Finite Group Databasebeta

← #86985

Subgroups — #86985

Order 384.

1104 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#556C211yesmaximal, characteristic, complemented
2#5726C211yesmaximal, characteristic, complemented
3D3×C4⋊Q8C211yesmaximal, characteristic, complemented
4#5710C214yesmaximal, complemented
5#5709C214yesmaximal, complemented
6#5677C212yesmaximal, complemented
7#5668C212yesmaximal, complemented
8#5635C212yesmaximal, complemented
9#5629C212yesmaximal, complemented
10#5584C211yesmaximal, characteristic, complemented
11#5578C212yesmaximal, complemented
12#5576C212yesmaximal, complemented
13C4×#239C211yesmaximal, characteristic, complemented
14#5555C211yesmaximal, characteristic, complemented
15#5548C211yesmaximal, characteristic, complemented
16#5532C212yesmaximal, complemented
17#5523C212yesmaximal, complemented
18#3511—31nomaximal, Sylow
19C2×#239C2211yescharacteristic
20C2×#239C2212yes—
21C2×D3×Q8C2212yes—
22C2×#237C2212yes—
23C2×#235C2212yes—
24C3×C4⋊Q8C2211yescharacteristic
25C3×#126C2214yes—
26C3×#124C2212yes—
27C3×C22⋊Q8C2212yes—
28C3×C4×Q8C2211yescharacteristic
29C3×C4×Q8C2212yes—
30C3×C4×D4C2211yescharacteristic
31C3×#117C2212yes—
32#946C2212yes—
33#946C2212yes—
34Q8×Dic3C2211yescharacteristic
35#944C2212yes—
36#940C2211yescharacteristic
37#938C2211yescharacteristic
38#930C2212yes—
39C4×#67C2212yes—
40C2×C4⋊Dic3C2212yes—
41#898C2214yes—
42#897C2214yes—
43#897C2212yes—
44#896C2214yes—
45#895C2214yes—
46#895C2212yes—
47#895C2211yescharacteristic
48#894C2214yes—
49#893C2212yes—
50#893C2211yescharacteristic
51#892C2212yes—
52#892C2211yescharacteristic
53#892C2212yes—
54D3×C4⋊C4C2212yes—
55D3×C4⋊C4C2212yes—
56D3×C4⋊C4C2212yes—
57#889C2214yes—
58#888C2212yes—
59#888C2212yes—
60#888C2211yescharacteristic
61#886C2214yes—
62#885C2214yes—
63#885C2212yes—
64#884C2214yes—
65#884C2214yes—
66#884C2214yes—
67#884C2212yes—
68#67⋊C4C2214yes—
69#67⋊C4C2214yes—
70#67⋊C4C2214yes—
71#878C2214yes—
72#875C2212yes—
73#875C2214yes—
74C4×D12C2212yes—
75C4×D12C2214yes—
76C4×D12C2211yescharacteristic
77C4×D12C2211yescharacteristic
78#872C2214yes—
79C42×D3C2212yes—
80C42×D3C2211yescharacteristic
81#869C2211yescharacteristic
82C4×Dic6C2212yes—
83C4×Dic6C2214yes—
84#556—31no—
85#556D311yescharacteristic, complemented
86#555—32no—
87#554—32no—
88#553—34no—
89#548—32no—
90#548—32no—
91#547—31no—
92#547—31no—
93#542—34no—
94#537—32no—
95#533—32no—
96#532—32no—
97#532—31no—
98C2×C4⋊Q8—31no—
99#520—32no—
100C4×#41—31no—
101C2×C3×Q8C2312yes—
102C2×C3×Q8C2311yescharacteristic
103C2×C3×D4C2311yescharacteristic
104C22×C3×C4C2312yes—
105C2×#67C2312yes—
106C2×#67C2312yes—
107C2×#67C2312yes—
108C2×#67C2314yes—
109C22×Dic3C2314yes—
110#239—24no—
111#239—44no—
112D3×Q8—44no—
113#237—44no—
114#235—28no—
115C2×D12C2312yes—
116C2×D12C2311yescharacteristic
117C2×D12C2312yes—
118C2×D12C2311yescharacteristic
119C2×D12C2314yes—
120C2×C4×D3C2312yes—
121C2×C4×D3C2311yescharacteristic
122C2×C4×D3C2312yes—
123C2×C4×D3C2311yescharacteristic
124C2×C4×D3C2312yes—
125C2×C4×D3C2312yes—
126C2×C4×D3C2314yes—
127C2×C4×D3C2314yes—
128C2×C4×D3C2311yescharacteristic
129C2×Dic6C2312yes—
130C2×Dic6C2312yes—
131C2×Dic6C2314yes—
132C3×C4⋊C4C2312yes—
133C3×C4⋊C4C2312yes—
134C3×C4⋊C4C2311yescharacteristic
135C3×C4⋊C4C2314yes—
136C3×C4⋊C4C2311yescharacteristic
137C3×C4⋊C4C2312yes—
138C3×C22⋊C4C2314yes—
139C3×C22⋊C4C2314yes—
140C3×C22⋊C4C2312yes—
141C3×C42C2312yes—
142C3×C42C2314yes—
143C3×C42C2311yescharacteristic
144#217C2312yes—
145(C2×D3)⋊C4C2314yes—
146(C2×D3)⋊C4C2314yes—
147(C2×D3)⋊C4C2312yes—
148(C2×D3)⋊C4C2314yes—
149(C2×D3)⋊C4C2312yes—
150(C2×D3)⋊C4C2312yes—
151(C2×D3)⋊C4C2312yes—
152(C2×D3)⋊C4C2312yes—
153(C2×D3)⋊C4C2314yes—
154C4⋊Dic3C2311yescharacteristic
155C4⋊Dic3C2314yes—
156C4⋊Dic3C2312yes—
157C4⋊Dic3C2312yes—
158C4⋊Dic3C2312yes—
159C4⋊Dic3C2311yescharacteristic
160Dic3⋊C4C2314yes—
161Dic3⋊C4C2314yes—
162Dic3⋊C4C2314yes—
163Dic3⋊C4C2312yes—
164Dic3⋊C4C2314yes—
165C4×Dic3C2312yes—
166C4×Dic3C2314yes—
167C4×Dic3C2311yescharacteristic
168C4×Dic3C2312yes—
169C4×Dic3C2311yescharacteristic
170C4×Dic3C2311yescharacteristic
171C2×#41—31no—
172C2×#41—32no—
173C2×#41—32no—
174C2×#41—32no—
175C22×Q8—32no—
176C4⋊Q8—32no—
177C4⋊Q8—32no—
178C4⋊Q8—32no—
179C4⋊Q8—31no—
180C4⋊Q8—31no—
181C4⋊Q8C2×D311yescharacteristic
182#127—31no—
183#126—34no—
184#126C2×D314yes—
185#125—34no—
186#124—32no—
187#124—34no—
188#124—32no—
189#124—34no—
190#124C2×D312yes—
191#123—34no—
192#123—34no—
193C22⋊Q8—32no—
194C22⋊Q8—32no—
195C22⋊Q8—34no—
196C22⋊Q8—32no—
197C22⋊Q8—34no—
198C22⋊Q8—34no—
199C22⋊Q8C2×D312yes—
200#121—31no—
201#121—32no—
202#121—31no—
203#121—32no—
204#121—34no—
205#121—34no—
206#121—32no—
207#121—34no—
208#121—34no—
209C4×Q8—31no—
210C4×Q8—34no—
211C4×Q8—32no—
212C4×Q8C2×D311yescharacteristic
213C4×Q8C2×D312yes—
214C4×D4—32no—
215C4×D4—32no—
216C4×D4—31no—
217C4×D4—34no—
218C4×D4—34no—
219C4×D4—34no—
220C4×D4—31no—
221C4×D4—32no—
222C4×D4—34no—
223C4×D4—31no—
224C4×D4C2×D311yescharacteristic
225#117—31no—
226#117—32no—
227#117—32no—
228#117—34no—
229#117C2×D312yes—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—32no—
233C2×C4⋊C4—32no—
234C2×C42—31no—
235C2×C42—32no—
236C23×C3C2412yes—
237C22×D3C2412yes—
238C22×D3C2412yes—
239C22×D3C2411yescharacteristic
240C3×Q8—22no—
241C3×Q8—24no—
242C3×D4—22no—
243C2×C3×C4—24no—
244C2×C3×C4C2412yes—
245C2×C3×C4C2412yes—
246C2×C3×C4C2411yescharacteristic
247C2×C3×C4C2414yes—
248C2×C3×C4C2412yes—
249C2×C3×C4C2411yescharacteristic
250C2×C3×C4C2411yescharacteristic
251#67—24no—
252#67—24no—
253#67—24no—
254#67—44no—
255C2×Dic3—44no—
256C2×Dic3C2412yes—
257C2×Dic3C2412yes—
258C2×Dic3C2411yescharacteristic
259C2×Dic3C2414yes—
260C2×Dic3C2411yescharacteristic
261C2×Dic3C2411yescharacteristic
262D12—24no—
263D12—24no—
264D12—22no—
265D12—22no—
266D12—44no—
267C4×D3—24no—
268C4×D3—24no—
269C4×D3—22no—
270C4×D3—24no—
271C4×D3—22no—
272C4×D3—24no—
273C4×D3—44no—
274C4×D3—44no—
275C4×D3#4114yescomplemented
276Dic6—24no—
277Dic6—24no—
278Dic6—44no—
279#41—68no—
280#41—64no—
281#41—124no—
282#41—124no—
283C2×Q8—32no—
284C2×Q8—32no—
285C2×Q8—34no—
286C2×Q8—124no—
287C2×Q8C22×D311yescharacteristic
288C2×Q8C22×D312yes—
289C2×D4—32no—
290C2×D4—32no—
291C2×D4—32no—
292C2×D4—34no—
293C2×D4—32no—
294C2×D4—32no—
295C2×D4—31no—
296C2×D4—31no—
297C2×D4—34no—
298C2×D4C22×D311yescharacteristic
299C22×C4—31no—
300C22×C4—34no—
301C22×C4—31no—
302C22×C4—32no—
303C22×C4—32no—
304C22×C4—31no—
305C22×C4—32no—
306C22×C4—32no—
307C22×C4—34no—
308C22×C4—34no—
309C22×C4C22×D312yes—
310C4⋊C4—31no—
311C4⋊C4—32no—
312C4⋊C4—34no—
313C4⋊C4—34no—
314C4⋊C4—34no—
315C4⋊C4—34no—
316C4⋊C4—32no—
317C4⋊C4—32no—
318C4⋊C4—32no—
319C4⋊C4—31no—
320C4⋊C4—34no—
321C4⋊C4C22×D312yes—
322C4⋊C4C22×D312yes—
323C4⋊C4C22×D311yescharacteristic
324C4⋊C4C22×D312yes—
325C4⋊C4C22×D311yescharacteristic
326C4⋊C4C22×D314yes—
327C22⋊C4—32no—
328C22⋊C4—32no—
329C22⋊C4—34no—
330C22⋊C4—32no—
331C22⋊C4—32no—
332C22⋊C4—34no—
333C22⋊C4—34no—
334C22⋊C4—34no—
335C22⋊C4—32no—
336C22⋊C4—32no—
337C22⋊C4C22×D314yes—
338C22⋊C4C22×D314yes—
339C22⋊C4C22×D312yes—
340C42—31no—
341C42—32no—
342C42—31no—
343C42—32no—
344C42—31no—
345C42—34no—
346C42C22×D311yescharacteristic
347C42C22×D312yes—
348C42C22×D314yes—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3C2511yescharacteristic
353C2×D3—21no—
354C2×D3—21no—
355C2×D3—22no—
356C2×D3—22no—
357C2×D3—22no—
358C2×D3—22no—
359C2×D3—22no—
360C2×D3—22no—
361C2×D3C2×#4112yes—
362C3×C4—22no—
363C3×C4—22no—
364C3×C4—24no—
365C3×C4—22no—
366C3×C4—21no—
367C3×C4C2×#4112yes—
368C3×C4H(5,𝔽2)12yes—
369Dic3—22no—
370Dic3—24no—
371Dic3—21no—
372Dic3—22no—
373Dic3C2×#4112yes—
374Dic32−1+412yes—
375C23—31no—
376C23—32no—
377C23—32no—
378C23C23×D312yes—
379Q8—22no—
380Q8—24no—
381Q8—64no—
382Q8—64no—
383Q8—124no—
384D4—22no—
385D4—64no—
386D4—64no—
387D4—64no—
388D4—64no—
389D4—64no—
390D4—62no—
391D4—62no—
392D4—124no—
393D4—124no—
394C2×C4—24no—
395C2×C4—31no—
396C2×C4—31no—
397C2×C4—34no—
398C2×C4—31no—
399C2×C4—32no—
400C2×C4—32no—
401C2×C4—34no—
402C2×C4—64no—
403C2×C4—62no—
404C2×C4—64no—
405C2×C4—64no—
406C2×C4—62no—
407C2×C4—64no—
408C2×C4—124no—
409C2×C4—124no—
410C2×C4—124no—
411C2×C4C23×D312yes—
412C2×C4C23×D311yescharacteristic
413C2×C4C23×D312yes—
414C2×C4C23×D312yes—
415C2×C4C23×D314yes—
416C2×C4C23×D311yescharacteristic
417C2×C4C23×D311yescharacteristic
418C2×C3—42no—
419C2×C3C22×#4111yescharacteristic
420C2×C3C2×H(5,𝔽2)11yescharacteristic
421C2×C3C2×2−1+411yescharacteristic
422D3—22no—
423D3—42no—
424D3—42no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—32no—
429C22—61no—
430C22—61no—
431C22—62no—
432C22—62no—
433C22—62no—
434C22—62no—
435C22—62no—
436C22—62no—
437C22C24×D311yescharacteristic
438C4—22no—
439C4—22no—
440C4—24no—
441C4—22no—
442C4—21no—
443C4—32no—
444C4—32no—
445C4—62no—
446C4—62no—
447C4—61no—
448C4—64no—
449C4D3×#4112yes—
450C4#100912yes—
451C3#351111yescharacteristic, complemented, Sylow
452C2—42no—
453C2—62no—
454C2—122no—
455C2—122no—
456C2C2×D3×#4111yescharacteristic
457C2C2×#100911yescharacteristic
458C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#556C2
hypercenter#556D3
hypocenterC3#3511
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. #5726C2semidirect
3. D3×C4⋊Q8C2semidirect
4. #5710C2semidirect
5. #5709C2semidirect
6. #5677C2semidirect
7. #5668C2semidirect
8. #5635C2semidirect
9. #5629C2semidirect
10. #5584C2semidirect
11. #5578C2semidirect
12. #5576C2semidirect
13. C4×#239C2semidirect
14. #5555C2semidirect
15. #5548C2semidirect
16. #5532C2semidirect
17. #5523C2semidirect
85. #556D3semidirect
275. C4×D3#41semidirect
451. C3#3511semidirect

Maximal subgroups

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

subgroupindex
C3×#5562
#57262
D3×C4⋊Q82
#57102
#57092
#56772
#56682
#56352
#56292
#55842
#55782
#55762
C4×#2392
#55552
#55482
#55322
#55232
#35113