Finite Group Databasebeta

← #87006

Subgroups — #87006

Order 384.

1012 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#556C211yesmaximal, characteristic, complemented
2#5732C211yesmaximal, characteristic, complemented
3#5728C211yesmaximal, characteristic, complemented
4#5707C214yesmaximal, complemented
5#5702C214yesmaximal, complemented
6#5681C212yesmaximal, complemented
7#5680C212yesmaximal, complemented
8#5642C212yesmaximal, complemented
9#5628C212yesmaximal, complemented
10#5580C212yesmaximal, complemented
11#5579C211yesmaximal, characteristic, complemented
12C4×D3×Q8C211yesmaximal, characteristic, complemented
13#5569C212yesmaximal, complemented
14#5548C211yesmaximal, characteristic
15#5543C211yesmaximal, characteristic, complemented
16#5529C212yesmaximal, complemented
17#5523C212yesmaximal, complemented
18#3507—31nomaximal, Sylow
19C2×D3×Q8C2211yescharacteristic
20C2×#235C2212yes—
21C3×C4⋊Q8C2211yescharacteristic
22C3×#126C2214yes—
23C3×#124C2212yes—
24C3×C22⋊Q8C2212yes—
25C3×C4×Q8C2212yes—
26C3×C4×Q8C2211yescharacteristic
27C3×C4×D4C2211yescharacteristic
28C3×#117C2212yes—
29#947C2212yes—
30#946C2212yes—
31#946C2212yes—
32Q8×Dic3C2211yescharacteristic
33Q8×Dic3C2212yes—
34Q8×Dic3C2212yes—
35#940C2211yescharacteristic
36#938C2211yescharacteristic
37D4×Dic3C2212yes—
38C4×#67C2212yes—
39C2×C4⋊Dic3C2212yes—
40#924C2212yes—
41#898C2214yes—
42#898C2214yes—
43#897C2212yes—
44#897C2212yes—
45#896C2214yes—
46#895C2211yescharacteristic
47#894C2214yes—
48#892C2212yes—
49#892C2212yes—
50D3×C4⋊C4C2212yes—
51D3×C4⋊C4C2211yescharacteristic
52D3×C4⋊C4C2212yes—
53#890C2214yes—
54#890C2212yes—
55#890C2212yes—
56#888C2214yes—
57#888C2211yescharacteristic
58#887C2212yes—
59#887C2212yes—
60#887C2211yescharacteristic
61#886C2214yes—
62#885C2214yes—
63#885C2212yes—
64#884C2212yes—
65#883C2214yes—
66#883C2214yes—
67#67⋊C4C2214yes—
68#67⋊C4C2214yes—
69#879C2214yes—
70#878C2214yes—
71#877C2214yes—
72C4×D12C2212yes—
73C4×D12C2214yes—
74C4×D12C2211yescharacteristic
75#872C2214yes—
76C42×D3C2212yes—
77C42×D3C2211yescharacteristic
78#870C2212yes—
79#870C2214yes—
80#869C2211yescharacteristic
81C4×Dic6C2212yes—
82C4×Dic6C2214yes—
83C4×Dic6C2211yescharacteristic
84#559—32no—
85#557—32no—
86#556—31no—
87#556—34no—
88#556D311yescharacteristic, complemented
89#554—31no—
90#554—31no—
91#552—32no—
92#552—32no—
93#539—34no—
94#536—32no—
95#533—31no—
96#532—31no—
97#529—32no—
98#529—32no—
99#520—32no—
100C2×C4×Q8—31no—
101C2×C3×Q8C2312yes—
102C2×C3×Q8C2311yescharacteristic
103C2×C3×D4C2311yescharacteristic
104C22×C3×C4C2312yes—
105C2×#67C2312yes—
106C2×#67C2312yes—
107C2×#67C2312yes—
108C22×Dic3C2314yes—
109D3×Q8—24no—
110#235—28no—
111C2×D12C2311yescharacteristic
112C2×D12C2312yes—
113C2×C4×D3C2311yescharacteristic
114C2×C4×D3C2312yes—
115C2×C4×D3C2312yes—
116C2×C4×D3C2312yes—
117C2×C4×D3C2311yescharacteristic
118C2×C4×D3C2314yes—
119C2×C4×D3C2311yescharacteristic
120C2×Dic6C2312yes—
121C2×Dic6C2312yes—
122C2×Dic6C2311yescharacteristic
123C2×Dic6C2312yes—
124C3×C4⋊C4C2311yescharacteristic
125C3×C4⋊C4C2312yes—
126C3×C4⋊C4C2312yes—
127C3×C4⋊C4C2311yescharacteristic
128C3×C4⋊C4C2314yes—
129C3×C4⋊C4C2312yes—
130C3×C22⋊C4C2314yes—
131C3×C22⋊C4C2312yes—
132C3×C22⋊C4C2314yes—
133C3×C42C2311yescharacteristic
134C3×C42C2312yes—
135C3×C42C2314yes—
136#217C2312yes—
137#217C2314yes—
138(C2×D3)⋊C4C2312yes—
139(C2×D3)⋊C4C2314yes—
140(C2×D3)⋊C4C2314yes—
141(C2×D3)⋊C4C2314yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2314yes—
144C4⋊Dic3C2314yes—
145C4⋊Dic3C2312yes—
146C4⋊Dic3C2311yescharacteristic
147C4⋊Dic3C2312yes—
148C4⋊Dic3C2312yes—
149C4⋊Dic3C2311yescharacteristic
150C4⋊Dic3C2314yes—
151C4⋊Dic3C2314yes—
152Dic3⋊C4C2312yes—
153Dic3⋊C4C2314yes—
154Dic3⋊C4C2312yes—
155Dic3⋊C4C2312yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2314yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2314yes—
160C4×Dic3C2311yescharacteristic
161C4×Dic3C2312yes—
162C4×Dic3C2314yes—
163C4×Dic3C2314yes—
164C4×Dic3C2312yes—
165C4×Dic3C2312yes—
166C4×Dic3C2311yescharacteristic
167C4×Dic3C2311yescharacteristic
168C2×#41—32no—
169C22×Q8—31no—
170C4⋊Q8—31no—
171C4⋊Q8—34no—
172C4⋊Q8—31no—
173C4⋊Q8C2×D311yescharacteristic
174#127—31no—
175#126—34no—
176#126—34no—
177#126—34no—
178#126C2×D314yes—
179#125—32no—
180#125—32no—
181#125—34no—
182#125—32no—
183#125—34no—
184#124—32no—
185#124—32no—
186#124—34no—
187#124C2×D312yes—
188#123—34no—
189#123—34no—
190#123—34no—
191#123—34no—
192C22⋊Q8—32no—
193C22⋊Q8—32no—
194C22⋊Q8—32no—
195C22⋊Q8—32no—
196C22⋊Q8—32no—
197C22⋊Q8—34no—
198C22⋊Q8—34no—
199C22⋊Q8C2×D312yes—
200#121—31no—
201#121—31no—
202#121—32no—
203C4×Q8—31no—
204C4×Q8—32no—
205C4×Q8—32no—
206C4×Q8—31no—
207C4×Q8—32no—
208C4×Q8—32no—
209C4×Q8—31no—
210C4×Q8—32no—
211C4×Q8—34no—
212C4×Q8C2×D311yescharacteristic
213C4×Q8C2×D312yes—
214C4×D4—32no—
215C4×D4—32no—
216C4×D4—34no—
217C4×D4—34no—
218C4×D4—34no—
219C4×D4—32no—
220C4×D4—31no—
221C4×D4C2×D311yescharacteristic
222#117—32no—
223#117—32no—
224#117—34no—
225#117—34no—
226#117C2×D312yes—
227C2×C4⋊C4—32no—
228C2×C4⋊C4—31no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—32no—
231C2×C42—32no—
232C2×C42—31no—
233C23×C3C2412yes—
234C22×D3C2412yes—
235C22×D3C2411yescharacteristic
236C3×Q8—22no—
237C3×Q8—24no—
238C3×D4—22no—
239C2×C3×C4—24no—
240C2×C3×C4C2414yes—
241C2×C3×C4C2412yes—
242C2×C3×C4C2412yes—
243C2×C3×C4C2411yescharacteristic
244C2×C3×C4C2412yes—
245C2×C3×C4C2411yescharacteristic
246C2×C3×C4C2411yescharacteristic
247#67—24no—
248#67—24no—
249#67—42no—
250C2×Dic3—44no—
251C2×Dic3C2411yescharacteristic
252C2×Dic3C2412yes—
253C2×Dic3C2412yes—
254C2×Dic3C2411yescharacteristic
255C2×Dic3C2412yes—
256C2×Dic3C2414yes—
257C2×Dic3C2411yescharacteristic
258D12—24no—
259D12—22no—
260C4×D3—24no—
261C4×D3—24no—
262C4×D3—22no—
263C4×D3—44no—
264C4×D3—42no—
265C4×D3—41no—
266C4×D3#4114yes—
267Dic6—22no—
268Dic6—24no—
269Dic6—24no—
270Dic6—42no—
271#41—68no—
272C2×Q8—31no—
273C2×Q8—32no—
274C2×Q8—32no—
275C2×Q8—32no—
276C2×Q8—64no—
277C2×Q8C22×D312yes—
278C2×Q8C22×D311yescharacteristic
279C2×D4—32no—
280C2×D4—32no—
281C2×D4—32no—
282C2×D4—31no—
283C2×D4—32no—
284C2×D4C22×D311yescharacteristic
285C22×C4—31no—
286C22×C4—34no—
287C22×C4—31no—
288C22×C4—32no—
289C22×C4—32no—
290C22×C4—32no—
291C22×C4—31no—
292C22×C4—34no—
293C22×C4C22×D312yes—
294C4⋊C4—31no—
295C4⋊C4—32no—
296C4⋊C4—32no—
297C4⋊C4—34no—
298C4⋊C4—34no—
299C4⋊C4—32no—
300C4⋊C4—32no—
301C4⋊C4—32no—
302C4⋊C4—34no—
303C4⋊C4—34no—
304C4⋊C4—34no—
305C4⋊C4—32no—
306C4⋊C4—31no—
307C4⋊C4—32no—
308C4⋊C4—34no—
309C4⋊C4—34no—
310C4⋊C4C22×D311yescharacteristic
311C4⋊C4C22×D312yes—
312C4⋊C4C22×D312yes—
313C4⋊C4C22×D311yescharacteristic
314C4⋊C4C22×D312yes—
315C4⋊C4C22×D314yes—
316C22⋊C4—32no—
317C22⋊C4—34no—
318C22⋊C4—32no—
319C22⋊C4—34no—
320C22⋊C4—34no—
321C22⋊C4—34no—
322C22⋊C4—32no—
323C22⋊C4—34no—
324C22⋊C4C22×D314yes—
325C22⋊C4C22×D314yes—
326C22⋊C4C22×D312yes—
327C42—32no—
328C42—32no—
329C42—31no—
330C42—32no—
331C42—31no—
332C42—31no—
333C42—34no—
334C42—34no—
335C42C22×D311yescharacteristic
336C42C22×D312yes—
337C42C22×D314yes—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3C2511yescharacteristic
342C2×D3—21no—
343C2×D3—21no—
344C2×D3—22no—
345C2×D3—22no—
346C2×D3—22no—
347C2×D3C2×#4112yes—
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—24no—
351C3×C4—21no—
352C3×C4—22no—
353C3×C4C2×#4112yes—
354C3×C42−1+412yes—
355Dic3—22no—
356Dic3—22no—
357Dic3—21no—
358Dic3—24no—
359Dic3—22no—
360Dic3—21no—
361Dic3C2×#4112yes—
362C23—31no—
363C23—32no—
364C23C23×D312yes—
365Q8—22no—
366Q8—24no—
367Q8—64no—
368Q8—64no—
369Q8—62no—
370Q8—122no—
371D4—22no—
372D4—64no—
373D4—64no—
374D4—64no—
375D4—62no—
376D4—122no—
377C2×C4—24no—
378C2×C4—31no—
379C2×C4—34no—
380C2×C4—31no—
381C2×C4—32no—
382C2×C4—32no—
383C2×C4—32no—
384C2×C4—31no—
385C2×C4—34no—
386C2×C4—64no—
387C2×C4—64no—
388C2×C4—62no—
389C2×C4—122no—
390C2×C4—124no—
391C2×C4—121no—
392C2×C4—124no—
393C2×C4C23×D314yes—
394C2×C4C23×D312yes—
395C2×C4C23×D312yes—
396C2×C4C23×D311yescharacteristic
397C2×C4C23×D312yes—
398C2×C4C23×D311yescharacteristic
399C2×C4C23×D311yescharacteristic
400C2×C3—42no—
401C2×C3C22×#4111yescharacteristic
402C2×C3C2×2−1+411yescharacteristic
403C2×C3#58511yescharacteristic
404D3—22no—
405D3—42no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—32no—
410C22—61no—
411C22—61no—
412C22—62no—
413C22—62no—
414C22—62no—
415C22C24×D311yescharacteristic
416C4—22no—
417C4—22no—
418C4—24no—
419C4—22no—
420C4—21no—
421C4—32no—
422C4—62no—
423C4—62no—
424C4—61no—
425C4—62no—
426C4—64no—
427C4—61no—
428C4D3×#4112yes—
429C4#101012yes—
430C3#350711yescharacteristic, complemented, Sylow
431C2—42no—
432C2—62no—
433C2—122no—
434C2C2×D3×#4111yescharacteristic
435C2C2×#101011yescharacteristic
436C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#556C2
prime radical#5548C2
hypercenter#556D3
hypocenterC3#3507
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #556

Complements

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

NH ≅ G/Ntype
1. C3×#556C2semidirect
2. #5732C2semidirect
3. #5728C2semidirect
4. #5707C2semidirect
5. #5702C2semidirect
6. #5681C2semidirect
7. #5680C2semidirect
8. #5642C2semidirect
9. #5628C2semidirect
10. #5580C2semidirect
11. #5579C2semidirect
12. C4×D3×Q8C2semidirect
13. #5569C2semidirect
15. #5543C2semidirect
16. #5529C2semidirect
17. #5523C2semidirect
88. #556D3semidirect
430. C3#3507semidirect

Maximal subgroups

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

subgroupindex
C3×#5562
#57322
#57282
#57072
#57022
#56812
#56802
#56422
#56282
#55802
#55792
C4×D3×Q82
#55692
#55482
#55432
#55292
#55232
#35073