Finite Group Databasebeta

← #86230

Subgroups — #86230

Order 384.

1022 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#532C211yesmaximal, characteristic, complemented
2#5833C212yesmaximal, complemented
3#5825C212yesmaximal, complemented
4#5729C212yesmaximal, complemented
5#5719C212yesmaximal, complemented
6#5669C214yesmaximal, complemented
7#5621C214yesmaximal, complemented
8#5616C212yesmaximal, complemented
9#5614C212yesmaximal, complemented
10#5563C211yesmaximal, characteristic, complemented
11#5552C212yesmaximal, complemented
12#5551C211yesmaximal, characteristic, complemented
13#5543C211yesmaximal, characteristic, complemented
14#5541C212yesmaximal, complemented
15D4×Dic6C211yesmaximal, characteristic, complemented
16#5479C211yesmaximal, characteristic, complemented
17#5470C211yesmaximal, characteristic, complemented
18#3547—31nomaximal, Sylow
19C2×C3×#41C2212yes—
20C2×#235C2212yes—
21C22×Dic6C2212yes—
22C3×C4⋊Q8C2211yescharacteristic
23C3×#127C2211yescharacteristic
24C3×#124C2212yes—
25C3×#121C2211yescharacteristic
26C3×#121C2211yescharacteristic
27C3×#121C2212yes—
28C3×C4×D4C2211yescharacteristic
29C3×C4×D4C2212yes—
30C3×C4×D4C2211yescharacteristic
31C2×C3×C42C2211yescharacteristic
32#946C2214yes—
33#944C2214yes—
34#939C2214yes—
35#939C2214yes—
36#939C2214yes—
37#938C2212yes—
38#938C2212yes—
39#938C2214yes—
40#936C2214yes—
41#935C2214yes—
42#935C2214yes—
43#935C2214yes—
44D4×Dic3C2212yes—
45D4×Dic3C2212yes—
46#930C2211yescharacteristic
47#930C2212yes—
48#929C2212yes—
49C4×#67C2212yes—
50#926C2212yes—
51#926C2212yes—
52C2×C4⋊Dic3C2212yes—
53#924C2212yes—
54#924C2212yes—
55#924C2211yescharacteristic
56#924C2212yes—
57#924C2212yes—
58#924C2212yes—
59#896C2214yes—
60#892C2212yes—
61D3×C4⋊C4C2212yes—
62#890C2212yes—
63#889C2214yes—
64#888C2212yes—
65#885C2214yes—
66#885C2214yes—
67#883C2214yes—
68#67⋊C4C2214yes—
69#879C2214yes—
70#878C2214yes—
71#878C2214yes—
72#877C2214yes—
73#876C2212yes—
74#875C2211yescharacteristic
75C4×D12C2211yescharacteristic
76#872C2212yes—
77#870C2212yes—
78#870C2211yescharacteristic
79#869C2211yescharacteristic
80#869C2211yescharacteristic
81C4×Dic6C2212yes—
82C4×Dic6C2212yes—
83C4×Dic6C2211yescharacteristic
84#562—34no—
85#555—32no—
86#555—31no—
87#554—31no—
88#553—32no—
89#552—32no—
90D4×Q8—31no—
91#548—32no—
92#547—31no—
93#544—32no—
94#542—34no—
95#540—32no—
96#537—32no—
97#536—32no—
98#533—31no—
99#532D311yescharacteristic, complemented
100#529—31no—
101C3×#41—44no—
102C2×C3×Q8C2312yes—
103C2×C3×D4C2311yescharacteristic
104C2×C3×D4C2311yescharacteristic
105C2×C3×D4C2312yes—
106C2×C3×D4C2312yes—
107C2×C3×D4C2312yes—
108C2×C3×D4C2312yes—
109C22×C3×C4C2311yescharacteristic
110C22×C3×C4C2312yes—
111C22×C3×C4C2312yes—
112C22×C3×C4C2311yescharacteristic
113C22×C3×C4C2311yescharacteristic
114C2×#67C2314yes—
115C2×#67C2314yes—
116C2×#67C2312yes—
117C22×Dic3C2314yes—
118C22×Dic3C2314yes—
119#235—44no—
120C2×D12C2311yescharacteristic
121C2×C4×D3C2312yes—
122C2×C4×D3C2314yes—
123C2×Dic6—44no—
124C2×Dic6C2312yes—
125C2×Dic6C2311yescharacteristic
126C2×Dic6C2312yes—
127C2×Dic6C2314yes—
128C3×C4⋊C4C2311yescharacteristic
129C3×C4⋊C4C2312yes—
130C3×C4⋊C4C2311yescharacteristic
131C3×C22⋊C4C2312yes—
132C3×C22⋊C4C2312yes—
133C3×C22⋊C4C2312yes—
134C3×C22⋊C4C2312yes—
135C3×C42C2311yescharacteristic
136C3×C42C2311yescharacteristic
137C3×C42C2311yescharacteristic
138C3×C42C2311yescharacteristic
139#217C2314yes—
140#217C2314yes—
141#217C2314yes—
142#217C2314yes—
143#217C2314yes—
144#217C2312yes—
145(C2×D3)⋊C4C2312yes—
146(C2×D3)⋊C4C2312yes—
147(C2×D3)⋊C4C2312yes—
148(C2×D3)⋊C4C2312yes—
149(C2×D3)⋊C4C2314yes—
150C4⋊Dic3C2312yes—
151C4⋊Dic3C2312yes—
152C4⋊Dic3C2311yescharacteristic
153C4⋊Dic3C2312yes—
154C4⋊Dic3C2312yes—
155C4⋊Dic3C2311yescharacteristic
156C4⋊Dic3C2314yes—
157Dic3⋊C4C2314yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2312yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2314yes—
164Dic3⋊C4C2314yes—
165Dic3⋊C4C2314yes—
166C4×Dic3C2314yes—
167C4×Dic3C2312yes—
168C4×Dic3C2314yes—
169C2×#41—32no—
170C2×#41C2×D312yes—
171C22×Q8—32no—
172C4⋊Q8—34no—
173C4⋊Q8—32no—
174C4⋊Q8—31no—
175C4⋊Q8—31no—
176C4⋊Q8C2×D311yescharacteristic
177#127C2×D311yescharacteristic
178#126—34no—
179#126—32no—
180#125—34no—
181#125—32no—
182#125—31no—
183#125—32no—
184#124—34no—
185#124—34no—
186#124—34no—
187#124—31no—
188#124C2×D312yes—
189#123—34no—
190#123—34no—
191#123—34no—
192#123—32no—
193#123—34no—
194C22⋊Q8—34no—
195C22⋊Q8—31no—
196C22⋊Q8—32no—
197C22⋊Q8—32no—
198C22⋊Q8—32no—
199C22⋊Q8—32no—
200C22⋊Q8—32no—
201C22⋊Q8—34no—
202C22⋊Q8—34no—
203C22⋊Q8—34no—
204#121—34no—
205#121—34no—
206#121—34no—
207#121—32no—
208#121—34no—
209#121—32no—
210#121—31no—
211#121—32no—
212#121C2×D311yescharacteristic
213#121C2×D311yescharacteristic
214#121C2×D312yes—
215C4×Q8—31no—
216C4×Q8—32no—
217C4×Q8—32no—
218C4×D4—32no—
219C4×D4—32no—
220C4×D4—32no—
221C4×D4—34no—
222C4×D4—31no—
223C4×D4C2×D311yescharacteristic
224C4×D4C2×D312yes—
225C4×D4C2×D311yescharacteristic
226#117—32no—
227#117—32no—
228#117—32no—
229#117—34no—
230#117—32no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—32no—
233C2×C42C2×D311yescharacteristic
234C23×C3C2411yescharacteristic
235C23×C3C2412yes—
236C23×C3C2412yes—
237C22×D3C2412yes—
238C3×Q8—42no—
239C3×D4—22no—
240C3×D4—22no—
241C3×D4—42no—
242C3×D4—42no—
243C3×D4—42no—
244C3×D4—42no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—41no—
250C2×C3×C4C2411yescharacteristic
251C2×C3×C4C2412yes—
252C2×C3×C4C2411yescharacteristic
253C2×C3×C4C2411yescharacteristic
254C2×C3×C4C2412yes—
255C2×C3×C4C2411yescharacteristic
256C2×C3×C4C2411yescharacteristic
257C2×C3×C4C2411yescharacteristic
258#67—44no—
259#67—44no—
260#67—42no—
261C2×Dic3—44no—
262C2×Dic3—44no—
263C2×Dic3C2412yes—
264C2×Dic3C2414yes—
265C2×Dic3C2412yes—
266C2×Dic3C2414yes—
267C2×Dic3C2412yes—
268D12—22no—
269C4×D3—24no—
270C4×D3—44no—
271Dic6—22no—
272Dic6—24no—
273Dic6—44no—
274Dic6—42no—
275#41—44no—
276#41—124no—
277C2×Q8—31no—
278C2×Q8—32no—
279C2×Q8—32no—
280C2×Q8—34no—
281C2×Q8—124no—
282C2×Q8C22×D312yes—
283C2×D4—32no—
284C2×D4—34no—
285C2×D4—34no—
286C2×D4—31no—
287C2×D4C22×D311yescharacteristic
288C2×D4C22×D311yescharacteristic
289C2×D4C22×D312yes—
290C2×D4C22×D312yes—
291C2×D4C22×D312yes—
292C2×D4C22×D312yes—
293C22×C4—34no—
294C22×C4—34no—
295C22×C4—32no—
296C22×C4—34no—
297C22×C4C22×D311yescharacteristic
298C22×C4C22×D311yescharacteristic
299C22×C4C22×D312yes—
300C22×C4C22×D312yes—
301C22×C4C22×D311yescharacteristic
302C4⋊C4—31no—
303C4⋊C4—32no—
304C4⋊C4—32no—
305C4⋊C4—34no—
306C4⋊C4—34no—
307C4⋊C4—32no—
308C4⋊C4—31no—
309C4⋊C4—32no—
310C4⋊C4—34no—
311C4⋊C4—32no—
312C4⋊C4—32no—
313C4⋊C4—34no—
314C4⋊C4—34no—
315C4⋊C4—34no—
316C4⋊C4—32no—
317C4⋊C4—32no—
318C4⋊C4C22×D311yescharacteristic
319C4⋊C4C22×D311yescharacteristic
320C4⋊C4C22×D312yes—
321C22⋊C4—34no—
322C22⋊C4—34no—
323C22⋊C4—34no—
324C22⋊C4—34no—
325C22⋊C4—34no—
326C22⋊C4—32no—
327C22⋊C4—32no—
328C22⋊C4—32no—
329C22⋊C4—34no—
330C22⋊C4—32no—
331C22⋊C4—32no—
332C22⋊C4C22×D312yes—
333C22⋊C4C22×D312yes—
334C22⋊C4C22×D312yes—
335C22⋊C4C22×D312yes—
336C42—34no—
337C42—34no—
338C42—32no—
339C42C22×D311yescharacteristic
340C42C22×D311yescharacteristic
341C42C22×D311yescharacteristic
342C42C22×D311yescharacteristic
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3—21no—
351C22×C3—22no—
352C22×C3C2511yescharacteristic
353C2×D3—22no—
354C2×D3—22no—
355C2×D3—22no—
356C3×C4—22no—
357C3×C4—22no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4H(5,𝔽2)12yes—
363C3×C42−1+412yes—
364Dic3—22no—
365Dic3—24no—
366Dic3—22no—
367Dic3—24no—
368Dic3—22no—
369C23—32no—
370C23C23×D311yescharacteristic
371C23C23×D312yes—
372C23C23×D312yes—
373Q8—42no—
374Q8—62no—
375Q8—64no—
376Q8—124no—
377Q8—122no—
378D4—22no—
379D4—22no—
380D4—42no—
381D4—42no—
382D4—42no—
383D4—42no—
384D4—62no—
385D4—124no—
386D4—122no—
387D4—124no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—32no—
393C2×C4—34no—
394C2×C4—32no—
395C2×C4—32no—
396C2×C4—34no—
397C2×C4—41no—
398C2×C4—64no—
399C2×C4—124no—
400C2×C4—124no—
401C2×C4—124no—
402C2×C4C23×D311yescharacteristic
403C2×C4C23×D312yes—
404C2×C4C23×D311yescharacteristic
405C2×C4C23×D311yescharacteristic
406C2×C4C23×D311yescharacteristic
407C2×C4C23×D312yes—
408C2×C4C23×D311yescharacteristic
409C2×C4C23×D311yescharacteristic
410C2×C3—42no—
411C2×C3—42no—
412C2×C3—41no—
413C2×C3C2×H(5,𝔽2)11yescharacteristic
414C2×C3C2×2−1+411yescharacteristic
415C2×C3#58511yescharacteristic
416D3—42no—
417C22—22no—
418C22—21no—
419C22—22no—
420C22—21no—
421C22—22no—
422C22—22no—
423C22—21no—
424C22—22no—
425C22—22no—
426C22—62no—
427C22—62no—
428C22—62no—
429C22C24×D311yescharacteristic
430C4—22no—
431C4—21no—
432C4—22no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—64no—
437C4—62no—
438C4—62no—
439C4—62no—
440C4—64no—
441C4#100412yes—
442C4#101012yes—
443C3#354711yescharacteristic, complemented, Sylow
444C2—42no—
445C2—41no—
446C2—42no—
447C2—122no—
448C2C2×#100411yescharacteristic
449C2C2×#101011yescharacteristic
450C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#532C2
hypercenter#532D3
hypocenterC3#3547
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C3×#532C2semidirect
2. #5833C2semidirect
3. #5825C2semidirect
4. #5729C2semidirect
5. #5719C2semidirect
6. #5669C2semidirect
7. #5621C2semidirect
8. #5616C2semidirect
9. #5614C2semidirect
10. #5563C2semidirect
11. #5552C2semidirect
12. #5551C2semidirect
13. #5543C2semidirect
14. #5541C2semidirect
15. D4×Dic6C2semidirect
16. #5479C2semidirect
17. #5470C2semidirect
99. #532D3semidirect
443. C3#3547semidirect

Maximal subgroups

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

subgroupindex
C3×#5322
#58332
#58252
#57292
#57192
#56692
#56212
#56162
#56142
#55632
#55522
#55512
#55432
#55412
D4×Dic62
#54792
#54702
#35473