Finite Group Databasebeta

← #29834967

Subgroups — #29834967

Order 15616.

590 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456396C211yesmaximal, characteristic
2C4×Q16×D61C211yesmaximal, characteristic, complemented
3#21448003C212yesmaximal, complemented
4#21447962C211yesmaximal, characteristic, complemented
5#21447961C211yesmaximal, characteristic, complemented
6#21447856C211yesmaximal, characteristic, complemented
7#21447734C212yesmaximal, complemented
8#21447674C212yesmaximal, complemented
9SD32×Dic61C212yesmaximal, complemented
10#21447617C211yesmaximal, characteristic, complemented
11C61×#2207C211yesmaximal, characteristic, complemented
12C2×Q16×D61C2211yescharacteristic
13#16840310—21no—
14C4×Q8×D61—21no—
15C2×#14519721C2212yes—
16#16840167C2211yescharacteristic
17#16840108—21no—
18#16839995—21no—
19#16839995C2212yes—
20C4×Dic244C2211yescharacteristic
21Q16×Dic61C2211yescharacteristic
22#16839856C2211yescharacteristic
23C4×#11758202—21no—
24#16839854—21no—
25D8×Dic61—21no—
26D8×Dic61C2212yes—
27#107×D61C2211yescharacteristic
28#103×D61—21no—
29C4×C8×D61C2211yescharacteristic
30#16839590C2212yes—
31#16839589C2212yes—
32C4⋊#11758185C2211yescharacteristic
33#16839567C2212yes—
34#16839566C2212yes—
35#16839563C2212yes—
36#16839541C2212yes—
37C16×Dic61C2212yes—
38C2×SD32×C61C2212yes—
39C61×#493C2211yescharacteristic
40C4×Q16×C61C2211yescharacteristic
41C61×C4⋊C16C2211yescharacteristic
42C61×#357C2212yes—
43C2×#11758224—41no—
44C2×Q8×D61—21no—
45C2×Dic244C2311yescharacteristic
46C2×#11758202—21no—
47Q16×D61—22no—
48Q16×D61—22no—
49C2×C8×D61D411yescharacteristic
50C2×C8×D61C2311yescharacteristic
51#14519777—21no—
52#14519764—41no—
53#14519748—22no—
54#14519748—41no—
55Q8×Dic61—21no—
56C4×Dic122—21no—
57C4×Dic122—41no—
58#14519742—21no—
59#14519742—41no—
60D4×Dic61—22no—
61D4×Dic61—41no—
62C4⋊C4×D61—21no—
63C4⋊C4×D61—41no—
64C42×D61—41no—
65C42×D61D411yescharacteristic
66#14519721—24no—
67C2×#11758185C2312yes—
68#14519711—21no—
69#14519711C2312yes—
70#14519711C2311yescharacteristic
71#14519710—21no—
72#14519702—21no—
73#14519701—21no—
74#14519700—21no—
75#14519698—22no—
76#14519698—22no—
77C4×#92418D411yescharacteristic
78C8×Dic61D411yescharacteristic
79C8×Dic61C2311yescharacteristic
80C2×Q16×C61C2311yescharacteristic
81C2×D8×C61—21no—
82C2×D8×C61C2312yes—
83#127×C61—21no—
84C4×Q8×C61—21no—
85SD32×C61—24no—
86C2×C16×C61C2312yes—
87#107×C61C2311yescharacteristic
88#103×C61—21no—
89C4×C8×C61C2311yescharacteristic
90C2×Q8×C61—21no—
91C2×D4×C61—22no—
92C2×D4×C61—41no—
93C2×D4×C61—41no—
94C2×C22⋊D61—42no—
95C22×Dic61—42no—
96Q8×D61—24no—
97Q8×D61—42no—
98#11758224—84no—
99C2×C4×D61—41no—
100C2×C4×D61—41no—
101C2×C4×D61D811yescharacteristic
102C2×C4×D61D811yescharacteristic
103C2×C4×D61C2×D411yescharacteristic
104C2×Dic122—21no—
105C2×Dic122—41no—
106C2×Dic122—41no—
107Q16×C61—21no—
108Q16×C61#4112yescomplemented
109D8×C61—22no—
110D8×C61—22no—
111D8×C61—42no—
112C2×C8×C61C2×D411yescharacteristic
113C2×C8×C61C2411yescharacteristic
114C4⋊C4×C61—21no—
115C4⋊C4×C61—41no—
116C42×C61—41no—
117C42×C61C2×D411yescharacteristic
118C22⋊Dic61—42no—
119C22⋊Dic61—42no—
120#11758202—24no—
121C4⋊Dic61—22no—
122C4⋊Dic61—21no—
123C4⋊Dic61—41no—
124C4⋊Dic61—41no—
125#11758196—41no—
126#11758196—41no—
127#11758196—41no—
128C4×Dic61—41no—
129C4×Dic61—41no—
130C4×Dic61—41no—
131C4×Dic61D811yescharacteristic
132C4×Dic61D811yescharacteristic
133C4×Dic61C2×D411yescharacteristic
134C2×#92418C2×D411yescharacteristic
135C2×#92418C2×D411yescharacteristic
136Dic244—21no—
137Dic244#4112yescomplemented
138C8×D61—21no—
139C8×D61—21no—
140C8×D61—41no—
141C16×C61—22no—
142#11758185—22no—
143C23×C61—42no—
144C22×D61C2×D811yescharacteristic
145Q8×C61—22no—
146Q8×C61—41no—
147D4×C61—42no—
148D4×C61—42no—
149D4×C61—82no—
150D4×C61—82no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61C2×D811yescharacteristic
154C2×C4×C61C2×D811yescharacteristic
155C2×C4×C61C22×D411yescharacteristic
156C22⋊D61—84no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—42no—
160C2×Dic61—84no—
161C2×Dic61C2×D811yescharacteristic
162C2×Dic61C2×D811yescharacteristic
163C2×Dic61C2×D811yescharacteristic
164C4×D61—21no—
165C4×D61—21no—
166C4×D61—41no—
167C4×D61—44no—
168C4×D61—82no—
169C4×D61SD3214yescomplemented
170Dic122—22no—
171Dic122—44no—
172Dic122—41no—
173Dic122—82no—
174C8×C61—21no—
175C8×C61C2×#4111yescharacteristic
176C8×C61C2×#4111yescharacteristic
177#92418—21no—
178#92418—21no—
179#33663—611nomaximal, Sylow
180C22×C61—82no—
181C22×C61—82no—
182C22×C61—82no—
183C22×C61C22×D811yescharacteristic
184C2×D61—21no—
185C2×D61—21no—
186C2×D61C2×SD3212yes—
187C4×C61—21no—
188C4×C61—42no—
189C4×C61—81no—
190C4×C61C2×SD3212yes—
191C4×C61#52511yescharacteristic
192C4×C61#57511yescharacteristic
193Dic61—21no—
194Dic61—21no—
195Dic61—42no—
196Dic61—82no—
197Dic61—81no—
198Dic61C2×SD3212yes—
199#3169—611no—
200C2×C4×Q16—611no—
201#2233—612no—
202#2225—611no—
203#2225—611no—
204#2207—611no—
205#2207D6111yescharacteristic, complemented
206#2185—612no—
207C4×SD32—612no—
208C4×SD32—612no—
209#2162—611no—
210C2×C61—162no—
211C2×C61#310011yescharacteristic
212C2×C61C22×SD3211yescharacteristic
213C2×C61#341411yescharacteristic
214D61—22no—
215C22×Q16—611no—
216#532—1221no—
217C2×C4×Q8—1221no—
218C2×SD32—612no—
219C2×SD32C2×D6112yes—
220#500—611no—
221#493C2×D6111yescharacteristic
222#486—1221no—
223#466—612no—
224#466—1221no—
225C4×Q16—611no—
226C4×Q16—611no—
227C4×Q16—611no—
228C4×Q16—1221no—
229C4×Q16—1221no—
230C4×Q16C2×D6111yescharacteristic
231C4×D8—612no—
232C4×D8—1221no—
233C2×#107—611no—
234C2×#103—1221no—
235C2×C4×C8—611no—
236#367—612no—
237#367—612no—
238C4⋊C16—611no—
239C4⋊C16C2×D6111yescharacteristic
240#358—612no—
241#358—612no—
242#357—612no—
243#357C2×D6112yes—
244C22⋊C16—612no—
245C4×C16—612no—
246C61#3366311yescharacteristic, complemented, Sylow
247C2×#41—2441no—
248C22×Q8—1221no—
249C2×Q16—611no—
250C2×Q16—1221no—
251C2×Q16—1222no—
252C2×Q16—1222no—
253C2×Q16C22×D6111yescharacteristic
254C2×D8—21no—
255C2×D8C22×D6112yes—
256C22×C8—611no—
257C22×C8—611no—
258C4⋊Q8—1221no—
259#127—21no—
260#124—2441no—
261#121—1222no—
262#121—2441no—
263C4×Q8—21no—
264C4×Q8—1221no—
265C4×Q8—1221no—
266C4×Q8—1221no—
267C4×Q8—2441no—
268C4×Q8—2441no—
269C4×D4—1222no—
270C4×D4—2441no—
271C2×C4⋊C4—1221no—
272C2×C4⋊C4—2441no—
273C2×C42—611no—
274C2×C42—2441no—
275SD32—24no—
276SD32—1224no—
277C2×C16—612no—
278C2×C16C22×D6112yes—
279#107—611no—
280#107—612no—
281#107—1221no—
282#107—1221no—
283#107C22×D6111yescharacteristic
284#103—21no—
285#103—1221no—
286#103—1221no—
287#103—1221no—
288#102—1222no—
289#102—1222no—
290C4×C8—611no—
291C4×C8—611no—
292C4×C8—611no—
293C4×C8C22×D6111yescharacteristic
294#41—4884no—
295C2×Q8—21no—
296C2×Q8—1224no—
297C2×Q8—1221no—
298C2×Q8—2441no—
299C2×Q8—2442no—
300C2×Q8—2441no—
301C2×D4—22no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—2442no—
305C22×C4—611no—
306C22×C4—611no—
307C22×C4—611no—
308C22×C4—2441no—
309C22×C4—2441no—
310C22×C4—2442no—
311Q16—21no—
312Q16—612no—
313Q16—1224no—
314Q16—1221no—
315Q16#1175822412yescomplemented
316D8—22no—
317D8—22no—
318D8—42no—
319C2×C8—611no—
320C2×C8—611no—
321C2×C8—1221no—
322C2×C8—1221no—
323C2×C8—2441no—
324C2×C8D4×D6111yescharacteristic
325C2×C8C23×D6111yescharacteristic
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—1222no—
329C4⋊C4—1221no—
330C4⋊C4—2441no—
331C4⋊C4—2441no—
332C4⋊C4—2441no—
333C4⋊C4—2441no—
334C4⋊C4—2441no—
335C22⋊C4—2442no—
336C22⋊C4—2442no—
337C42—41no—
338C42—611no—
339C42—611no—
340C42—611no—
341C42—2441no—
342C42—2441no—
343C42—2441no—
344C42D4×D6111yescharacteristic
345C16—22no—
346C16—1222no—
347C23—42no—
348C23—611no—
349Q8—22no—
350Q8—41no—
351Q8—1222no—
352Q8—4882no—
353Q8—2444no—
354Q8—2441no—
355D4—42no—
356D4—42no—
357D4—82no—
358D4—82no—
359D4—4884no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—611no—
363C2×C4—614no—
364C2×C4—611no—
365C2×C4—611no—
366C2×C4—1221no—
367C2×C4—1221no—
368C2×C4—4882no—
369C2×C4—4884no—
370C2×C4—2441no—
371C2×C4—2444no—
372C2×C4—2441no—
373C2×C4—2442no—
374C2×C4—2441no—
375C2×C4D8×D6111yescharacteristic
376C2×C4D8×D6111yescharacteristic
377C2×C4C2×D4×D6111yescharacteristic
378C8—21no—
379C8—1221no—
380C8—1221no—
381C8#41×D6111yescharacteristic
382C8C2×#1175822411yescharacteristic
383C22—82no—
384C22—82no—
385C22—82no—
386C22—612no—
387C22—1221no—
388C22—1221no—
389C22C2×D8×D6111yescharacteristic
390C4—21no—
391C4—42no—
392C4—81no—
393C4—612no—
394C4—1221no—
395C4—1221no—
396C4—4881no—
397C4—4882no—
398C4—2442no—
399C4SD32×D6112yes—
400C4#1684027211yescharacteristic
401C4#1684055811yescharacteristic
402C2—162no—
403C2—1222no—
404C2#2145581811yescharacteristic
405C2C2×SD32×D6111yescharacteristic
406C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2207C2
socleC22×C61C22×D8
prime radical#21456396C2
hypercenter#2207D61
hypocenterC61#33663
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2207

Complements

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

NH ≅ G/Ntype
2. C4×Q16×D61C2semidirect
3. #21448003C2semidirect
4. #21447962C2semidirect
5. #21447961C2semidirect
6. #21447856C2semidirect
7. #21447734C2semidirect
8. #21447674C2semidirect
9. SD32×Dic61C2semidirect
10. #21447617C2semidirect
11. C61×#2207C2semidirect
108. Q16×C61#41semidirect
137. Dic244#41semidirect
169. C4×D61SD32semidirect
205. #2207D61semidirect
246. C61#33663semidirect
315. Q16#11758224semidirect

Maximal subgroups

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

subgroupindex
#214563962
C4×Q16×D612
#214480032
#214479622
#214479612
#214478562
#214477342
#214476742
SD32×Dic612
#214476172
C61×#22072
#3366361