Finite Group Databasebeta

← #86128

Subgroups — #86128

Order 384.

1104 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×#529C211yesmaximal, characteristic, complemented
2#5710C214yesmaximal, complemented
3#5690C212yesmaximal, complemented
4#5681C212yesmaximal, complemented
5#5655C214yesmaximal, complemented
6#5624C212yesmaximal, complemented
7#5596C212yesmaximal, complemented
8#5585C211yesmaximal, characteristic, complemented
9Q8×Dic6C211yesmaximal, characteristic, complemented
10#5555C212yesmaximal, complemented
11#5553C211yesmaximal, characteristic, complemented
12#5552C212yesmaximal, complemented
13#5543C211yesmaximal, characteristic, complemented
14#5534C212yesmaximal, complemented
15#5523C212yesmaximal, complemented
16#5477C211yesmaximal, characteristic, complemented
17C2×C4×Dic6C211yesmaximal, characteristic, complemented
18#3522—31nomaximal, Sylow
19C2×#237C2214yes—
20C2×#235C2212yes—
21C2×#235C2212yes—
22C22×Dic6C2211yescharacteristic
23C3×#126C2212yes—
24C3×#125C2211yescharacteristic
25C3×#124C2211yescharacteristic
26C3×#123C2212yes—
27C3×C22⋊Q8C2211yescharacteristic
28C3×#121C2211yescharacteristic
29C3×C4×Q8C2211yescharacteristic
30C3×C4×D4C2211yescharacteristic
31C3×C4×D4C2212yes—
32C3×#117C2212yes—
33C2×C3×C42C2211yescharacteristic
34#947C2212yes—
35Q8×Dic3C2212yes—
36#940C2212yes—
37#939C2214yes—
38D4×Dic3C2212yes—
39#930C2212yes—
40#930C2212yes—
41#930C2212yes—
42C2×C4⋊Dic3C2211yescharacteristic
43C2×C4⋊Dic3C2212yes—
44#924C2212yes—
45C2×Dic3⋊C4C2212yes—
46C2×C4×Dic3C2212yes—
47#898C2214yes—
48#897C2214yes—
49#896C2214yes—
50#896C2214yes—
51#894C2214yes—
52#893C2212yes—
53#893C2212yes—
54#892C2214yes—
55D3×C4⋊C4C2214yes—
56#890C2212yes—
57#888C2214yes—
58#888C2212yes—
59#886C2214yes—
60#886C2214yes—
61#885C2214yes—
62#885C2214yes—
63#884C2214yes—
64#883C2214yes—
65#67⋊C4C2214yes—
66#67⋊C4C2214yes—
67#67⋊C4C2214yes—
68#878C2214yes—
69#875C2212yes—
70#875C2212yes—
71#874C2211yescharacteristic
72C4×D12C2212yes—
73C4×D12C2212yes—
74C4×D12C2212yes—
75#869C2211yescharacteristic
76#869C2212yes—
77C4×Dic6C2212yes—
78C4×Dic6C2212yes—
79C4×Dic6C2211yescharacteristic
80C4×Dic6C2211yescharacteristic
81C4×Dic6C2211yescharacteristic
82C4×Dic6C2211yescharacteristic
83C4×Dic6C2212yes—
84#559—32no—
85#559—31no—
86Q82—31no—
87#554—32no—
88#554—31no—
89#550—31no—
90#548—32no—
91#547—34no—
92#547—32no—
93#542—34no—
94#542—32no—
95#532—32no—
96#532—31no—
97#529—32no—
98#529D311yescharacteristic, complemented
99#520—32no—
100C2×C4×Q8—31no—
101C2×C3×Q8C2311yescharacteristic
102C2×C3×D4C2311yescharacteristic
103C2×C3×D4C2312yes—
104C22×C3×C4C2311yescharacteristic
105C22×C3×C4C2311yescharacteristic
106C22×C3×C4C2312yes—
107C22×C3×C4C2311yescharacteristic
108C2×#67C2314yes—
109C2×#67C2314yes—
110C2×#67C2314yes—
111C22×Dic3C2312yes—
112C22×Dic3C2312yes—
113C22×Dic3C2314yes—
114#237—48no—
115#235—28no—
116#235—44no—
117C2×D12C2312yes—
118C2×D12C2312yes—
119C2×D12C2312yes—
120C2×C4×D3C2314yes—
121C2×C4×D3C2314yes—
122C2×C4×D3C2314yes—
123C2×Dic6—24no—
124C2×Dic6C2311yescharacteristic
125C2×Dic6C2312yes—
126C2×Dic6C2312yes—
127C2×Dic6C2314yes—
128C2×Dic6C2311yescharacteristic
129C3×C4⋊C4C2311yescharacteristic
130C3×C4⋊C4C2312yes—
131C3×C4⋊C4C2311yescharacteristic
132C3×C4⋊C4C2312yes—
133C3×C4⋊C4C2312yes—
134C3×C4⋊C4C2312yes—
135C3×C22⋊C4C2312yes—
136C3×C22⋊C4C2312yes—
137C3×C22⋊C4C2312yes—
138C3×C22⋊C4C2312yes—
139C3×C22⋊C4C2312yes—
140C3×C42C2311yescharacteristic
141C3×C42C2311yescharacteristic
142C3×C42C2311yescharacteristic
143C3×C42C2311yescharacteristic
144C3×C42C2312yes—
145#217C2314yes—
146(C2×D3)⋊C4C2314yes—
147(C2×D3)⋊C4C2314yes—
148(C2×D3)⋊C4C2314yes—
149(C2×D3)⋊C4C2314yes—
150(C2×D3)⋊C4C2314yes—
151(C2×D3)⋊C4C2314yes—
152C4⋊Dic3C2311yescharacteristic
153C4⋊Dic3C2311yescharacteristic
154C4⋊Dic3C2312yes—
155C4⋊Dic3C2312yes—
156C4⋊Dic3C2312yes—
157C4⋊Dic3C2314yes—
158C4⋊Dic3C2314yes—
159Dic3⋊C4C2312yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2314yes—
164Dic3⋊C4C2314yes—
165C4×Dic3C2312yes—
166C4×Dic3C2312yes—
167C4×Dic3C2312yes—
168C4×Dic3C2312yes—
169C4×Dic3C2314yes—
170C2×#41—34no—
171C2×#41—32no—
172C2×#41—32no—
173C22×Q8—31no—
174C4⋊Q8—32no—
175C4⋊Q8—34no—
176C4⋊Q8—31no—
177C4⋊Q8—32no—
178#127—32no—
179#127—31no—
180#126—34no—
181#126C2×D312yes—
182#125—32no—
183#125C2×D311yescharacteristic
184#124—32no—
185#124—34no—
186#124—34no—
187#124—32no—
188#124—32no—
189#124C2×D311yescharacteristic
190#123—34no—
191#123—34no—
192#123—34no—
193#123—34no—
194#123C2×D312yes—
195C22⋊Q8—32no—
196C22⋊Q8—34no—
197C22⋊Q8—34no—
198C22⋊Q8—34no—
199C22⋊Q8—34no—
200C22⋊Q8C2×D311yescharacteristic
201#121—34no—
202#121—32no—
203#121—32no—
204#121—32no—
205#121—34no—
206#121C2×D311yescharacteristic
207C4×Q8—32no—
208C4×Q8—31no—
209C4×Q8—32no—
210C4×Q8—31no—
211C4×Q8—31no—
212C4×Q8—31no—
213C4×Q8—32no—
214C4×Q8—32no—
215C4×Q8C2×D311yescharacteristic
216C4×D4—32no—
217C4×D4—32no—
218C4×D4—32no—
219C4×D4—34no—
220C4×D4—34no—
221C4×D4—34no—
222C4×D4—32no—
223C4×D4—32no—
224C4×D4—32no—
225C4×D4C2×D312yes—
226C4×D4C2×D311yescharacteristic
227#117—34no—
228#117C2×D312yes—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—31no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—34no—
233C2×C42—32no—
234C2×C42C2×D311yescharacteristic
235C23×C3C2412yes—
236C23×C3C2411yescharacteristic
237C22×D3C2414yes—
238C3×Q8—22no—
239C3×D4—24no—
240C3×D4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—24no—
244C2×C3×C4—22no—
245C2×C3×C4C2411yescharacteristic
246C2×C3×C4C2412yes—
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2411yescharacteristic
249C2×C3×C4C2412yes—
250C2×C3×C4C2412yes—
251C2×C3×C4C2411yescharacteristic
252C2×C3×C4C2411yescharacteristic
253C2×C3×C4C2411yescharacteristic
254#67—28no—
255#67—28no—
256#67—44no—
257C2×Dic3—24no—
258C2×Dic3—24no—
259C2×Dic3—44no—
260C2×Dic3C2412yes—
261C2×Dic3C2412yes—
262C2×Dic3C2412yes—
263C2×Dic3C2414yes—
264C2×Dic3C2412yes—
265D12—24no—
266D12—24no—
267D12—42no—
268C4×D3—28no—
269C4×D3—44no—
270C4×D3—44no—
271Dic6—22no—
272Dic6—24no—
273Dic6—42no—
274Dic6—44no—
275Dic6#4114yescomplemented
276#41—68no—
277#41—124no—
278#41—128no—
279C2×Q8—31no—
280C2×Q8—32no—
281C2×Q8—31no—
282C2×Q8—34no—
283C2×Q8—32no—
284C2×Q8—64no—
285C2×Q8C22×D311yescharacteristic
286C2×D4—34no—
287C2×D4—34no—
288C2×D4—34no—
289C2×D4—32no—
290C2×D4—32no—
291C2×D4—32no—
292C2×D4C22×D311yescharacteristic
293C2×D4C22×D312yes—
294C22×C4—32no—
295C22×C4—32no—
296C22×C4—34no—
297C22×C4—34no—
298C22×C4—34no—
299C22×C4—34no—
300C22×C4C22×D311yescharacteristic
301C22×C4C22×D311yescharacteristic
302C22×C4C22×D311yescharacteristic
303C22×C4C22×D312yes—
304C4⋊C4—32no—
305C4⋊C4—31no—
306C4⋊C4—34no—
307C4⋊C4—34no—
308C4⋊C4—32no—
309C4⋊C4—32no—
310C4⋊C4—31no—
311C4⋊C4—32no—
312C4⋊C4—32no—
313C4⋊C4—32no—
314C4⋊C4—32no—
315C4⋊C4—34no—
316C4⋊C4—34no—
317C4⋊C4C22×D312yes—
318C4⋊C4C22×D312yes—
319C4⋊C4C22×D311yescharacteristic
320C4⋊C4C22×D311yescharacteristic
321C4⋊C4C22×D312yes—
322C4⋊C4C22×D312yes—
323C22⋊C4—34no—
324C22⋊C4—34no—
325C22⋊C4—34no—
326C22⋊C4—34no—
327C22⋊C4—34no—
328C22⋊C4—34no—
329C22⋊C4—34no—
330C22⋊C4C22×D312yes—
331C22⋊C4C22×D312yes—
332C22⋊C4C22×D312yes—
333C22⋊C4C22×D312yes—
334C22⋊C4C22×D312yes—
335C42—32no—
336C42—32no—
337C42—32no—
338C42—32no—
339C42—34no—
340C42C22×D311yescharacteristic
341C42C22×D311yescharacteristic
342C42C22×D311yescharacteristic
343C42C22×D311yescharacteristic
344C42C22×D312yes—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3H(5,𝔽2)12yes—
351C22×C3C2511yescharacteristic
352C2×D3—24no—
353C2×D3—24no—
354C2×D3—24no—
355C3×C4—21no—
356C3×C4—22no—
357C3×C4—22no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—22no—
362C3×C4C2×#4112yes—
363C3×C42−1+412yes—
364Dic3—22no—
365Dic3—24no—
366Dic3—22no—
367Dic3—22no—
368Dic3C2×#4114yes—
369C23—34no—
370C23C23×D312yes—
371C23C23×D311yescharacteristic
372Q8—22no—
373Q8—34no—
374Q8—64no—
375Q8—62no—
376Q8—122no—
377Q8—124no—
378D4—24no—
379D4—22no—
380D4—68no—
381D4—68no—
382D4—64no—
383D4—64no—
384D4—122no—
385D4—124no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—32no—
391C2×C4—32no—
392C2×C4—32no—
393C2×C4—32no—
394C2×C4—34no—
395C2×C4—68no—
396C2×C4—64no—
397C2×C4—64no—
398C2×C4—124no—
399C2×C4—124no—
400C2×C4—124no—
401C2×C4C23×D311yescharacteristic
402C2×C4C23×D312yes—
403C2×C4C23×D311yescharacteristic
404C2×C4C23×D311yescharacteristic
405C2×C4C23×D311yescharacteristic
406C2×C4C23×D312yes—
407C2×C4C23×D312yes—
408C2×C4C23×D311yescharacteristic
409C2×C4C23×D311yescharacteristic
410C2×C3—22no—
411C2×C3—42no—
412C2×C3C22×#4111yescharacteristic
413C2×C3C2×H(5,𝔽2)11yescharacteristic
414C2×C3C2×2−1+411yescharacteristic
415D3—44no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—64no—
422C22—64no—
423C22—64no—
424C22#100912yes—
425C22C24×D311yescharacteristic
426C4—22no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—22no—
431C4—21no—
432C4—22no—
433C4—34no—
434C4—62no—
435C4—62no—
436C4—62no—
437C4—64no—
438C4D3×#4112yes—
439C4#101012yes—
440C3#352211yescharacteristic, complemented, Sylow
441C2—22no—
442C2—42no—
443C2—124no—
444C2C2×D3×#4111yescharacteristic
445C2C2×#100911yescharacteristic
446C2C2×#101011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#529C2
hypercenter#529D3
hypocenterC3#3522
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C3×#529C2semidirect
2. #5710C2semidirect
3. #5690C2semidirect
4. #5681C2semidirect
5. #5655C2semidirect
6. #5624C2semidirect
7. #5596C2semidirect
8. #5585C2semidirect
9. Q8×Dic6C2semidirect
10. #5555C2semidirect
11. #5553C2semidirect
12. #5552C2semidirect
13. #5543C2semidirect
14. #5534C2semidirect
15. #5523C2semidirect
16. #5477C2semidirect
17. C2×C4×Dic6C2semidirect
98. #529D3semidirect
275. Dic6#41semidirect
440. C3#3522semidirect

Maximal subgroups

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

subgroupindex
C3×#5292
#57102
#56902
#56812
#56552
#56242
#55962
#55852
Q8×Dic62
#55552
#55532
#55522
#55432
#55342
#55232
#54772
C2×C4×Dic62
#35223