Finite Group Databasebeta

← #86953

Subgroups — #86953

Order 384.

1004 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#554C211yesmaximal, characteristic, complemented
2#5730C211yesmaximal, characteristic, complemented
3D3×C4⋊Q8C211yesmaximal, characteristic, complemented
4#5694C212yesmaximal, complemented
5#5689C212yesmaximal, complemented
6#5646C214yesmaximal, complemented
7#5643C212yesmaximal, complemented
8#5639C212yesmaximal, complemented
9#5628C214yesmaximal, complemented
10#5580C211yesmaximal, characteristic, complemented
11Q8×Dic6C211yesmaximal, characteristic, complemented
12#5563C211yesmaximal, characteristic, complemented
13#5561C212yesmaximal, complemented
14#5557C211yesmaximal, characteristic, complemented
15#5551C212yesmaximal, complemented
16#5507C212yesmaximal, complemented
17#5500C212yesmaximal, complemented
18#3544—31nomaximal, Sylow
19C2×D3×Q8C2212yes—
20C2×#235C2214yes—
21C3×C4⋊Q8C2211yescharacteristic
22C3×#125C2212yes—
23C3×C22⋊Q8C2212yes—
24C3×C22⋊Q8C2214yes—
25C3×C4×Q8C2211yescharacteristic
26C3×C4×D4C2212yes—
27C3×C4×D4C2211yescharacteristic
28C2×C3×C4⋊C4C2212yes—
29#947C2211yescharacteristic
30#946C2211yescharacteristic
31#946C2214yes—
32Q8×Dic3C2212yes—
33#944C2212yes—
34#938C2211yescharacteristic
35#938C2212yes—
36#936C2211yescharacteristic
37#930C2212yes—
38#929C2214yes—
39C4×#67C2212yes—
40#926C2212yes—
41#926C2212yes—
42#924C2214yes—
43#924C2214yes—
44#898C2214yes—
45#898C2212yes—
46#897C2214yes—
47#897C2211yescharacteristic
48#897C2212yes—
49#896C2214yes—
50#895C2211yescharacteristic
51#894C2214yes—
52#894C2212yes—
53#893C2212yes—
54#892C2214yes—
55#892C2212yes—
56D3×C4⋊C4C2212yes—
57D3×C4⋊C4C2212yes—
58D3×C4⋊C4C2212yes—
59#890C2212yes—
60#890C2214yes—
61#890C2211yescharacteristic
62#890C2211yescharacteristic
63#889C2214yes—
64#888C2212yes—
65#888C2214yes—
66#888C2212yes—
67#888C2214yes—
68#888C2212yes—
69#885C2214yes—
70#885C2214yes—
71#883C2214yes—
72#883C2214yes—
73#883C2212yes—
74#879C2214yes—
75#879C2212yes—
76C4×D12C2212yes—
77C4×D12C2211yescharacteristic
78#872C2212yes—
79C42×D3C2211yescharacteristic
80#869C2212yes—
81#869C2211yescharacteristic
82C4×Dic6C2212yes—
83C4×Dic6C2211yescharacteristic
84#563—34no—
85Q82—31no—
86#555—31no—
87#554—31no—
88#554D311yescharacteristic, complemented
89#552—32no—
90#552—32no—
91#547—32no—
92#547—32no—
93#544—32no—
94#544—32no—
95#542—32no—
96#542—32no—
97#536—34no—
98C2×C4⋊Q8—31no—
99#529—31no—
100#529—31no—
101C2×C3×Q8C2311yescharacteristic
102C2×C3×Q8C2312yes—
103C2×C3×D4C2311yescharacteristic
104C22×C3×C4C2312yes—
105C22×C3×C4C2314yes—
106C2×#67C2312yes—
107C2×#67C2314yes—
108D3×Q8—44no—
109#235—48no—
110C2×D12C2311yescharacteristic
111C2×D12C2312yes—
112C2×C4×D3C2311yescharacteristic
113C2×C4×D3C2311yescharacteristic
114C2×C4×D3C2311yescharacteristic
115C2×C4×D3C2312yes—
116C2×C4×D3C2314yes—
117C2×C4×D3C2314yes—
118C2×Dic6C2311yescharacteristic
119C2×Dic6C2312yes—
120C2×Dic6C2314yes—
121C2×Dic6C2314yes—
122C3×C4⋊C4C2311yescharacteristic
123C3×C4⋊C4C2312yes—
124C3×C4⋊C4C2312yes—
125C3×C4⋊C4C2314yes—
126C3×C4⋊C4C2312yes—
127C3×C4⋊C4C2314yes—
128C3×C4⋊C4C2311yescharacteristic
129C3×C22⋊C4C2314yes—
130C3×C22⋊C4C2312yes—
131C3×C42C2311yescharacteristic
132C3×C42C2312yes—
133#217C2312yes—
134#217C2312yes—
135#217C2314yes—
136#217C2312yes—
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—
145C4⋊Dic3C2312yes—
146C4⋊Dic3C2311yescharacteristic
147C4⋊Dic3C2314yes—
148C4⋊Dic3C2312yes—
149C4⋊Dic3C2312yes—
150C4⋊Dic3C2311yescharacteristic
151C4⋊Dic3C2314yes—
152C4⋊Dic3C2312yes—
153C4⋊Dic3C2312yes—
154Dic3⋊C4C2312yes—
155Dic3⋊C4C2314yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2312yes—
160Dic3⋊C4C2314yes—
161Dic3⋊C4C2314yes—
162C4×Dic3C2314yes—
163C4×Dic3C2312yes—
164C4×Dic3C2311yescharacteristic
165C4×Dic3C2311yescharacteristic
166C4×Dic3C2311yescharacteristic
167C4×Dic3C2314yes—
168C2×#41—34no—
169C22×Q8—32no—
170C4⋊Q8—32no—
171C4⋊Q8—32no—
172C4⋊Q8—32no—
173C4⋊Q8—34no—
174C4⋊Q8—34no—
175C4⋊Q8—32no—
176C4⋊Q8—31no—
177C4⋊Q8—32no—
178C4⋊Q8C2×D311yescharacteristic
179#126—32no—
180#126—34no—
181#126—32no—
182#126—34no—
183#125—31no—
184#125—31no—
185#125—32no—
186#125—34no—
187#125—34no—
188#125C2×D312yes—
189#124—31no—
190#124—31no—
191#124—34no—
192#124—34no—
193#123—34no—
194#123—32no—
195#123—34no—
196#123—32no—
197#123—34no—
198#123—34no—
199C22⋊Q8—31no—
200C22⋊Q8—34no—
201C22⋊Q8—34no—
202C22⋊Q8—34no—
203C22⋊Q8—31no—
204C22⋊Q8—34no—
205C22⋊Q8—32no—
206C22⋊Q8—34no—
207C22⋊Q8C2×D314yes—
208C22⋊Q8C2×D312yes—
209#121—31no—
210#121—32no—
211#121—32no—
212#121—31no—
213C4×Q8—32no—
214C4×Q8—31no—
215C4×Q8—32no—
216C4×Q8C2×D311yescharacteristic
217C4×D4—32no—
218C4×D4—32no—
219C4×D4—32no—
220C4×D4—31no—
221C4×D4C2×D311yescharacteristic
222C4×D4C2×D312yes—
223#117—32no—
224#117—32no—
225#117—34no—
226#117—32no—
227#117—32no—
228C2×C4⋊C4—32no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4C2×D312yes—
232C2×C42—31no—
233C23×C3C2412yes—
234C22×D3C2412yes—
235C22×D3C2411yescharacteristic
236C3×Q8—22no—
237C3×Q8—42no—
238C3×D4—22no—
239C2×C3×C4—24no—
240C2×C3×C4—44no—
241C2×C3×C4C2412yes—
242C2×C3×C4C2412yes—
243C2×C3×C4C2411yescharacteristic
244C2×C3×C4C2412yes—
245C2×C3×C4C2414yes—
246C2×C3×C4C2411yescharacteristic
247C2×C3×C4C2411yescharacteristic
248#67—24no—
249#67—44no—
250C2×Dic3C2412yes—
251C2×Dic3C2411yescharacteristic
252C2×Dic3C2411yescharacteristic
253C2×Dic3C2412yes—
254C2×Dic3C2414yes—
255C2×Dic3C2411yescharacteristic
256C2×Dic3C2412yes—
257D12—24no—
258D12—22no—
259C4×D3—28no—
260C4×D3—22no—
261C4×D3—22no—
262C4×D3—24no—
263C4×D3—22no—
264C4×D3—44no—
265Dic6—22no—
266Dic6—44no—
267Dic6—44no—
268Dic6—42no—
269#41—128no—
270C2×Q8—31no—
271C2×Q8—34no—
272C2×Q8—34no—
273C2×Q8—32no—
274C2×Q8—124no—
275C2×Q8C22×D311yescharacteristic
276C2×Q8C22×D312yes—
277C2×D4—32no—
278C2×D4—34no—
279C2×D4—31no—
280C2×D4—32no—
281C2×D4C22×D311yescharacteristic
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—34no—
286C22×C4—32no—
287C22×C4—34no—
288C22×C4C22×D312yes—
289C22×C4C22×D314yes—
290C4⋊C4—31no—
291C4⋊C4—32no—
292C4⋊C4—32no—
293C4⋊C4—34no—
294C4⋊C4—34no—
295C4⋊C4—34no—
296C4⋊C4—31no—
297C4⋊C4—34no—
298C4⋊C4—32no—
299C4⋊C4—32no—
300C4⋊C4—32no—
301C4⋊C4—32no—
302C4⋊C4—32no—
303C4⋊C4—32no—
304C4⋊C4—32no—
305C4⋊C4—34no—
306C4⋊C4—34no—
307C4⋊C4C22×D312yes—
308C4⋊C4C22×D312yes—
309C4⋊C4C22×D311yescharacteristic
310C4⋊C4C22×D314yes—
311C4⋊C4C22×D311yescharacteristic
312C4⋊C4C22×D314yes—
313C4⋊C4C22×D312yes—
314C22⋊C4—32no—
315C22⋊C4—34no—
316C22⋊C4—32no—
317C22⋊C4—32no—
318C22⋊C4—32no—
319C22⋊C4—32no—
320C22⋊C4—32no—
321C22⋊C4—34no—
322C22⋊C4—32no—
323C22⋊C4—34no—
324C22⋊C4—32no—
325C22⋊C4—32no—
326C22⋊C4C22×D314yes—
327C22⋊C4C22×D312yes—
328C42—31no—
329C42—31no—
330C42—31no—
331C42—34no—
332C42—32no—
333C42—34no—
334C42C22×D311yescharacteristic
335C42C22×D312yes—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—22no—
339C22×C3C2511yescharacteristic
340C2×D3—21no—
341C2×D3—21no—
342C2×D3—22no—
343C2×D3—22no—
344C2×D3—22no—
345C2×D3H(5,𝔽2)12yes—
346C3×C4—21no—
347C3×C4—22no—
348C3×C4—24no—
349C3×C4—22no—
350C3×C4—22no—
351C3×C42−1+412yes—
352C3×C42−1+412yes—
353Dic3—21no—
354Dic3—24no—
355Dic3—22no—
356Dic3—22no—
357Dic3—21no—
358Dic3—22no—
359Dic3—21no—
360C23—31no—
361C23—32no—
362C23C23×D312yes—
363Q8—22no—
364Q8—42no—
365Q8—62no—
366Q8—124no—
367Q8—122no—
368Q8—124no—
369D4—22no—
370D4—64no—
371D4—64no—
372D4—62no—
373D4—124no—
374C2×C4—24no—
375C2×C4—32no—
376C2×C4—31no—
377C2×C4—32no—
378C2×C4—31no—
379C2×C4—31no—
380C2×C4—34no—
381C2×C4—32no—
382C2×C4—44no—
383C2×C4—62no—
384C2×C4—68no—
385C2×C4—62no—
386C2×C4—64no—
387C2×C4—62no—
388C2×C4—124no—
389C2×C4C23×D312yes—
390C2×C4C23×D312yes—
391C2×C4C23×D311yescharacteristic
392C2×C4C23×D312yes—
393C2×C4C23×D314yes—
394C2×C4C23×D311yescharacteristic
395C2×C4C23×D311yescharacteristic
396C2×C3—42no—
397C2×C3C2×H(5,𝔽2)11yescharacteristic
398C2×C3C2×2−1+411yescharacteristic
399C2×C3C2×2−1+411yescharacteristic
400D3—22no—
401D3—42no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—32no—
406C22—61no—
407C22—61no—
408C22—62no—
409C22—62no—
410C22—62no—
411C22C24×D311yescharacteristic
412C4—22no—
413C4—22no—
414C4—22no—
415C4—24no—
416C4—21no—
417C4—61no—
418C4—62no—
419C4—62no—
420C4—61no—
421C4—64no—
422C4—61no—
423C4—62no—
424C4#100712yes—
425C4#101012yes—
426C3#354411yescharacteristic, complemented, Sylow
427C2—42no—
428C2—62no—
429C2—122no—
430C2C2×#100711yescharacteristic
431C2C2×#101011yescharacteristic
432C2D3×H(5,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#554C2
hypercenter#554D3
hypocenterC3#3544
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #554

Complements

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

NH ≅ G/Ntype
1. C3×#554C2semidirect
2. #5730C2semidirect
3. D3×C4⋊Q8C2semidirect
4. #5694C2semidirect
5. #5689C2semidirect
6. #5646C2semidirect
7. #5643C2semidirect
8. #5639C2semidirect
9. #5628C2semidirect
10. #5580C2semidirect
11. Q8×Dic6C2semidirect
12. #5563C2semidirect
13. #5561C2semidirect
14. #5557C2semidirect
15. #5551C2semidirect
16. #5507C2semidirect
17. #5500C2semidirect
88. #554D3semidirect
426. C3#3544semidirect

Maximal subgroups

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

subgroupindex
C3×#5542
#57302
D3×C4⋊Q82
#56942
#56892
#56462
#56432
#56392
#56282
#55802
Q8×Dic62
#55632
#55612
#55572
#55512
#55072
#55002
#35443