Finite Group Databasebeta

← #86229

Subgroups — #86229

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×#532C211yesmaximal, characteristic, complemented
2#5824C214yesmaximal, complemented
3#5730C211yesmaximal, characteristic, complemented
4#5722C211yesmaximal, characteristic, complemented
5#5718C211yesmaximal, characteristic, complemented
6#5713C211yesmaximal, characteristic, complemented
7#5669C214yesmaximal, complemented
8#5611C212yesmaximal, complemented
9#5600C212yesmaximal, complemented
10#5598C212yesmaximal, complemented
11#5596C212yesmaximal, complemented
12#5561C212yesmaximal, complemented
13#5559C212yesmaximal, complemented
14#5546C212yesmaximal, complemented
15#5539C212yesmaximal, complemented
16#5479C211yesmaximal, characteristic, complemented
17C2×C4×Dic6C211yesmaximal, characteristic, complemented
18#3522—31nomaximal, Sylow
19C2×C3×#41C2212yes—
20C2×#237C2212yes—
21C2×#237C2214yes—
22C22×Dic6C2211yescharacteristic
23C3×C4⋊Q8C2211yescharacteristic
24C3×#127C2211yescharacteristic
25C3×#124C2212yes—
26C3×#121C2212yes—
27C3×#121C2212yes—
28C3×C4×D4C2212yes—
29C3×C4×D4C2212yes—
30C2×C3×C42C2211yescharacteristic
31#946C2214yes—
32#944C2214yes—
33#940C2212yes—
34#939C2214yes—
35#939C2214yes—
36#938C2212yes—
37#936C2212yes—
38#936C2212yes—
39#935C2214yes—
40#935C2214yes—
41D4×Dic3C2212yes—
42D4×Dic3C2212yes—
43D4×Dic3C2212yes—
44D4×Dic3C2214yes—
45#930C2211yescharacteristic
46#929C2212yes—
47#929C2214yes—
48C4×#67C2212yes—
49C4×#67C2214yes—
50C2×C4⋊Dic3C2211yescharacteristic
51#924C2211yescharacteristic
52#924C2214yes—
53C2×Dic3⋊C4C2212yes—
54C2×Dic3⋊C4C2214yes—
55C2×C4×Dic3C2212yes—
56#897C2212yes—
57#893C2212yes—
58D3×C4⋊C4C2212yes—
59#890C2212yes—
60#888C2212yes—
61#888C2212yes—
62#887C2212yes—
63#887C2212yes—
64#885C2214yes—
65#885C2214yes—
66#883C2214yes—
67#67⋊C4C2214yes—
68#879C2214yes—
69#878C2214yes—
70#878C2214yes—
71#877C2214yes—
72#876C2212yes—
73#875C2211yescharacteristic
74C4×D12C2211yescharacteristic
75#872C2212yes—
76#870C2211yescharacteristic
77C4×Dic6C2212yes—
78C4×Dic6C2211yescharacteristic
79C4×Dic6C2211yescharacteristic
80C4×Dic6C2211yescharacteristic
81C4×Dic6C2212yes—
82C4×Dic6C2211yescharacteristic
83C4×Dic6C2211yescharacteristic
84#559—31no—
85#559—32no—
86Q82—31no—
87#554—31no—
88#554—32no—
89#550—31no—
90#548—32no—
91#547—34no—
92#547—32no—
93#542—34no—
94#542—32no—
95#532—32no—
96#532D311yescharacteristic, complemented
97#529—32no—
98#529—31no—
99#520—32no—
100C2×C4×Q8—31no—
101C3×#41—44no—
102C2×C3×Q8C2312yes—
103C2×C3×D4C2312yes—
104C2×C3×D4C2314yes—
105C2×C3×D4C2312yes—
106C2×C3×D4C2312yes—
107C22×C3×C4C2311yescharacteristic
108C22×C3×C4C2311yescharacteristic
109C22×C3×C4C2311yescharacteristic
110C22×C3×C4C2314yes—
111C2×#67C2314yes—
112C2×#67C2312yes—
113C2×#67C2314yes—
114C22×Dic3C2314yes—
115C22×Dic3C2312yes—
116C22×Dic3C2312yes—
117C22×Dic3C2314yes—
118#237—28no—
119#237—48no—
120C2×D12C2311yescharacteristic
121C2×C4×D3C2312yes—
122C2×C4×D3C2314yes—
123C2×Dic6—24no—
124C2×Dic6C2312yes—
125C2×Dic6C2311yescharacteristic
126C2×Dic6C2311yescharacteristic
127C2×Dic6C2314yes—
128C2×Dic6C2311yescharacteristic
129C3×C4⋊C4C2312yes—
130C3×C4⋊C4C2312yes—
131C3×C22⋊C4C2314yes—
132C3×C22⋊C4C2314yes—
133C3×C42C2311yescharacteristic
134C3×C42C2311yescharacteristic
135C3×C42C2311yescharacteristic
136C3×C42C2311yescharacteristic
137#217C2314yes—
138#217C2314yes—
139#217C2314yes—
140#217C2314yes—
141#217C2312yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2312yes—
145(C2×D3)⋊C4C2312yes—
146(C2×D3)⋊C4C2314yes—
147C4⋊Dic3C2312yes—
148C4⋊Dic3C2311yescharacteristic
149C4⋊Dic3C2311yescharacteristic
150C4⋊Dic3C2311yescharacteristic
151C4⋊Dic3C2311yescharacteristic
152C4⋊Dic3C2314yes—
153Dic3⋊C4C2312yes—
154Dic3⋊C4C2312yes—
155Dic3⋊C4C2312yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2312yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2314yes—
162Dic3⋊C4C2314yes—
163Dic3⋊C4C2314yes—
164C4×Dic3C2312yes—
165C4×Dic3C2312yes—
166C4×Dic3C2312yes—
167C4×Dic3C2312yes—
168C4×Dic3C2312yes—
169C4×Dic3C2314yes—
170C2×#41—32no—
171C2×#41—34no—
172C2×#41C2×D312yes—
173C22×Q8—31no—
174C4⋊Q8—34no—
175C4⋊Q8—32no—
176C4⋊Q8—32no—
177C4⋊Q8C2×D311yescharacteristic
178#127—32no—
179#127C2×D311yescharacteristic
180#126—34no—
181#126—32no—
182#125—32no—
183#125—31no—
184#124—32no—
185#124—32no—
186#124—34no—
187#124—34no—
188#124—31no—
189#124C2×D312yes—
190#123—34no—
191#123—34no—
192#123—32no—
193#123—34no—
194#123—34no—
195C22⋊Q8—34no—
196C22⋊Q8—31no—
197C22⋊Q8—34no—
198C22⋊Q8—32no—
199C22⋊Q8—34no—
200C22⋊Q8—34no—
201#121—34no—
202#121—32no—
203#121—34no—
204#121—31no—
205#121C2×D312yes—
206#121C2×D312yes—
207C4×Q8—32no—
208C4×Q8—32no—
209C4×Q8—31no—
210C4×Q8—32no—
211C4×Q8—31no—
212C4×Q8—31no—
213C4×Q8—31no—
214C4×Q8—32no—
215C4×Q8—31no—
216C4×D4—32no—
217C4×D4—34no—
218C4×D4—32no—
219C4×D4—32no—
220C4×D4—34no—
221C4×D4—32no—
222C4×D4—32no—
223C4×D4—34no—
224C4×D4—31no—
225C4×D4C2×D312yes—
226C4×D4C2×D312yes—
227#117—34no—
228#117—32no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—31no—
231C2×C4⋊C4—34no—
232C2×C4⋊C4—32no—
233C2×C42—32no—
234C2×C42C2×D311yescharacteristic
235C23×C3C2414yes—
236C23×C3C2411yescharacteristic
237C22×D3C2412yes—
238C3×Q8—42no—
239C3×D4—28no—
240C3×D4—24no—
241C3×D4—24no—
242C3×D4—42no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—44no—
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2414yes—
249C2×C3×C4C2411yescharacteristic
250C2×C3×C4C2411yescharacteristic
251C2×C3×C4C2411yescharacteristic
252C2×C3×C4C2411yescharacteristic
253C2×C3×C4C2411yescharacteristic
254#67—24no—
255#67—28no—
256#67—44no—
257C2×Dic3—24no—
258C2×Dic3—24no—
259C2×Dic3—28no—
260C2×Dic3—44no—
261C2×Dic3C2412yes—
262C2×Dic3C2412yes—
263C2×Dic3C2412yes—
264C2×Dic3C2412yes—
265C2×Dic3C2412yes—
266C2×Dic3C2412yes—
267C2×Dic3C2412yes—
268D12—22no—
269C4×D3—24no—
270C4×D3—44no—
271Dic6—22no—
272Dic6—22no—
273Dic6—24no—
274Dic6—44no—
275Dic6#4114yescomplemented
276#41—44no—
277#41—68no—
278#41—128no—
279C2×Q8—31no—
280C2×Q8—31no—
281C2×Q8—32no—
282C2×Q8—31no—
283C2×Q8—34no—
284C2×Q8—64no—
285C2×Q8C22×D312yes—
286C2×D4—34no—
287C2×D4—32no—
288C2×D4—34no—
289C2×D4—31no—
290C2×D4C22×D312yes—
291C2×D4C22×D312yes—
292C2×D4C22×D314yes—
293C2×D4C22×D312yes—
294C22×C4—32no—
295C22×C4—32no—
296C22×C4—34no—
297C22×C4—34no—
298C22×C4—32no—
299C22×C4—34no—
300C22×C4C22×D311yescharacteristic
301C22×C4C22×D311yescharacteristic
302C22×C4C22×D311yescharacteristic
303C22×C4C22×D314yes—
304C4⋊C4—32no—
305C4⋊C4—31no—
306C4⋊C4—34no—
307C4⋊C4—32no—
308C4⋊C4—34no—
309C4⋊C4—32no—
310C4⋊C4—31no—
311C4⋊C4—32no—
312C4⋊C4—32no—
313C4⋊C4—32no—
314C4⋊C4—32no—
315C4⋊C4—31no—
316C4⋊C4—31no—
317C4⋊C4—32no—
318C4⋊C4—34no—
319C4⋊C4—34no—
320C4⋊C4—32no—
321C4⋊C4C22×D312yes—
322C4⋊C4C22×D312yes—
323C22⋊C4—34no—
324C22⋊C4—32no—
325C22⋊C4—34no—
326C22⋊C4—34no—
327C22⋊C4—34no—
328C22⋊C4—32no—
329C22⋊C4—32no—
330C22⋊C4—32no—
331C22⋊C4—34no—
332C22⋊C4—32no—
333C22⋊C4C22×D314yes—
334C22⋊C4C22×D314yes—
335C42—32no—
336C42—32no—
337C42—32no—
338C42—32no—
339C42—34no—
340C42—32no—
341C42C22×D311yescharacteristic
342C42C22×D311yescharacteristic
343C42C22×D311yescharacteristic
344C42C22×D311yescharacteristic
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—24no—
348C22×C3—24no—
349C22×C3—24no—
350C22×C3H(5,𝔽2)12yes—
351C22×C3C2511yescharacteristic
352C2×D3—22no—
353C2×D3—22no—
354C2×D3—22no—
355C3×C4—21no—
356C3×C4—24no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4C2×#4112yes—
361C3×C42−1+412yes—
362Dic3—22no—
363Dic3—22no—
364Dic3—22no—
365Dic3—22no—
366Dic3—22no—
367Dic3—22no—
368Dic3C2×#4114yes—
369C23—32no—
370C23C23×D314yes—
371C23C23×D311yescharacteristic
372Q8—34no—
373Q8—42no—
374Q8—64no—
375Q8—62no—
376Q8—62no—
377Q8—124no—
378D4—28no—
379D4—24no—
380D4—24no—
381D4—42no—
382D4—68no—
383D4—64no—
384D4—62no—
385D4—124no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—32no—
390C2×C4—32no—
391C2×C4—32no—
392C2×C4—32no—
393C2×C4—32no—
394C2×C4—32no—
395C2×C4—32no—
396C2×C4—44no—
397C2×C4—64no—
398C2×C4—68no—
399C2×C4—64no—
400C2×C4—64no—
401C2×C4—124no—
402C2×C4—124no—
403C2×C4C23×D311yescharacteristic
404C2×C4C23×D314yes—
405C2×C4C23×D311yescharacteristic
406C2×C4C23×D311yescharacteristic
407C2×C4C23×D311yescharacteristic
408C2×C4C23×D311yescharacteristic
409C2×C4C23×D311yescharacteristic
410C2×C3—22no—
411C2×C3—44no—
412C2×C3C22×#4111yescharacteristic
413C2×C3C2×H(5,𝔽2)11yescharacteristic
414C2×C3C2×2−1+411yescharacteristic
415D3—42no—
416C22—21no—
417C22—21no—
418C22—24no—
419C22—24no—
420C22—24no—
421C22—62no—
422C22—62no—
423C22—62no—
424C22#100412yes—
425C22C24×D311yescharacteristic
426C4—21no—
427C4—21no—
428C4—24no—
429C4—21no—
430C4—21no—
431C4—34no—
432C4—62no—
433C4—62no—
434C4—62no—
435C4—62no—
436C4—62no—
437C4—62no—
438C4D3×#4112yes—
439C4#101012yes—
440C3#352211yescharacteristic, complemented, Sylow
441C2—22no—
442C2—44no—
443C2—122no—
444C2C2×#100411yescharacteristic
445C2C2×D3×#4111yescharacteristic
446C2C2×#101011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#532C2
hypercenter#532D3
hypocenterC3#3522
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. #5824C2semidirect
3. #5730C2semidirect
4. #5722C2semidirect
5. #5718C2semidirect
6. #5713C2semidirect
7. #5669C2semidirect
8. #5611C2semidirect
9. #5600C2semidirect
10. #5598C2semidirect
11. #5596C2semidirect
12. #5561C2semidirect
13. #5559C2semidirect
14. #5546C2semidirect
15. #5539C2semidirect
16. #5479C2semidirect
17. C2×C4×Dic6C2semidirect
96. #532D3semidirect
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×#5322
#58242
#57302
#57222
#57182
#57132
#56692
#56112
#56002
#55982
#55962
#55612
#55592
#55462
#55392
#54792
C2×C4×Dic62
#35223