Finite Group Databasebeta

← #29833394

Subgroups — #29833394

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840192C211yesmaximal, characteristic
2#21454695C211yesmaximal, characteristic, complemented
3#21454693C211yesmaximal, characteristic, complemented
4#21447686C212yesmaximal, complemented
5#21447680C212yesmaximal, complemented
6#21447675C212yesmaximal, complemented
7#21447663C212yesmaximal, complemented
8C2×#16839587C211yesmaximal, characteristic, complemented
9#21447532C211yesmaximal, characteristic, complemented
10#21447418C211yesmaximal, characteristic, complemented
11C61×#2139C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13#16840245—21no—
14#16840243—21no—
15#16840192C418yescomplemented
16C2×#14519723C2211yescharacteristic
17C2×#14519721C2211yescharacteristic
18C2×#14519720C2211yescharacteristic
19C2×#14519718C2211yescharacteristic
20C22×#11758185C2211yescharacteristic
21#16839863—22no—
22#16839861—22no—
23Q16×Dic61C2212yes—
24#16839856C2212yes—
25D8×Dic61C2212yes—
26#16839844C2212yes—
27C2×#14519711C2211yescharacteristic
28#16839763—21no—
29#16839761—21no—
30#16839717C2211yescharacteristic
31#16839587C2212yes—
32#16839587C2212yes—
33#16839567C2212yes—
34#16839564C2212yes—
35#16839537C2212yes—
36C16×Dic61C2212yes—
37C2×#135×C61C2211yescharacteristic
38C2×M32×C61C2211yescharacteristic
39C2×#107×C61C2211yescharacteristic
40C61×#358C2212yes—
41C61×#357C2212yes—
42C2×#11758222—21no—
43#14519799—22no—
44#14519799C2×C414yescomplemented
45C2×Dic244C2311yescharacteristic
46C2×#11758190—21no—
47C2×D488C2311yescharacteristic
48C2×#11758194D411yescharacteristic
49Q8×Dic61—22no—
50#14519742—22no—
51D4×Dic61—22no—
52D4×Dic61—42no—
53#14519739—22no—
54C4×C22⋊D61—42no—
55#14519736—41no—
56#14519736D411yescharacteristic
57#14519735—41no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—41no—
60C4⋊C4×D61—41no—
61#14519723C2×C414yescomplemented
62#14519721C2×C414yescomplemented
63#14519720C2×C414yescomplemented
64#14519718C2×C414yescomplemented
65C2×#11758185C2×C414yescomplemented
66C2×#11758185C2311yescharacteristic
67C2×#11758185C2311yescharacteristic
68#14519711—21no—
69#14519711C2312yes—
70#14519701—22no—
71#14519700—22no—
72#14519698—22no—
73#14519697—22no—
74#14519687D412yes—
75C8×Dic61C2312yes—
76C2×#41×C61—21no—
77#135×C61—22no—
78#135×C61C2×C414yescomplemented
79C2×Q16×C61C2311yescharacteristic
80C2×SD16×C61—21no—
81C2×D8×C61C2311yescharacteristic
82C22×C8×C61C2311yescharacteristic
83C2×C4⋊C4×C61—21no—
84M32×C61—22no—
85C2×C16×C61C2312yes—
86#107×C61—21no—
87#107×C61C2312yes—
88#41×C61—24no—
89#41×C61—42no—
90C2×Q8×C61—21no—
91C2×D4×C61—21no—
92C2×D4×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61C2×D411yescharacteristic
96C2×C22⋊D61—41no—
97C22×Dic61—41no—
98C22×Dic61—42no—
99#11758222—24no—
100#11758222—42no—
101C2×D244—21no—
102C2×C4×D61—41no—
103C2×C4×D61—42no—
104C2×Dic122—21no—
105Q16×C61—21no—
106Q16×C61C22×C412yes—
107SD16×C61—24no—
108D8×C61—21no—
109D8×C61C22×C412yes—
110C2×C8×C61C22×C412yes—
111C2×C8×C61C2×D411yescharacteristic
112C2×C8×C61#4112yes—
113C2×C8×C61C2411yescharacteristic
114C4⋊C4×C61—22no—
115C4⋊C4×C61—41no—
116C22⋊Dic61—42no—
117C22⋊Dic61D812yes—
118#11758198—42no—
119C4⋊Dic61—22no—
120C4⋊Dic61—41no—
121C4⋊Dic61—42no—
122C4⋊Dic61D811yescharacteristic
123C4⋊Dic61D811yescharacteristic
124#11758196—42no—
125C4×Dic61—42no—
126C4×Dic61—42no—
127C4×Dic61C2×D412yes—
128#11758194—22no—
129C2×#92418C2×D412yes—
130Dic244—21no—
131Dic244C22×C412yes—
132D488—21no—
133D488C22×C412yes—
134#11758190—24no—
135C16×C61—22no—
136#11758185C22×C412yes—
137#11758185C22×C412yes—
138C23×C61—41no—
139C23×C61C2×D811yescharacteristic
140C22×D61—41no—
141Q8×C61—22no—
142Q8×C61—41no—
143D4×C61—22no—
144D4×C61—44no—
145D4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—44no—
148C2×C4×C61—42no—
149C2×C4×C61—161no—
150C2×C4×C61C4×D412yes—
151C2×C4×C61C2×D811yescharacteristic
152C2×C4×C61#13512yes—
153C2×C4×C61C22×D411yescharacteristic
154C22⋊D61—44no—
155C2×Dic61—41no—
156C2×Dic61—42no—
157C2×Dic61—162no—
158C2×Dic61—161no—
159C2×Dic61C2×D812yes—
160C2×Dic61C2×D812yes—
161D244—22no—
162D244—41no—
163C4×D61—44no—
164C4×D61—162no—
165Dic122—22no—
166Dic122—41no—
167C8×C61C4×D412yes—
168C8×C61C23×C411yescharacteristic
169C8×C61C2×#4111yescharacteristic
170#92418—22no—
171#33459—611nomaximal, Sylow
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—42no—
175C22×C61—81no—
176C22×C61—81no—
177C22×C61C4×D812yes—
178C22×C61C22×D811yescharacteristic
179C2×D61—42no—
180C2×D61—81no—
181C2×D61—81no—
182C4×C61—42no—
183C4×C61—82no—
184C4×C61C4×D812yes—
185C4×C61C2×C4×D411yescharacteristic
186C4×C61C2×#13511yescharacteristic
187Dic61—22no—
188Dic61—22no—
189Dic61—42no—
190Dic61—82no—
191C2×#508—611no—
192#2951—611no—
193#2951—611no—
194#2175—612no—
195#2175—612no—
196C4×Q32—612no—
197C4×D16—612no—
198C2×#366—611no—
199#2139—611no—
200#2139D6111yescharacteristic, complemented
201#2108—611no—
202C2×C61—22no—
203C2×C61—82no—
204C2×C61C2×C4×D811yescharacteristic
205C2×C61#341511yescharacteristic
206C2×C61#341711yescharacteristic
207D61—82no—
208C2×#135—611no—
209C2×#135C2×D6111yescharacteristic
210#520—1221no—
211#520—1221no—
212#508—618no—
213C2×Q32—611no—
214C2×SD32—611no—
215C2×SD32—611no—
216C2×D16—611no—
217C2×M32C2×D6111yescharacteristic
218C22×C16—611no—
219#440—1222no—
220#440—1222no—
221C4×Q16—612no—
222C4×Q16—612no—
223C4×D8—612no—
224C4×D8—612no—
225C2×#107—611no—
226C2×#107C2×D6111yescharacteristic
227#417—1221no—
228#417—1221no—
229#405—611no—
230#366—612no—
231#366—612no—
232#358—612no—
233#358C2×D6112yes—
234#357—612no—
235#357C2×D6112yes—
236#346—612no—
237C4×C16—612no—
238C61#3345911yescharacteristic, complemented, Sylow
239C2×#41—21no—
240C2×#41—1221no—
241#135—22no—
242#135—614no—
243#135—1222no—
244#135C4×D6114yescomplemented
245C2×Q16—611no—
246C2×Q16C22×D6111yescharacteristic
247C2×SD16—21no—
248C2×SD16—1221no—
249C2×D8—611no—
250C2×D8C22×D6111yescharacteristic
251C2×M16—611no—
252C22×C8C22×D6111yescharacteristic
253C4×Q8—1222no—
254C4×Q8—1222no—
255C4×D4—1222no—
256C4×D4—1222no—
257C4×D4—2442no—
258C4×D4—2442no—
259#117—611no—
260#117—2441no—
261#117—2441no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—2441no—
265C2×C4⋊C4—2441no—
266Q32—614no—
267SD32—614no—
268SD32—614no—
269D16—614no—
270M32—22no—
271C2×C16—614no—
272C2×C16—611no—
273C2×C16—611no—
274C2×C16C22×D6112yes—
275#107—21no—
276#107—612no—
277#107—1221no—
278#107C22×D6112yes—
279#103—1222no—
280#103—1222no—
281#102—1222no—
282#102—1222no—
283#97—612no—
284C4×C8—612no—
285#41—24no—
286#41—42no—
287#41—1224no—
288#41—2442no—
289C2×Q8—21no—
290C2×Q8—1221no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—1221no—
294C2×D4—2441no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—2442no—
298C22×C4—2442no—
299C22×C4—2441no—
300C22×C4—2441no—
301C22×C4D4×D6111yescharacteristic
302Q16—21no—
303Q16—612no—
304Q16—1221no—
305Q16C2×C4×D6112yes—
306SD16—24no—
307SD16—1224no—
308D8—21no—
309D8—612no—
310D8—1221no—
311D8C2×C4×D6112yes—
312M16—1222no—
313C2×C8—612no—
314C2×C8C2×C4×D6112yes—
315C2×C8D4×D6111yescharacteristic
316C2×C8#1175822412yes—
317C2×C8C23×D6111yescharacteristic
318C4⋊C4—22no—
319C4⋊C4—41no—
320C4⋊C4—611no—
321C4⋊C4—611no—
322C4⋊C4—1222no—
323C4⋊C4—2442no—
324C4⋊C4—2442no—
325C4⋊C4—2441no—
326C22⋊C4—612no—
327C22⋊C4—2442no—
328C22⋊C4—2442no—
329C42—612no—
330C42—2442no—
331C42—2442no—
332C16—22no—
333C16—612no—
334C16—612no—
335C23—41no—
336C23—2441no—
337C23D8×D6111yescharacteristic
338Q8—22no—
339Q8—41no—
340Q8—1222no—
341Q8—2441no—
342D4—22no—
343D4—44no—
344D4—41no—
345D4—1222no—
346D4—2444no—
347D4—2441no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—161no—
352C2×C4—612no—
353C2×C4—612no—
354C2×C4—9762no—
355C2×C4—9762no—
356C2×C4—9761no—
357C2×C4—2442no—
358C2×C4—2441no—
359C2×C4—2444no—
360C2×C4#1451973712yes—
361C2×C4D8×D6111yescharacteristic
362C2×C4#1451979812yes—
363C2×C4C2×D4×D6111yescharacteristic
364C8—1222no—
365C8#1451973712yes—
366C8C22×C4×D6111yescharacteristic
367C8C2×#1175822411yescharacteristic
368C22—21no—
369C22—21no—
370C22—42no—
371C22—81no—
372C22—81no—
373C22—4881no—
374C22—4881no—
375C22—2442no—
376C22#1683984212yes—
377C22C2×D8×D6111yescharacteristic
378C4—42no—
379C4—82no—
380C4—1222no—
381C4—1222no—
382C4—4882no—
383C4—2442no—
384C4#1683984212yes—
385C4C2×#1451973711yescharacteristic
386C4C2×#1451979811yescharacteristic
387C2—22no—
388C2—82no—
389C2—4882no—
390C2C2×#1683984211yescharacteristic
391C2#2145895911yescharacteristic
392C2#2145897111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2139C2
socleC22×C61C22×D8
prime radicalC2×#16840192C2
cosocleC2×C8×C61C24
hypercenter#2139D61
hypocenterC61#33459
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2139

Complements

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

NH ≅ G/Ntype
2. #21454695C2semidirect
3. #21454693C2semidirect
4. #21447686C2semidirect
5. #21447680C2semidirect
6. #21447675C2semidirect
7. #21447663C2semidirect
8. C2×#16839587C2semidirect
9. #21447532C2semidirect
10. #21447418C2semidirect
11. C61×#2139C2semidirect
15. #16840192C4semidirect
44. #14519799C2×C4semidirect
61. #14519723C2×C4semidirect
62. #14519721C2×C4semidirect
63. #14519720C2×C4semidirect
64. #14519718C2×C4semidirect
65. C2×#11758185C2×C4semidirect
78. #135×C61C2×C4semidirect
200. #2139D61semidirect
238. C61#33459semidirect
244. #135C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401922
#214546952
#214546932
#214476862
#214476802
#214476752
#214476632
C2×#168395872
#214475322
#214474182
C61×#21392
#3345961