Finite Group Databasebeta

← #86277

Subgroups — #86277

Order 384.

1068 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#533C211yesmaximal, characteristic, complemented
2#5730C211yesmaximal, characteristic, complemented
3#5727C211yesmaximal, characteristic, complemented
4#5722C211yesmaximal, characteristic, complemented
5#5721C211yesmaximal, characteristic, complemented
6#5690C212yesmaximal, complemented
7#5685C212yesmaximal, complemented
8#5636C214yesmaximal, complemented
9#5635C212yesmaximal, complemented
10#5632C212yesmaximal, complemented
11#5578C214yesmaximal, complemented
12#5578C212yesmaximal, complemented
13C4×D3×Q8C212yesmaximal, complemented
14#5522C212yesmaximal
15#5517C212yesmaximal, complemented
16C4×#235C211yesmaximal, characteristic
17C2×C4×Dic6C211yesmaximal, characteristic, complemented
18Q8×#41—31nomaximal, Sylow
19C2×D3×Q8C2212yes—
20C2×#235C2211yescharacteristic
21C22×Dic6C2211yescharacteristic
22C3×C4⋊Q8C2211yescharacteristic
23C3×C4⋊Q8C2211yescharacteristic
24C3×#125C2212yes—
25C3×C22⋊Q8C2212yes—
26C3×C22⋊Q8C2212yes—
27C3×C4×Q8C2212yes—
28C3×C4×Q8C2212yes—
29C3×#117C2212yes—
30C2×C3×C42C2211yescharacteristic
31#946C2212yes—
32#946C2214yes—
33Q8×Dic3C2212yes—
34#944C2212yes—
35#944C2214yes—
36C4×#67C2212yes—
37C4×#67C2212yes—
38#926C2211yescharacteristic
39C2×C4⋊Dic3C2211yescharacteristic
40C2×Dic3⋊C4C2212yes—
41C2×C4×Dic3C2212yes—
42#897C2212yes—
43#897C2214yes—
44#896C2214yes—
45#896C2214yes—
46#896C2214yes—
47#893C2212yes—
48#893C2212yes—
49D3×C4⋊C4C2212yes—
50D3×C4⋊C4C2214yes—
51#890C2212yes—
52#890C2212yes—
53#889C2214yes—
54#889C2214yes—
55#889C2214yes—
56#888C2214yes—
57#888C2212yes—
58#888C2212yes—
59#888C2212yes—
60#887C2212yes—
61#887C2214yes—
62#887C2212yes—
63#887C2212yes—
64#67⋊C4C2214yes—
65#67⋊C4C2214yes—
66#878C2214yes—
67#878C2214yes—
68C4×D12C2211yescharacteristic
69C4×D12C2211yescharacteristic
70#872C2212yes—
71#872C2214yes—
72C42×D3C2212yes—
73C42×D3C2214yes—
74C4×Dic6C2211yescharacteristic
75C4×Dic6C2212yes—
76C4×Dic6C2211yescharacteristic
77C4×Dic6C2211yescharacteristic
78C4×Dic6C2211yescharacteristic
79C4×Dic6C2211yescharacteristic
80C4×Dic6C2212yes—
81C4×Dic6C2211yescharacteristic
82C4×Dic6C2214yes—
83C4×Dic6C2214yes—
84Q82—31no—
85#557—31no—
86#557—32no—
87#554—31no—
88#554—32no—
89#554—32no—
90#554—34no—
91D4×Q8—31no—
92D4×Q8—32no—
93#533—34no—
94#533—32no—
95#533—32no—
96#533D311yescharacteristic, complemented
97C4×#41—32no—
98C4×#41—31no—
99C2×C4×Q8—32no—
100C2×C4×Q8—31no—
101C2×C3×Q8C2312yes—
102C2×C3×Q8C2312yes—
103C22×C3×C4C2311yescharacteristic
104C22×C3×C4C2311yescharacteristic
105C22×C3×C4C2311yescharacteristic
106C2×#67C2312yes—
107C22×Dic3C2312yes—
108C22×Dic3C2312yes—
109D3×Q8—28no—
110#235Q818yescomplemented
111C2×D12C2311yescharacteristic
112C2×C4×D3C2312yes—
113C2×C4×D3C2314yes—
114C2×C4×D3C2312yes—
115C2×C4×D3C2314yes—
116C2×C4×D3C2312yes—
117C2×Dic6—24no—
118C2×Dic6C2312yes—
119C2×Dic6C2311yescharacteristic
120C2×Dic6C2314yes—
121C2×Dic6C2314yes—
122C2×Dic6C2311yescharacteristic
123C2×Dic6C2311yescharacteristic
124C3×C4⋊C4C2312yes—
125C3×C4⋊C4C2312yes—
126C3×C4⋊C4C2312yes—
127C3×C4⋊C4C2314yes—
128C3×C4⋊C4C2314yes—
129C3×C4⋊C4C2312yes—
130C3×C22⋊C4C2314yes—
131C3×C42C2311yescharacteristic
132C3×C42C2311yescharacteristic
133C3×C42C2311yescharacteristic
134C3×C42C2314yes—
135C3×C42C2311yescharacteristic
136#217C2312yes—
137(C2×D3)⋊C4C2312yes—
138(C2×D3)⋊C4C2312yes—
139(C2×D3)⋊C4C2314yes—
140(C2×D3)⋊C4C2314yes—
141C4⋊Dic3C2311yescharacteristic
142C4⋊Dic3C2312yes—
143C4⋊Dic3C2311yescharacteristic
144C4⋊Dic3C2311yescharacteristic
145C4⋊Dic3C2311yescharacteristic
146C4⋊Dic3C2314yes—
147C4⋊Dic3C2314yes—
148Dic3⋊C4C2312yes—
149Dic3⋊C4C2312yes—
150Dic3⋊C4C2312yes—
151Dic3⋊C4C2312yes—
152Dic3⋊C4C2312yes—
153Dic3⋊C4C2312yes—
154Dic3⋊C4C2314yes—
155Dic3⋊C4C2314yes—
156Dic3⋊C4C2314yes—
157Dic3⋊C4C2314yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2314yes—
160C4×Dic3C2312yes—
161C4×Dic3C2312yes—
162C4×Dic3C2312yes—
163C4×Dic3C2312yes—
164C4×Dic3C2312yes—
165C4×Dic3C2312yes—
166C4×Dic3C2312yes—
167C4×Dic3C2314yes—
168C4×Dic3C2314yes—
169C2×#41—31no—
170C22×Q8—32no—
171C22×Q8—31no—
172C4⋊Q8—32no—
173C4⋊Q8—34no—
174C4⋊Q8—34no—
175C4⋊Q8—32no—
176C4⋊Q8—32no—
177C4⋊Q8—32no—
178C4⋊Q8C2×D311yescharacteristic
179C4⋊Q8C2×D311yescharacteristic
180#125—34no—
181#125—34no—
182#125—34no—
183#125—32no—
184#125—32no—
185#125C2×D312yes—
186C22⋊Q8—32no—
187C22⋊Q8—34no—
188C22⋊Q8—34no—
189C22⋊Q8—34no—
190C22⋊Q8—32no—
191C22⋊Q8—34no—
192C22⋊Q8—34no—
193C22⋊Q8—34no—
194C22⋊Q8—34no—
195C22⋊Q8C2×D312yes—
196C22⋊Q8C2×D312yes—
197C4×Q8—32no—
198C4×Q8—32no—
199C4×Q8—34no—
200C4×Q8—32no—
201C4×Q8—32no—
202C4×Q8—31no—
203C4×Q8—34no—
204C4×Q8—32no—
205C4×Q8—31no—
206C4×Q8—31no—
207C4×Q8—31no—
208C4×Q8—32no—
209C4×Q8—31no—
210C4×Q8—31no—
211C4×Q8—34no—
212C4×Q8C2×D312yes—
213C4×Q8C2×D312yes—
214C4×D4—32no—
215C4×D4—32no—
216C4×D4—32no—
217C4×D4—32no—
218C4×D4—34no—
219C4×D4—34no—
220C4×D4—31no—
221C4×D4—31no—
222#117—31no—
223#117—34no—
224#117—32no—
225#117C2×D312yes—
226C2×C4⋊C4—32no—
227C2×C4⋊C4—31no—
228C2×C4⋊C4—34no—
229C2×C4⋊C4—32no—
230C2×C42—32no—
231C2×C42—34no—
232C2×C42—32no—
233C2×C42C2×D311yescharacteristic
234C23×C3C2411yescharacteristic
235C22×D3C2412yes—
236C3×Q8—24no—
237C3×Q8—42no—
238C2×C3×C4—22no—
239C2×C3×C4—41no—
240C2×C3×C4C2×Q814yescomplemented
241C2×C3×C4C2411yescharacteristic
242C2×C3×C4C2414yes—
243C2×C3×C4C2414yes—
244C2×C3×C4C2411yescharacteristic
245C2×C3×C4C2411yescharacteristic
246C2×C3×C4C2411yescharacteristic
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2411yescharacteristic
249#67C2×Q818yescomplemented
250C2×Dic3—24no—
251C2×Dic3—42no—
252C2×Dic3C2412yes—
253C2×Dic3C2412yes—
254C2×Dic3C2412yes—
255C2×Dic3C2412yes—
256C2×Dic3C2412yes—
257C2×Dic3C2412yes—
258C2×Dic3C2412yes—
259D12C2×Q814yescomplemented
260C4×D3—24no—
261C4×D3—28no—
262C4×D3—42no—
263C4×D3—44no—
264C4×D3C2×Q818yescomplemented
265Dic6—22no—
266Dic6—24no—
267Dic6—28no—
268Dic6—44no—
269Dic6C2×Q814yescomplemented
270Dic6#4114yescomplemented
271#41—38no—
272C2×Q8—31no—
273C2×Q8—31no—
274C2×Q8—34no—
275C2×Q8—32no—
276C2×Q8—31no—
277C2×Q8—34no—
278C2×Q8—64no—
279C2×Q8—68no—
280C2×Q8C22×D312yes—
281C2×Q8C22×D312yes—
282C2×D4—32no—
283C2×D4—31no—
284C22×C4—32no—
285C22×C4—32no—
286C22×C4—32no—
287C22×C4—34no—
288C22×C4—32no—
289C22×C4—34no—
290C22×C4—32no—
291C22×C4C22×D311yescharacteristic
292C22×C4C22×D311yescharacteristic
293C22×C4C22×D311yescharacteristic
294C4⋊C4—32no—
295C4⋊C4—31no—
296C4⋊C4—34no—
297C4⋊C4—32no—
298C4⋊C4—32no—
299C4⋊C4—34no—
300C4⋊C4—32no—
301C4⋊C4—34no—
302C4⋊C4—32no—
303C4⋊C4—31no—
304C4⋊C4—34no—
305C4⋊C4—34no—
306C4⋊C4—32no—
307C4⋊C4—32no—
308C4⋊C4—31no—
309C4⋊C4—31no—
310C4⋊C4—34no—
311C4⋊C4—34no—
312C4⋊C4—34no—
313C4⋊C4C22×D314yes—
314C4⋊C4C22×D312yes—
315C4⋊C4C22×D312yes—
316C4⋊C4C22×D312yes—
317C4⋊C4C22×D312yes—
318C4⋊C4C22×D314yes—
319C22⋊C4—32no—
320C22⋊C4—34no—
321C22⋊C4—34no—
322C22⋊C4—32no—
323C22⋊C4—32no—
324C22⋊C4C22×D314yes—
325C42—34no—
326C42—32no—
327C42—32no—
328C42—34no—
329C42—32no—
330C42—32no—
331C42—32no—
332C42—32no—
333C42—32no—
334C42C22×D311yescharacteristic
335C42C22×D311yescharacteristic
336C42C22×D311yescharacteristic
337C42C22×D314yes—
338C42C22×D311yescharacteristic
339C22×C3—21no—
340C22×C3—21no—
341C22×C3C22×Q812yes—
342C22×C3C2511yescharacteristic
343C2×D3—22no—
344C2×D3—22no—
345C2×D3C22×Q814yes—
346C3×C4—21no—
347C3×C4—24no—
348C3×C4—24no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4C22×Q812yes—
352C3×C4C22×Q812yes—
353C3×C4C2×#4112yes—
354Dic3—22no—
355Dic3—22no—
356Dic3—22no—
357Dic3—22no—
358Dic3—22no—
359Dic3C22×Q814yes—
360Dic3C2×#4114yes—
361C23—32no—
362C23C23×D311yescharacteristic
363Q8—24no—
364Q8—34no—
365Q8—34no—
366Q8—42no—
367Q8—64no—
368Q8—68no—
369Q8—62no—
370Q8—124no—
371D4—38no—
372D4—34no—
373C2×C4—22no—
374C2×C4—32no—
375C2×C4—32no—
376C2×C4—38no—
377C2×C4—32no—
378C2×C4—32no—
379C2×C4—32no—
380C2×C4—32no—
381C2×C4—32no—
382C2×C4—41no—
383C2×C4—68no—
384C2×C4—64no—
385C2×C4—64no—
386C2×C4—122no—
387C2×C4—124no—
388C2×C4—122no—
389C2×C4D3×Q814yescomplemented
390C2×C4C23×D311yescharacteristic
391C2×C4C23×D314yes—
392C2×C4C23×D314yes—
393C2×C4C23×D311yescharacteristic
394C2×C4C23×D311yescharacteristic
395C2×C4C23×D311yescharacteristic
396C2×C4C23×D311yescharacteristic
397C2×C4C23×D311yescharacteristic
398C2×C3—22no—
399C2×C3C23×Q811yescharacteristic
400C2×C3C22×#4111yescharacteristic
401C2×C3#58511yescharacteristic
402D3—24no—
403C22—21no—
404C22—21no—
405C22—34no—
406C22—62no—
407C22—62no—
408C22C2×D3×Q812yes—
409C22C24×D311yescharacteristic
410C4—21no—
411C4—24no—
412C4—21no—
413C4—21no—
414C4—24no—
415C4—34no—
416C4—34no—
417C4—62no—
418C4—62no—
419C4—62no—
420C4—62no—
421C4—62no—
422C4C2×D3×Q812yes—
423C4C2×D3×Q812yes—
424C4D3×#4112yes—
425C3Q8×#4111yescharacteristic, complemented, Sylow
426C2—22no—
427C2—64no—
428C2C22×D3×Q811yescharacteristic
429C2C2×D3×#4111yescharacteristic
430C2#596411yescharacteristic
subgroupG/H
centerC2×C4C23×D3
commutator, socle, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC3×#533C2
prime radicalC2×#235C22
hypercenter#533D3
hypocenterC3Q8×#41
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #533

Complements

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

NH ≅ G/Ntype
1. C3×#533C2semidirect
2. #5730C2semidirect
3. #5727C2semidirect
4. #5722C2semidirect
5. #5721C2semidirect
6. #5690C2semidirect
7. #5685C2semidirect
8. #5636C2semidirect
9. #5635C2semidirect
10. #5632C2semidirect
11. #5578C2semidirect
12. #5578C2semidirect
13. C4×D3×Q8C2semidirect
15. #5517C2semidirect
17. C2×C4×Dic6C2semidirect
96. #533D3semidirect
110. #235Q8semidirect
240. C2×C3×C4C2×Q8semidirect
249. #67C2×Q8semidirect
259. D12C2×Q8semidirect
264. C4×D3C2×Q8semidirect
269. Dic6C2×Q8semidirect
270. Dic6#41semidirect
389. C2×C4D3×Q8semidirect
425. C3Q8×#41semidirect

Maximal subgroups

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

subgroupindex
C3×#5332
#57302
#57272
#57222
#57212
#56902
#56852
#56362
#56352
#56322
#55782
C4×D3×Q82
#55222
#55172
C4×#2352
C2×C4×Dic62
Q8×#413