Finite Group Databasebeta

← #29834970

Subgroups — #29834970

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#21456399C211yesmaximal, characteristic
2C4×Q16×D61C211yesmaximal, characteristic, complemented
3#21448006C212yesmaximal, complemented
4#21447958C211yesmaximal, characteristic, complemented
5#21447860C211yesmaximal, characteristic, complemented
6#21447859C211yesmaximal, characteristic, complemented
7#21447737C212yesmaximal, complemented
8#21447670C212yesmaximal, complemented
9#21447668C212yesmaximal, complemented
10#21447617C211yesmaximal, characteristic, complemented
11C61×#2225C211yesmaximal, characteristic, complemented
12C2×Q16×D61C2211yescharacteristic
13#16840313—21no—
14C4×Q8×D61—21no—
15C2×#14519722C2212yes—
16C2×#14519720C2212yes—
17#16840139C2211yescharacteristic
18#16840110—21no—
19#16839999—21no—
20#16839999C2212yes—
21C4×Dic244C2211yescharacteristic
22Q16×Dic61C2211yescharacteristic
23#16839856C2211yescharacteristic
24C4×#11758202—21no—
25#16839854—21no—
26#16839844—21no—
27#16839844C2212yes—
28#107×D61C2211yescharacteristic
29#103×D61—21no—
30C4×C8×D61C2211yescharacteristic
31#16839589C2212yes—
32C4⋊#11758185C2211yescharacteristic
33#16839565C2212yes—
34#16839564C2212yes—
35#16839562C2212yes—
36#16839541C2212yes—
37C16×Dic61C2212yes—
38C61×#500C2211yescharacteristic
39C4×Q16×C61C2211yescharacteristic
40C61×#367C2212yes—
41C61×C4⋊C16C2211yescharacteristic
42C61×#358C2212yes—
43C2×#11758226—41no—
44C2×Q8×D61—21no—
45C2×Dic244C2311yescharacteristic
46C2×#11758202—21no—
47Q16×D61—22no—
48Q16×D61—22no—
49C2×D488—21no—
50C2×D488C2312yes—
51C2×C8×D61D411yescharacteristic
52C2×C8×D61C2311yescharacteristic
53#14519774—21no—
54#14519766—41no—
55#14519752—22no—
56#14519752—41no—
57Q8×Dic61—21no—
58C4×Dic122—21no—
59C4×Dic122—41no—
60#14519742—21no—
61#14519742—41no—
62#14519739—22no—
63#14519739—41no—
64C4⋊C4×D61—21no—
65C4⋊C4×D61—41no—
66C42×D61—41no—
67C42×D61D411yescharacteristic
68#14519722—24no—
69#14519720—24no—
70C2×#11758185C2312yes—
71#14519711C2311yescharacteristic
72#14519710—21no—
73#14519702—21no—
74#14519701—21no—
75#14519700—21no—
76#14519697—22no—
77#14519697—22no—
78C4×#92418D411yescharacteristic
79C8×Dic61D411yescharacteristic
80C8×Dic61C2311yescharacteristic
81C2×Q16×C61C2311yescharacteristic
82C4⋊Q8×C61—21no—
83C4×Q8×C61—21no—
84C2×C16×C61C2312yes—
85#107×C61—21no—
86#107×C61C2312yes—
87#107×C61C2311yescharacteristic
88#103×C61—21no—
89C4×C8×C61C2311yescharacteristic
90C2×Q8×C61—21no—
91C2×Q8×C61—41no—
92#11758226—84no—
93Q8×D61—24no—
94Q8×D61—42no—
95C2×D244—22no—
96C2×D244—42no—
97C2×D244—41no—
98C2×D244—41no—
99C2×C4×D61—41no—
100C2×C4×D61—41no—
101C2×C4×D61—42no—
102C2×C4×D61D811yescharacteristic
103C2×C4×D61D811yescharacteristic
104C2×C4×D61C2×D411yescharacteristic
105C2×Dic122—21no—
106C2×Dic122—41no—
107Q16×C61—21no—
108Q16×C61#4112yescomplemented
109C2×C8×C61C2×D411yescharacteristic
110C2×C8×C61C2411yescharacteristic
111C4⋊C4×C61—21no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C42×C61—41no—
116C42×C61C2×D411yescharacteristic
117#11758202—24no—
118#11758198—42no—
119#11758198—42no—
120C4⋊Dic61—21no—
121C4⋊Dic61—41no—
122#11758196—41no—
123#11758196—41no—
124#11758196—41no—
125C4×Dic61—41no—
126C4×Dic61—41no—
127C4×Dic61—41no—
128C4×Dic61D811yescharacteristic
129C4×Dic61D811yescharacteristic
130C4×Dic61C2×D411yescharacteristic
131C2×#92418C2×D411yescharacteristic
132C2×#92418C2×D411yescharacteristic
133Dic244—21no—
134Dic244#4112yescomplemented
135D488—22no—
136D488—22no—
137D488—42no—
138C8×D61—21no—
139C8×D61—21no—
140C8×D61—41no—
141C16×C61—22no—
142#11758185—22no—
143C22×D61—42no—
144C22×D61C2×D811yescharacteristic
145Q8×C61—22no—
146Q8×C61—41no—
147Q8×C61—82no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61C2×D811yescharacteristic
152C2×C4×C61C2×D811yescharacteristic
153C2×C4×C61C22×D411yescharacteristic
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61C2×D811yescharacteristic
157C2×Dic61C2×D811yescharacteristic
158C2×Dic61C2×D811yescharacteristic
159D244—42no—
160D244—42no—
161D244—82no—
162D244—84no—
163D244—82no—
164C4×D61—21no—
165C4×D61—21no—
166C4×D61—41no—
167C4×D61—44no—
168C4×D61—82no—
169C4×D61—84no—
170C4×D61SD3214yescomplemented
171Dic122—22no—
172Dic122—44no—
173Dic122—41no—
174C8×C61—21no—
175C8×C61C2×#4111yescharacteristic
176C8×C61C2×#4111yescharacteristic
177#92418—21no—
178#92418—21no—
179#33663—611nomaximal, Sylow
180C22×C61C22×D811yescharacteristic
181C2×D61—21no—
182C2×D61—21no—
183C2×D61—82no—
184C2×D61—82no—
185C2×D61—82no—
186C2×D61C2×SD3212yes—
187C4×C61—21no—
188C4×C61—42no—
189C4×C61—82no—
190C4×C61—81no—
191C4×C61C2×SD3212yes—
192C4×C61#52511yescharacteristic
193C4×C61#57511yescharacteristic
194Dic61—21no—
195Dic61—21no—
196Dic61—42no—
197Dic61—81no—
198Dic61C2×SD3212yes—
199#3169—611no—
200C2×C4×Q16—611no—
201#2233—612no—
202#2225—611no—
203#2225D6111yescharacteristic, complemented
204#2207—611no—
205#2207—611no—
206#2185—612no—
207C4×SD32—612no—
208C4×SD32—612no—
209#2162—611no—
210C2×C61#310011yescharacteristic
211C2×C61C22×SD3211yescharacteristic
212C2×C61#341411yescharacteristic
213D61—22no—
214D61—162no—
215C22×Q16—611no—
216#532—1221no—
217C2×C4×Q8—1221no—
218C2×SD32—612no—
219C2×SD32—612no—
220#500C2×D6111yescharacteristic
221#493—611no—
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#367C2×D6112yes—
238C4⋊C16—611no—
239C4⋊C16C2×D6111yescharacteristic
240#358—612no—
241#358C2×D6112yes—
242#357—612no—
243#357—612no—
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—612no—
255C2×D8—1221no—
256C22×C8—611no—
257C22×C8—611no—
258C4⋊Q8—21no—
259#127—1221no—
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—1224no—
276SD32—1224no—
277C2×C16—612no—
278C2×C16C22×D6112yes—
279#107—21no—
280#107—611no—
281#107—1221no—
282#107C22×D6111yescharacteristic
283#107C22×D6112yes—
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—41no—
297C2×Q8—1224no—
298C2×Q8—1221no—
299C2×Q8—2441no—
300C2×Q8—2442no—
301C2×D4—1222no—
302C2×D4—2442no—
303C2×D4—2441no—
304C2×D4—2441no—
305C22×C4—611no—
306C22×C4—611no—
307C22×C4—611no—
308C22×C4—2442no—
309C22×C4—2441no—
310C22×C4—2441no—
311Q16—21no—
312Q16—612no—
313Q16—1224no—
314Q16—1221no—
315Q16#1175822612yescomplemented
316D8—1222no—
317D8—1222no—
318D8—2442no—
319C2×C8—611no—
320C2×C8—611no—
321C2×C8—1221no—
322C2×C8—1221no—
323C2×C8—2441no—
324C2×C8D4×D6111yescharacteristic
325C2×C8C23×D6111yescharacteristic
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—1221no—
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—611no—
348C23—2442no—
349Q8—22no—
350Q8—41no—
351Q8—82no—
352Q8—1222no—
353Q8—2444no—
354Q8—2441no—
355D4—4882no—
356D4—4884no—
357D4—4882no—
358D4—2442no—
359D4—2442no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—611no—
364C2×C4—614no—
365C2×C4—611no—
366C2×C4—611no—
367C2×C4—1221no—
368C2×C4—1221no—
369C2×C4—4884no—
370C2×C4—4882no—
371C2×C4—2441no—
372C2×C4—2444no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4D8×D6111yescharacteristic
376C2×C4D8×D6111yescharacteristic
377C2×C4C2×D4×D6111yescharacteristic
378C8—21no—
379C8—1221no—
380C8—1221no—
381C8#41×D6111yescharacteristic
382C8C2×#1175822611yescharacteristic
383C22—612no—
384C22—1221no—
385C22—1221no—
386C22—4882no—
387C22—4882no—
388C22—4882no—
389C22C2×D8×D6111yescharacteristic
390C4—21no—
391C4—42no—
392C4—81no—
393C4—82no—
394C4—612no—
395C4—1221no—
396C4—1221no—
397C4—4881no—
398C4—2442no—
399C4SD32×D6112yes—
400C4#1684027511yescharacteristic
401C4#1684056111yescharacteristic
402C2—1222no—
403C2—9762no—
404C2#2145582111yescharacteristic
405C2C2×SD32×D6111yescharacteristic
406C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2225C2
socleC22×C61C22×D8
prime radical#21456399C2
hypercenter#2225D61
hypocenterC61#33663
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2225

Complements

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

NH ≅ G/Ntype
2. C4×Q16×D61C2semidirect
3. #21448006C2semidirect
4. #21447958C2semidirect
5. #21447860C2semidirect
6. #21447859C2semidirect
7. #21447737C2semidirect
8. #21447670C2semidirect
9. #21447668C2semidirect
10. #21447617C2semidirect
11. C61×#2225C2semidirect
108. Q16×C61#41semidirect
134. Dic244#41semidirect
170. C4×D61SD32semidirect
203. #2225D61semidirect
246. C61#33663semidirect
315. Q16#11758226semidirect

Maximal subgroups

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

subgroupindex
#214563992
C4×Q16×D612
#214480062
#214479582
#214478602
#214478592
#214477372
#214476702
#214476682
#214476172
C61×#22252
#3366361