C2214yes—
41D4×Dic3C2212yes—
42C4×#67C2212yes—
43C4×#67C2214yes—
44C4×#67C2212yes—
45#926C2212yes—
46#926C2212yes—
47#926C2212yes—
48#924C2214yes—
49#924C2214yes—
50#924C2214yes—
51#924C2212yes—
52#924C2212yes—
53#897C2211yescharacteristic
54#897C2211yescharacteristic
55#896C2214yes—
56D3×C4⋊C4C2212yes—
57D3×C4⋊C4C2212yes—
58#890C2214yes—
59#890C2212yes—
60#890C2211yescharacteristic
61#890C2211yescharacteristic
62#889C2214yes—
63#889C2214yes—
64#888C2214yes—
65#888C2212yes—
66#888C2212yes—
67#888C2212yes—
68#887C2212yes—
69#887C2212yes—
70#883C2212yes—
71#883C2212yes—
72#67⋊C4C2214yes—
73#879C2212yes—
74#879C2212yes—
75#879C2214yes—
76#879C2214yes—
77#879C2214yes—
78#878C2214yes—
79#878C2214yes—
80C42×D3C2211yescharacteristic
81#869C2211yescharacteristic
82C4×Dic6C2211yescharacteristic
83C4×Dic6C2211yescharacteristic
84#563—34no—
85#557—31no—
86#556—31no—
87#554—32no—
88#553—32no—
89D4×Q8—32no—
90D4×Q8—31no—
91#548—32no—
92#547D311yescharacteristic, complemented
93#544—32no—
94#542—32no—
95#541—32no—
96#539—34no—
97#536—32no—
98C2×C4⋊Q8—31no—
99#529—31no—
100#529—31no—
101C3×#41—44no—
102C2×C3×Q8C2312yes—
103C2×C3×D4C2311yescharacteristic
104C2×C3×D4C2311yescharacteristic
105C2×C3×D4C2314yes—
106C22×C3×C4C2312yes—
107C22×C3×C4C2312yes—
108C22×C3×C4C2314yes—
109C2×#67C2312yes—
110C2×#67C2312yes—
111C22×Dic3C2314yes—
112D3×Q8—44no—
113C2×C4×D3C2311yescharacteristic
114C2×C4×D3C2311yescharacteristic
115C2×C4×D3C2311yescharacteristic
116C2×C4×D3C2314yes—
117C2×Dic6—44no—
118C2×Dic6C2311yescharacteristic
119C2×Dic6C2311yescharacteristic
120C2×Dic6C2312yes—
121C2×Dic6C2314yes—
122C2×Dic6C2314yes—
123C3×C4⋊C4C2311yescharacteristic
124C3×C4⋊C4C2312yes—
125C3×C4⋊C4C2311yescharacteristic
126C3×C4⋊C4C2312yes—
127C3×C4⋊C4C2314yes—
128C3×C22⋊C4C2312yes—
129C3×C22⋊C4C2314yes—
130C3×C22⋊C4C2312yes—
131C3×C42C2311yescharacteristic
132#217C2312yes—
133#217C2314yes—
134#217C2312yes—
135#217C2312yes—
136#217C2314yes—
137#217C2312yes—
138#217C2312yes—
139#217C2314yes—
140#217C2312yes—
141(C2×D3)⋊C4C2312yes—
142(C2×D3)⋊C4C2312yes—
143C4⋊Dic3C2311yescharacteristic
144C4⋊Dic3C2311yescharacteristic
145C4⋊Dic3C2312yes—
146C4⋊Dic3C2312yes—
147C4⋊Dic3C2312yes—
148C4⋊Dic3C2312yes—
149C4⋊Dic3C2312yes—
150C4⋊Dic3C2314yes—
151C4⋊Dic3C2312yes—
152Dic3⋊C4C2312yes—
153Dic3⋊C4C2312yes—
154Dic3⋊C4C2314yes—
155Dic3⋊C4C2314yes—
156Dic3⋊C4C2314yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2314yes—
160Dic3⋊C4C2314yes—
161Dic3⋊C4C2314yes—
162C4×Dic3C2312yes—
163C4×Dic3C2312yes—
164C4×Dic3C2311yescharacteristic
165C4×Dic3C2311yescharacteristic
166C4×Dic3C2311yescharacteristic
167C4×Dic3C2314yes—
168C4×Dic3C2314yes—
169C2×#41C2×D312yes—
170C22×Q8—32no—
171C22×Q8—32no—
172C4⋊Q8—32no—
173C4⋊Q8—32no—
174C4⋊Q8—32no—
175C4⋊Q8—32no—
176C4⋊Q8—32no—
177C4⋊Q8—34no—
178C4⋊Q8—31no—
179C4⋊Q8C2×D311yescharacteristic
180#126—32no—
181#126—32no—
182#126—34no—
183#126—34no—
184#126—34no—
185#125—31no—
186#125—31no—
187#125—34no—
188#125—34no—
189#125—34no—
190#125—32no—
191#124—31no—
192#124—31no—
193#124—32no—
194#124—34no—
195#123—34no—
196#123—34no—
197#123—32no—
198#123—32no—
199#123C2×D314yes—
200C22⋊Q8—34no—
201C22⋊Q8—34no—
202C22⋊Q8—32no—
203C22⋊Q8—32no—
204C22⋊Q8—34no—
205C22⋊Q8—31no—
206C22⋊Q8—31no—
207C22⋊Q8—34no—
208C22⋊Q8—34no—
209C22⋊Q8—34no—
210C22⋊Q8C2×D312yes—
211#121—31no—
212#121—31no—
213#121—34no—
214#121C2×D312yes—
215C4×Q8—32no—
216C4×Q8—32no—
217C4×Q8—32no—
218C4×Q8—31no—
219C4×Q8—31no—
220C4×D4—32no—
221C4×D4—32no—
222C4×D4—32no—
223C4×D4—34no—
224C4×D4—34no—
225C4×D4C2×D311yescharacteristic
226C4×D4C2×D311yescharacteristic
227#117—32no—
228#117—32no—
229#117—32no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4C2×D312yes—
233C2×C42—31no—
234C23×C3C2412yes—
235C23×C3C2412yes—
236C22×D3C2411yescharacteristic
237C3×Q8—24no—
238C3×D4—22no—
239C3×D4—22no—
240C3×D4—44no—
241C2×C3×C4—24no—
242C2×C3×C4—24no—
243C2×C3×C4—44no—
244C2×C3×C4C2412yes—
245C2×C3×C4C2411yescharacteristic
246C2×C3×C4C2414yes—
247C2×C3×C4C2412yes—
248C2×C3×C4C2411yescharacteristic
249C2×C3×C4C2411yescharacteristic
250#67—24no—
251#67—24no—
252C2×Dic3—44no—
253C2×Dic3C2412yes—
254C2×Dic3C2412yes—
255C2×Dic3C2412yes—
256C2×Dic3C2411yescharacteristic
257C2×Dic3C2411yescharacteristic
258C2×Dic3C2411yescharacteristic
259C2×Dic3C2412yes—
260C2×Dic3C2414yes—
261C4×D3—22no—
262C4×D3—22no—
263C4×D3—22no—
264C4×D3—44no—
265Dic6—22no—
266Dic6—24no—
267Dic6—22no—
268Dic6—44no—
269Dic6—44no—
270#41—44no—
271C2×Q8—31no—
272C2×Q8—32no—
273C2×Q8—31no—
274C2×Q8—34no—
275C2×Q8—34no—
276C2×Q8—124no—
277C2×Q8—124no—
278C2×Q8C22×D312yes—
279C2×D4—32no—
280C2×D4—32no—
281C2×D4C22×D311yescharacteristic
282C2×D4C22×D311yescharacteristic
283C2×D4C22×D314yes—
284C22×C4—34no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—31no—
288C22×C4—34no—
289C22×C4C22×D312yes—
290C22×C4C22×D312yes—
291C22×C4C22×D314yes—
292C4⋊C4—32no—
293C4⋊C4—32no—
294C4⋊C4—34no—
295C4⋊C4—34no—
296C4⋊C4—31no—
297C4⋊C4—32no—
298C4⋊C4—32no—
299C4⋊C4—32no—
300C4⋊C4—32no—
301C4⋊C4—34no—
302C4⋊C4—32no—
303C4⋊C4—34no—
304C4⋊C4—32no—
305C4⋊C4—31no—
306C4⋊C4—34no—
307C4⋊C4—34no—
308C4⋊C4—34no—
309C4⋊C4—32no—
310C4⋊C4—32no—
311C4⋊C4C22×D312yes—
312C4⋊C4C22×D312yes—
313C4⋊C4C22×D311yescharacteristic
314C4⋊C4C22×D314yes—
315C4⋊C4C22×D311yescharacteristic
316C22⋊C4—34no—
317C22⋊C4—32no—
318C22⋊C4—32no—
319C22⋊C4—32no—
320C22⋊C4—32no—
321C22⋊C4—34no—
322C22⋊C4—32no—
323C22⋊C4—34no—
324C22⋊C4—32no—
325C22⋊C4—32no—
326C22⋊C4—32no—
327C22⋊C4C22×D314yes—
328C22⋊C4C22×D312yes—
329C22⋊C4C22×D312yes—
330C42—31no—
331C42—31no—
332C42—31no—
333C42—32no—
334C42—32no—
335C42—34no—
336C42—34no—
337C42C22×D311yescharacteristic
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3C2511yescharacteristic
345C2×D3—21no—
346C2×D3—21no—
347C2×D32−1+412yes—
348C3×C4—22no—
349C3×C4—24no—
350C3×C4—21no—
351C3×C4—22no—
352C3×C4H(5,𝔽2)12yes—
353C3×C42−1+412yes—
354Dic3—24no—
355Dic3—22no—
356Dic3—21no—
357Dic3—22no—
358Dic3—21no—
359Dic3—21no—
360Dic3—22no—
361Dic3—22no—
362C23—31no—
363C23C23×D312yes—
364C23C23×D312yes—
365Q8—24no—
366Q8—62no—
367Q8—64no—
368Q8—62no—
369Q8—124no—
370Q8—124no—
371D4—22no—
372D4—22no—
373D4—44no—
374D4—64no—
375D4—64no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—32no—
379C2×C4—32no—
380C2×C4—32no—
381C2×C4—31no—
382C2×C4—31no—
383C2×C4—31no—
384C2×C4—32no—
385C2×C4—34no—
386C2×C4—44no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—62no—
390C2×C4—124no—
391C2×C4—124no—
392C2×C4C23×D312yes—
393C2×C4C23×D311yescharacteristic
394C2×C4C23×D314yes—
395C2×C4C23×D312yes—
396C2×C4C23×D311yescharacteristic
397C2×C4C23×D311yescharacteristic
398C2×C3—42no—
399C2×C3—42no—
400C2×C3C2×H(5,𝔽2)11yescharacteristic
401C2×C3C2×2−1+411yescharacteristic
402C2×C3C2×2−1+411yescharacteristic
403D3—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—32no—
411C22—61no—
412C22—61no—
413C22C24×D311yescharacteristic
414C4—22no—
415C4—24no—
416C4—22no—
417C4—21no—
418C4—61no—
419C4—64no—
420C4—62no—
421C4—61no—
422C4—62no—
423C4—61no—
424C4—62no—
425C4—62no—
426C4#100412yes—
427C4#101012yes—
428C3#355211yescharacteristic, complemented, Sylow
429C2—42no—
430C2—42no—
431C2—62no—
432C2C2×#100411yescharacteristic
433C2C2×#101011yescharacteristic
434C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#547C2
hypercenter#547D3
hypocenterC3#3552
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C3×#547C2semidirect
2. #5831C2semidirect
3. #5825C2semidirect
4. D3×C4⋊Q8C2semidirect
5. #5721C2semidirect
6. #5660C2semidirect
7. #5651C2semidirect
8. #5632C2semidirect
9. #5625C2semidirect
10. #5614C2semidirect
11. #5603C2semidirect
12. #5557C2semidirect
13. #5557C2semidirect
14. #5542C2semidirect
15. D4×Dic6C2semidirect
16. #5511C2semidirect
17. #5500C2semidirect
92. #547D3semidirect
428. C3#3552semidirect

Maximal subgroups

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

subgroupindex
C3×#5472
#58312
#58252
D3×C4⋊Q82
#57212
#56602
#56512
#56322
#56252
#56142
#56032
#55572
#55422
D4×Dic62
#55112
#55002
#35523