Finite Group Databasebeta

← #87049

Subgroups — #87049

Order 384.

1112 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#559C211yesmaximal, characteristic, complemented
2#5718C211yesmaximal, characteristic, complemented
3#5716C211yesmaximal, characteristic, complemented
4#5680C212yesmaximal, complemented
5#5677C212yesmaximal, complemented
6#5671C212yesmaximal, complemented
7#5668C212yesmaximal, complemented
8#5661C214yesmaximal, complemented
9#5649C214yesmaximal, complemented
10#5584C211yesmaximal, characteristic, complemented
11C4×#239C211yesmaximal, characteristic, complemented
12C4×D3×Q8C211yesmaximal, characteristic, complemented
13#5567C211yesmaximal, characteristic, complemented
14#5559C212yesmaximal, complemented
15#5558C212yesmaximal, complemented
16C4×#237C212yesmaximal, complemented
17C4×#237C212yesmaximal, complemented
18#3480—31nomaximal, Sylow
19C2×#239C2211yescharacteristic
20C2×D3×Q8C2211yescharacteristic
21C2×#237C2212yes—
22C2×#237C2212yes—
23C3×#127C2211yescharacteristic
24C3×#124C2212yes—
25C3×#124C2212yes—
26C3×#123C2214yes—
27C3×C4×Q8C2211yescharacteristic
28C3×C4×Q8C2211yescharacteristic
29C3×C4×D4C2212yes—
30C3×C4×D4C2212yes—
31#946C2212yes—
32Q8×Dic3C2211yescharacteristic
33Q8×Dic3C2211yescharacteristic
34#944C2212yes—
35#940C2212yes—
36#939C2214yes—
37#938C2212yes—
38#938C2212yes—
39#938C2212yes—
40#936C2212yes—
41D4×Dic3C2212yes—
42D4×Dic3C2212yes—
43D4×Dic3C2212yes—
44#929C2214yes—
45C4×#67C2214yes—
46C4×#67C2214yes—
47C2×C4×Dic3C2214yes—
48#898C2214yes—
49#893C2212yes—
50#893C2211yescharacteristic
51#892C2212yes—
52#892C2211yescharacteristic
53#892C2212yes—
54D3×C4⋊C4C2212yes—
55D3×C4⋊C4C2211yescharacteristic
56#889C2214yes—
57#887C2212yes—
58#887C2211yescharacteristic
59#887C2212yes—
60#885C2214yes—
61#884C2214yes—
62#884C2214yes—
63#883C2214yes—
64#67⋊C4C2214yes—
65#67⋊C4C2214yes—
66#67⋊C4C2214yes—
67#879C2214yes—
68#879C2214yes—
69#878C2214yes—
70#877C2214yes—
71#877C2214yes—
72#875C2212yes—
73C4×D12C2212yes—
74C4×D12C2211yescharacteristic
75C42×D3C2212yes—
76C42×D3C2212yes—
77C42×D3C2211yescharacteristic
78#870C2212yes—
79#869C2211yescharacteristic
80C4×Dic6C2212yes—
81C4×Dic6C2212yes—
82C4×Dic6C2212yes—
83C4×Dic6C2211yescharacteristic
84#559—32no—
85#559D311yescharacteristic, complemented
86#557—31no—
87#556—32no—
88#556—31no—
89#553—34no—
90#553—32no—
91#550—31no—
92#548—32no—
93#532—31no—
94#532—32no—
95#529—32no—
96#529—34no—
97C4×#41—31no—
98C4×#41—32no—
99C4×#41—32no—
100C2×C4×Q8—31no—
101C2×C3×Q8C2311yescharacteristic
102C2×C3×Q8C2311yescharacteristic
103C2×C3×D4C2312yes—
104C2×C3×D4C2312yes—
105C2×C3×D4C2312yes—
106C22×C3×C4C2314yes—
107C2×#67C2314yes—
108C2×#67C2314yes—
109C2×#67C2314yes—
110C22×Dic3C2314yes—
111C22×Dic3C2314yes—
112#239—24no—
113D3×Q8—24no—
114#237—28no—
115#237—28no—
116C2×D12C2311yescharacteristic
117C2×D12C2312yes—
118C2×C4×D3C2311yescharacteristic
119C2×C4×D3C2312yes—
120C2×C4×D3C2312yes—
121C2×C4×D3C2312yes—
122C2×C4×D3C2312yes—
123C2×C4×D3C2312yes—
124C2×C4×D3C2311yescharacteristic
125C2×C4×D3C2311yescharacteristic
126C2×Dic6C2312yes—
127C2×Dic6C2312yes—
128C2×Dic6C2311yescharacteristic
129C3×C4⋊C4C2311yescharacteristic
130C3×C4⋊C4C2311yescharacteristic
131C3×C4⋊C4C2312yes—
132C3×C4⋊C4C2312yes—
133C3×C22⋊C4C2314yes—
134C3×C22⋊C4C2314yes—
135C3×C22⋊C4C2314yes—
136C3×C42C2312yes—
137C3×C42C2311yescharacteristic
138C3×C42C2312yes—
139#217C2314yes—
140#217C2314yes—
141#217C2314yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2312yes—
145(C2×D3)⋊C4C2314yes—
146(C2×D3)⋊C4C2314yes—
147C4⋊Dic3C2311yescharacteristic
148C4⋊Dic3C2312yes—
149C4⋊Dic3C2311yescharacteristic
150C4⋊Dic3C2312yes—
151C4⋊Dic3C2312yes—
152C4⋊Dic3C2312yes—
153Dic3⋊C4C2312yes—
154Dic3⋊C4C2314yes—
155Dic3⋊C4C2314yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2314yes—
160C4×Dic3C2312yes—
161C4×Dic3C2311yescharacteristic
162C4×Dic3C2314yes—
163C4×Dic3C2314yes—
164C4×Dic3C2311yescharacteristic
165C4×Dic3C2312yes—
166C4×Dic3C2312yes—
167C4×Dic3C2312yes—
168C4×Dic3C2312yes—
169C4×Dic3C2311yescharacteristic
170C2×#41—31no—
171C2×#41—32no—
172C2×#41—32no—
173C22×Q8—31no—
174C4⋊Q8—32no—
175C4⋊Q8—31no—
176#127—32no—
177#127C2×D311yescharacteristic
178#126—34no—
179#126—34no—
180#126—34no—
181#125—34no—
182#125—32no—
183#124—32no—
184#124—34no—
185#124—32no—
186#124C2×D312yes—
187#124C2×D312yes—
188#123—34no—
189#123—34no—
190#123C2×D314yes—
191C22⋊Q8—32no—
192C22⋊Q8—34no—
193#121—32no—
194#121—34no—
195#121—32no—
196#121—32no—
197#121—34no—
198#121—34no—
199C4×Q8—31no—
200C4×Q8—31no—
201C4×Q8—31no—
202C4×Q8—32no—
203C4×Q8—32no—
204C4×Q8—31no—
205C4×Q8—32no—
206C4×Q8—32no—
207C4×Q8—32no—
208C4×Q8C2×D311yescharacteristic
209C4×Q8C2×D311yescharacteristic
210C4×D4—32no—
211C4×D4—32no—
212C4×D4—32no—
213C4×D4—34no—
214C4×D4—34no—
215C4×D4—32no—
216C4×D4—31no—
217C4×D4—34no—
218C4×D4—34no—
219C4×D4—34no—
220C4×D4—32no—
221C4×D4—31no—
222C4×D4C2×D312yes—
223C4×D4C2×D312yes—
224#117—31no—
225#117—32no—
226#117—32no—
227#117—34no—
228#117—34no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—31no—
231C2×C42—34no—
232C2×C42—31no—
233C2×C42—32no—
234C2×C42—32no—
235C23×C3C2414yes—
236C22×D3C2412yes—
237C22×D3C2411yescharacteristic
238C3×Q8—22no—
239C3×Q8—22no—
240C3×D4—24no—
241C3×D4—24no—
242C3×D4—42no—
243C2×C3×C4—44no—
244C2×C3×C4C2412yes—
245C2×C3×C4C2412yes—
246C2×C3×C4C2412yes—
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2412yes—
249C2×C3×C4C2411yescharacteristic
250C2×C3×C4C2411yescharacteristic
251#67—28no—
252#67—28no—
253#67—44no—
254C2×Dic3—28no—
255C2×Dic3—28no—
256C2×Dic3C2412yes—
257C2×Dic3C2411yescharacteristic
258C2×Dic3C2412yes—
259C2×Dic3C2414yes—
260C2×Dic3C2411yescharacteristic
261C2×Dic3C2411yescharacteristic
262C2×Dic3C2412yes—
263D12—24no—
264D12—22no—
265C4×D3—24no—
266C4×D3—22no—
267C4×D3—24no—
268C4×D3—24no—
269C4×D3—42no—
270C4×D3—41no—
271C4×D3—42no—
272C4×D3#4114yescomplemented
273Dic6—24no—
274Dic6—24no—
275Dic6#4114yescomplemented
276#41—68no—
277#41—64no—
278#41—68no—
279C2×Q8—31no—
280C2×Q8—32no—
281C2×Q8—32no—
282C2×Q8—64no—
283C2×Q8C22×D311yescharacteristic
284C2×Q8C22×D311yescharacteristic
285C2×D4—34no—
286C2×D4—34no—
287C2×D4—34no—
288C2×D4—32no—
289C2×D4—31no—
290C2×D4C22×D312yes—
291C2×D4C22×D312yes—
292C2×D4C22×D312yes—
293C22×C4—31no—
294C22×C4—34no—
295C22×C4—34no—
296C22×C4—32no—
297C22×C4—31no—
298C22×C4—32no—
299C22×C4—32no—
300C22×C4—31no—
301C22×C4—32no—
302C22×C4—32no—
303C22×C4C22×D314yes—
304C4⋊C4—32no—
305C4⋊C4—32no—
306C4⋊C4—31no—
307C4⋊C4—32no—
308C4⋊C4—32no—
309C4⋊C4—31no—
310C4⋊C4—34no—
311C4⋊C4—32no—
312C4⋊C4—34no—
313C4⋊C4—32no—
314C4⋊C4—34no—
315C4⋊C4—34no—
316C4⋊C4—32no—
317C4⋊C4C22×D311yescharacteristic
318C4⋊C4C22×D312yes—
319C4⋊C4C22×D311yescharacteristic
320C4⋊C4C22×D312yes—
321C22⋊C4—34no—
322C22⋊C4—34no—
323C22⋊C4—34no—
324C22⋊C4—32no—
325C22⋊C4—32no—
326C22⋊C4—34no—
327C22⋊C4—32no—
328C22⋊C4—34no—
329C22⋊C4C22×D314yes—
330C22⋊C4C22×D314yes—
331C22⋊C4C22×D314yes—
332C42—32no—
333C42—32no—
334C42—31no—
335C42—31no—
336C42—32no—
337C42—32no—
338C42—34no—
339C42—34no—
340C42—31no—
341C42—32no—
342C42C22×D312yes—
343C42C22×D311yescharacteristic
344C42C22×D312yes—
345C22×C3—24no—
346C22×C3—24no—
347C22×C3—24no—
348C22×C3C2511yescharacteristic
349C2×D3—21no—
350C2×D3—21no—
351C2×D3—22no—
352C2×D3—22no—
353C2×D3—22no—
354C2×D3C2×#4112yes—
355C3×C4—22no—
356C3×C4—22no—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—21no—
360C3×C4C2×#4112yes—
361C3×C4C2×#4112yes—
362Dic3—21no—
363Dic3—24no—
364Dic3—22no—
365Dic3—21no—
366Dic3—22no—
367Dic3C2×#4114yes—
368Dic3C2×#4112yes—
369C23—31no—
370C23—32no—
371C23C23×D314yes—
372Q8—22no—
373Q8—22no—
374Q8—34no—
375Q8—64no—
376Q8—64no—
377D4—24no—
378D4—24no—
379D4—42no—
380D4—64no—
381D4—68no—
382D4—68no—
383D4—62no—
384D4—124no—
385C2×C4—31no—
386C2×C4—32no—
387C2×C4—34no—
388C2×C4—32no—
389C2×C4—31no—
390C2×C4—31no—
391C2×C4—34no—
392C2×C4—32no—
393C2×C4—44no—
394C2×C4—64no—
395C2×C4—64no—
396C2×C4—68no—
397C2×C4—68no—
398C2×C4—62no—
399C2×C4—64no—
400C2×C4—122no—
401C2×C4—122no—
402C2×C4—121no—
403C2×C4C23×D312yes—
404C2×C4C23×D312yes—
405C2×C4C23×D312yes—
406C2×C4C23×D311yescharacteristic
407C2×C4C23×D312yes—
408C2×C4C23×D311yescharacteristic
409C2×C4C23×D311yescharacteristic
410C2×C3—44no—
411C2×C3C22×#4111yescharacteristic
412C2×C3C22×#4111yescharacteristic
413C2×C3#58511yescharacteristic
414D3—22no—
415D3—42no—
416C22—24no—
417C22—24no—
418C22—24no—
419C22—32no—
420C22—61no—
421C22—61no—
422C22—62no—
423C22—62no—
424C22—62no—
425C22C24×D311yescharacteristic
426C4—22no—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—21no—
431C4—32no—
432C4—34no—
433C4—62no—
434C4—64no—
435C4—61no—
436C4—61no—
437C4—62no—
438C4D3×#4112yes—
439C4D3×#4112yes—
440C3#348011yescharacteristic, complemented, Sylow
441C2—44no—
442C2—62no—
443C2—122no—
444C2C2×D3×#4111yescharacteristic
445C2C2×D3×#4111yescharacteristic
446C2#596611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#559C2
hypercenter#559D3
hypocenterC3#3480
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #559

Complements

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

NH ≅ G/Ntype
1. C3×#559C2semidirect
2. #5718C2semidirect
3. #5716C2semidirect
4. #5680C2semidirect
5. #5677C2semidirect
6. #5671C2semidirect
7. #5668C2semidirect
8. #5661C2semidirect
9. #5649C2semidirect
10. #5584C2semidirect
11. C4×#239C2semidirect
12. C4×D3×Q8C2semidirect
13. #5567C2semidirect
14. #5559C2semidirect
15. #5558C2semidirect
16. C4×#237C2semidirect
17. C4×#237C2semidirect
85. #559D3semidirect
272. C4×D3#41semidirect
275. Dic6#41semidirect
440. C3#3480semidirect

Maximal subgroups

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

subgroupindex
C3×#5592
#57182
#57162
#56802
#56772
#56712
#56682
#56612
#56492
#55842
C4×#2392
C4×D3×Q82
#55672
#55592
#55582
C4×#2372
#34803