Finite Group Databasebeta

← #86218

Subgroups — #86218

Order 384.

1004 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2#5817C212yesmaximal, complemented
3Q8×#67C212yesmaximal, complemented
4#5735C214yesmaximal, complemented
5#5733C212yesmaximal, complemented
6#5732C211yesmaximal, characteristic, complemented
7#5731C212yesmaximal, complemented
8#5729C212yesmaximal, complemented
9#5727C211yesmaximal, characteristic, complemented
10#5726C211yesmaximal, characteristic, complemented
11#5724C212yesmaximal, complemented
12#5721C211yesmaximal, characteristic, complemented
13#5515C214yesmaximal, complemented
14#5514C212yesmaximal, complemented
15#5511C212yesmaximal, complemented
16#5479C211yesmaximal, characteristic, complemented
17#5479C211yesmaximal, characteristic, complemented
18#3552—31nomaximal, Sylow
19C22×C3×Q8C2212yes—
20C2×#239C2212yes—
21C2×D3×Q8C2212yes—
22C3×C4⋊Q8C2211yescharacteristic
23C3×C4⋊Q8C2212yes—
24C3×C4⋊Q8C2211yescharacteristic
25C3×C4⋊Q8C2212yes—
26C3×C4⋊Q8C2212yes—
27C2×C3×C4⋊C4C2212yes—
28C2×C3×C4⋊C4C2212yes—
29C2×C3×C42C2211yescharacteristic
30#947C2212yes—
31#947C2214yes—
32#946C2212yes—
33#946C2212yes—
34#946C2214yes—
35#946C2214yes—
36Q8×Dic3C2212yes—
37Q8×Dic3C2212yes—
38#944C2212yes—
39#944C2214yes—
40#930C2211yescharacteristic
41#930C2211yescharacteristic
42#930C2214yes—
43#929C2212yes—
44#929C2212yes—
45C4×#67C2212yes—
46C4×#67C2212yes—
47C4×#67C2214yes—
48C4×#67C2214yes—
49#924C2211yescharacteristic
50#924C2211yescharacteristic
51#924C2214yes—
52#898C2214yes—
53#898C2214yes—
54#898C2214yes—
55#897C2212yes—
56#896C2214yes—
57#896C2214yes—
58#896C2214yes—
59#895C2212yes—
60#894C2214yes—
61#894C2214yes—
62#894C2214yes—
63#893C2212yes—
64#892C2212yes—
65D3×C4⋊C4C2212yes—
66#890C2212yes—
67#889C2214yes—
68#889C2214yes—
69#889C2214yes—
70#888C2212yes—
71#887C2212yes—
72#876C2212yes—
73#876C2212yes—
74#875C2211yescharacteristic
75#875C2211yescharacteristic
76C4×D12C2211yescharacteristic
77C4×D12C2211yescharacteristic
78#872C2212yes—
79#872C2212yes—
80#870C2211yescharacteristic
81#870C2211yescharacteristic
82C4×Dic6C2211yescharacteristic
83C4×Dic6C2211yescharacteristic
84#563—34no—
85#557—31no—
86#556—31no—
87#554—32no—
88#553—32no—
89D4×Q8—32no—
90D4×Q8—31no—
91#548—32no—
92#547—31no—
93#544—32no—
94#542—32no—
95#541—32no—
96#539—34no—
97#536—32no—
98C2×C4⋊Q8D311yescharacteristic, complemented
99#529—31no—
100#529—31no—
101C2×C3×Q8—44no—
102C2×C3×Q8C2312yes—
103C2×C3×Q8C2312yes—
104C2×C3×Q8C2314yes—
105C22×C3×C4C2311yescharacteristic
106C22×C3×C4C2311yescharacteristic
107C22×C3×C4C2311yescharacteristic
108C22×C3×C4C2314yes—
109C2×#67C2312yes—
110C2×#67C2312yes—
111#239—44no—
112D3×Q8—44no—
113C2×D12C2311yescharacteristic
114C2×D12C2311yescharacteristic
115C2×D12C2314yes—
116C2×C4×D3C2312yes—
117C2×C4×D3C2312yes—
118C2×C4×D3C2314yes—
119C2×C4×D3C2314yes—
120C2×Dic6C2311yescharacteristic
121C2×Dic6C2311yescharacteristic
122C2×Dic6C2314yes—
123C3×C4⋊C4C2312yes—
124C3×C4⋊C4C2312yes—
125C3×C4⋊C4C2312yes—
126C3×C4⋊C4C2314yes—
127C3×C4⋊C4C2314yes—
128C3×C4⋊C4C2312yes—
129C3×C42C2311yescharacteristic
130C3×C42C2311yescharacteristic
131C3×C42C2311yescharacteristic
132C3×C42C2311yescharacteristic
133#217C2312yes—
134#217C2312yes—
135(C2×D3)⋊C4C2312yes—
136(C2×D3)⋊C4C2312yes—
137(C2×D3)⋊C4C2312yes—
138(C2×D3)⋊C4C2312yes—
139(C2×D3)⋊C4C2312yes—
140(C2×D3)⋊C4C2312yes—
141(C2×D3)⋊C4C2312yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2314yes—
144(C2×D3)⋊C4C2314yes—
145(C2×D3)⋊C4C2314yes—
146(C2×D3)⋊C4C2314yes—
147C4⋊Dic3C2311yescharacteristic
148C4⋊Dic3C2311yescharacteristic
149C4⋊Dic3C2311yescharacteristic
150C4⋊Dic3C2311yescharacteristic
151C4⋊Dic3C2314yes—
152C4⋊Dic3C2314yes—
153Dic3⋊C4C2312yes—
154Dic3⋊C4C2312yes—
155Dic3⋊C4C2312yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2312yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2314yes—
162Dic3⋊C4C2314yes—
163Dic3⋊C4C2314yes—
164Dic3⋊C4C2314yes—
165C4×Dic3C2312yes—
166C4×Dic3C2312yes—
167C4×Dic3C2314yes—
168C4×Dic3C2314yes—
169C2×#41—32no—
170C22×Q8—32no—
171C22×Q8C2×D312yes—
172C4⋊Q8—32no—
173C4⋊Q8—34no—
174C4⋊Q8—32no—
175C4⋊Q8C2×D312yes—
176C4⋊Q8C2×D312yes—
177C4⋊Q8C2×D311yescharacteristic
178C4⋊Q8C2×D311yescharacteristic
179C4⋊Q8C2×D312yes—
180#126—34no—
181#126—34no—
182#126—34no—
183#126—32no—
184#126—32no—
185#125—34no—
186#125—34no—
187#125—34no—
188#125—32no—
189#125—31no—
190#125—31no—
191#124—32no—
192#124—34no—
193#124—31no—
194#124—31no—
195#123—32no—
196#123—32no—
197#123—34no—
198#123—34no—
199#123—34no—
200C22⋊Q8—32no—
201C22⋊Q8—32no—
202C22⋊Q8—34no—
203C22⋊Q8—34no—
204C22⋊Q8—31no—
205C22⋊Q8—31no—
206C22⋊Q8—34no—
207C22⋊Q8—34no—
208C22⋊Q8—34no—
209C22⋊Q8—34no—
210C22⋊Q8—32no—
211#121—31no—
212#121—31no—
213#121—34no—
214#121—32no—
215C4×Q8—32no—
216C4×Q8—32no—
217C4×Q8—32no—
218C4×Q8—31no—
219C4×Q8—31no—
220C4×D4—32no—
221C4×D4—32no—
222C4×D4—34no—
223C4×D4—34no—
224C4×D4—32no—
225C4×D4—31no—
226C4×D4—31no—
227#117—32no—
228#117—32no—
229#117—32no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4C2×D312yes—
232C2×C4⋊C4C2×D312yes—
233C2×C42C2×D311yescharacteristic
234C23×C3C2411yescharacteristic
235C22×D3C2412yes—
236C22×D3C2412yes—
237C3×Q8—24no—
238C3×Q8—24no—
239C3×Q8—44no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—44no—
244C2×C3×C4C2414yes—
245C2×C3×C4C2414yes—
246C2×C3×C4C2411yescharacteristic
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2411yescharacteristic
249C2×C3×C4C2411yescharacteristic
250C2×C3×C4C2411yescharacteristic
251C2×C3×C4C2411yescharacteristic
252#67—24no—
253#67—24no—
254C2×Dic3C2412yes—
255C2×Dic3C2412yes—
256C2×Dic3C2412yes—
257C2×Dic3C2412yes—
258C2×Dic3C2412yes—
259C2×Dic3C2412yes—
260D12—22no—
261D12—22no—
262D12—44no—
263C4×D3—24no—
264C4×D3—24no—
265C4×D3—44no—
266C4×D3—44no—
267Dic6—22no—
268Dic6—22no—
269Dic6—44no—
270#41—124no—
271C2×Q8—31no—
272C2×Q8—31no—
273C2×Q8—34no—
274C2×Q8—44no—
275C2×Q8—124no—
276C2×Q8C22×D312yes—
277C2×Q8C22×D312yes—
278C2×Q8C22×D314yes—
279C2×D4—32no—
280C2×D4—32no—
281C2×D4—31no—
282C2×D4—31no—
283C2×D4—34no—
284C22×C4—32no—
285C22×C4—32no—
286C22×C4—34no—
287C22×C4—34no—
288C22×C4C22×D311yescharacteristic
289C22×C4C22×D311yescharacteristic
290C22×C4C22×D311yescharacteristic
291C22×C4C22×D314yes—
292C4⋊C4—31no—
293C4⋊C4—34no—
294C4⋊C4—31no—
295C4⋊C4—32no—
296C4⋊C4—32no—
297C4⋊C4—32no—
298C4⋊C4—32no—
299C4⋊C4—31no—
300C4⋊C4—34no—
301C4⋊C4—32no—
302C4⋊C4—34no—
303C4⋊C4—32no—
304C4⋊C4—31no—
305C4⋊C4—34no—
306C4⋊C4—34no—
307C4⋊C4—34no—
308C4⋊C4—32no—
309C4⋊C4—32no—
310C4⋊C4C22×D312yes—
311C4⋊C4C22×D312yes—
312C4⋊C4C22×D312yes—
313C4⋊C4C22×D314yes—
314C4⋊C4C22×D314yes—
315C4⋊C4C22×D312yes—
316C22⋊C4—32no—
317C22⋊C4—32no—
318C22⋊C4—34no—
319C22⋊C4—32no—
320C22⋊C4—32no—
321C22⋊C4—32no—
322C22⋊C4—32no—
323C22⋊C4—34no—
324C22⋊C4—34no—
325C22⋊C4—32no—
326C22⋊C4—32no—
327C22⋊C4—34no—
328C22⋊C4—32no—
329C22⋊C4—32no—
330C42—32no—
331C42—32no—
332C42—34no—
333C42—34no—
334C42C22×D311yescharacteristic
335C42C22×D311yescharacteristic
336C42C22×D311yescharacteristic
337C42C22×D311yescharacteristic
338C22×C3—21no—
339C22×C3—21no—
340C22×C32−1+412yes—
341C22×C3C2511yescharacteristic
342C2×D3—22no—
343C2×D3—22no—
344C2×D3—22no—
345C2×D3—22no—
346C2×D3—22no—
347C2×D3—22no—
348C3×C4—24no—
349C3×C4—24no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4H(5,𝔽2)12yes—
355C3×C42−1+412yes—
356Dic3—22no—
357Dic3—22no—
358Dic3—22no—
359Dic3—22no—
360Dic3—22no—
361Dic3—22no—
362C23—32no—
363C23—32no—
364C23C23×D311yescharacteristic
365Q8—24no—
366Q8—24no—
367Q8—44no—
368Q8—62no—
369Q8—62no—
370Q8—124no—
371D4—64no—
372D4—64no—
373D4—62no—
374D4—62no—
375D4—124no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—32no—
380C2×C4—32no—
381C2×C4—32no—
382C2×C4—32no—
383C2×C4—32no—
384C2×C4—32no—
385C2×C4—44no—
386C2×C4—64no—
387C2×C4—64no—
388C2×C4—124no—
389C2×C4—124no—
390C2×C4C23×D314yes—
391C2×C4C23×D314yes—
392C2×C4C23×D311yescharacteristic
393C2×C4C23×D311yescharacteristic
394C2×C4C23×D311yescharacteristic
395C2×C4C23×D311yescharacteristic
396C2×C4C23×D311yescharacteristic
397C2×C4C23×D311yescharacteristic
398C2×C3—22no—
399C2×C3C2×H(5,𝔽2)11yescharacteristic
400C2×C3C2×2−1+411yescharacteristic
401C2×C3C2×2−1+411yescharacteristic
402D3—42no—
403D3—42no—
404C22—21no—
405C22—21no—
406C22—62no—
407C22—62no—
408C22—62no—
409C22—62no—
410C22—62no—
411C22—62no—
412C22#100712yes—
413C22C24×D311yescharacteristic
414C4—21no—
415C4—24no—
416C4—24no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—62no—
421C4—62no—
422C4—62no—
423C4—62no—
424C4—62no—
425C4—62no—
426C4#100412yes—
427C4#100712yes—
428C3#355211yescharacteristic, complemented, Sylow
429C2—22no—
430C2—122no—
431C2—122no—
432C2C2×#100411yescharacteristic
433C2C2×#100711yescharacteristic
434C2C2×#100711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC2×C3×C4⋊Q8C2
hypercenterC2×C4⋊Q8D3
hypocenterC3#3552
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4⋊Q8C2semidirect
2. #5817C2semidirect
3. Q8×#67C2semidirect
4. #5735C2semidirect
5. #5733C2semidirect
6. #5732C2semidirect
7. #5731C2semidirect
8. #5729C2semidirect
9. #5727C2semidirect
10. #5726C2semidirect
11. #5724C2semidirect
12. #5721C2semidirect
13. #5515C2semidirect
14. #5514C2semidirect
15. #5511C2semidirect
16. #5479C2semidirect
17. #5479C2semidirect
98. C2×C4⋊Q8D3semidirect
428. C3#3552semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊Q82
#58172
Q8×#672
#57352
#57332
#57322
#57312
#57292
#57272
#57262
#57242
#57212
#55152
#55142
#55112
#54792
#35523