Finite Group Databasebeta

← #29834998

Subgroups — #29834998

Order 15616.

596 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456398C211yesmaximal, characteristic, complemented
2C4×#14519799C211yesmaximal, characteristic, complemented
3#21448011C212yesmaximal, complemented
4#21447972C211yesmaximal, characteristic, complemented
5#21447967C211yesmaximal, characteristic, complemented
6#21447860C211yesmaximal, characteristic, complemented
7#21447855C211yesmaximal, characteristic, complemented
8#21447744C212yesmaximal, complemented
9C4×#14519722C212yesmaximal, complemented
10C4×D976C212yesmaximal, complemented
11C61×#2162C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13#16840312—21no—
14C4×#11758222—21no—
15C2×#14519722C2212yes—
16C2×D976C2212yes—
17#16840167C2211yescharacteristic
18#16840139C2211yescharacteristic
19#16840109—21no—
20#16839998—21no—
21#16839998C2212yes—
22C4×Dic244C2211yescharacteristic
23C4×Dic244C2211yescharacteristic
24C4×#11758190—21no—
25C4×#11758190—21no—
26C4×D488—21no—
27C4×D488C2211yescharacteristic
28C4×D488C2211yescharacteristic
29C4×D488C2212yes—
30#16839800C2211yescharacteristic
31#16839759—21no—
32#16839590C2212yes—
33#16839588C2212yes—
34#16839567C2212yes—
35#16839564C2212yes—
36#16839564C2212yes—
37#16839564C2212yes—
38C2×C4×C8×C61C2211yescharacteristic
39C61×C4⋊C16C2211yescharacteristic
40C61×C4⋊C16C2211yescharacteristic
41C61×C22⋊C16C2212yes—
42C4×C16×C61C2212yes—
43C2×#11758222—21no—
44C2×#11758222—41no—
45#14519799—22no—
46#14519799—22no—
47C2×Dic244C2311yescharacteristic
48C2×#11758190—21no—
49C2×D488—21no—
50C2×D488C2312yes—
51C2×D488C2311yescharacteristic
52#14519777—21no—
53#14519774—21no—
54#14519765—41no—
55#14519751—22no—
56#14519751—41no—
57C4×Dic122—21no—
58C4×Dic122—21no—
59C4×D244—21no—
60C4×D244—21no—
61C4×D244—22no—
62C4×D244—41no—
63C4×C22⋊D61—41no—
64C4×C22⋊D61—41no—
65#14519736—21no—
66#14519734—41no—
67C42×D61—41no—
68#14519722—24no—
69D976—24no—
70#14519711—21no—
71#14519711C2312yes—
72#14519711C2311yescharacteristic
73#14519711C2311yescharacteristic
74#14519709—21no—
75#14519702—21no—
76#14519702—21no—
77#14519699—22no—
78#14519699—22no—
79#14519699—21no—
80#14519699—21no—
81C22×C8×C61D411yescharacteristic
82C22×C8×C61C2311yescharacteristic
83C2×C42×C61D411yescharacteristic
84C2×C16×C61C2312yes—
85C2×C16×C61C2312yes—
86C4×C8×C61D411yescharacteristic
87C4×C8×C61D411yescharacteristic
88C4×C8×C61C2311yescharacteristic
89C4×C8×C61C2311yescharacteristic
90C22×C4×C61D811yescharacteristic
91C22×C4×C61D811yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C2×C22⋊D61—41no—
94C2×C22⋊D61—42no—
95#11758222—24no—
96#11758222—42no—
97#11758222—84no—
98C2×D244—21no—
99C2×D244—22no—
100C2×D244—41no—
101C2×D244—41no—
102C2×C4×D61—41no—
103C2×C4×D61—42no—
104C2×C4×D61—41no—
105C2×C4×D61—41no—
106C2×Dic122—21no—
107C2×Dic122—41no—
108C2×C8×C61—21no—
109C2×C8×C61—21no—
110C2×C8×C61—41no—
111C2×C8×C61C2×D411yescharacteristic
112C2×C8×C61C2×D411yescharacteristic
113C2×C8×C61C2×D411yescharacteristic
114C2×C8×C61C2411yescharacteristic
115C42×C61D811yescharacteristic
116C42×C61D811yescharacteristic
117C42×C61C2×D411yescharacteristic
118C42×C61C2×D411yescharacteristic
119C22⋊Dic61—41no—
120#11758198—41no—
121#11758198—41no—
122#11758198—42no—
123#11758198—42no—
124C4⋊Dic61—22no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127C4⋊Dic61—41no—
128#11758196—41no—
129#11758196—41no—
130C4×Dic61—41no—
131C4×Dic61—41no—
132C4×Dic61—41no—
133Dic244—21no—
134Dic244#4112yescomplemented
135D488—21no—
136D488—22no—
137D488—22no—
138D488—42no—
139D488#4112yescomplemented
140#11758190—24no—
141C16×C61—22no—
142C16×C61—22no—
143C23×C61C2×D811yescharacteristic
144C22×D61—41no—
145C22×D61—42no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61C2×D811yescharacteristic
151C2×C4×C61C2×D811yescharacteristic
152C2×C4×C61C2×D811yescharacteristic
153C2×C4×C61C2×D811yescharacteristic
154C2×C4×C61C2×D811yescharacteristic
155C2×C4×C61C22×D411yescharacteristic
156C22⋊D61—44no—
157C22⋊D61—84no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—42no—
162D244—22no—
163D244—42no—
164D244—41no—
165D244—42no—
166D244—82no—
167D244—82no—
168C4×D61—44no—
169C4×D61—82no—
170C4×D61—82no—
171C4×D61—84no—
172Dic122—22no—
173Dic122—41no—
174Dic122—82no—
175C8×C61—21no—
176C8×C61—21no—
177C8×C61—21no—
178C8×C61C2×#4111yescharacteristic
179C8×C61C2×#4111yescharacteristic
180#33666—611nomaximal, Sylow
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61C22×D811yescharacteristic
185C2×D61—42no—
186C2×D61—81no—
187C2×D61—81no—
188C2×D61—82no—
189C2×D61—82no—
190C2×D61—82no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61#50812yes—
195C4×C61#50912yes—
196C4×C61#52511yescharacteristic
197C4×C61#57511yescharacteristic
198Dic61—42no—
199Dic61—81no—
200Dic61—82no—
201Dic61—81no—
202#3169—611no—
203C4×#135—611no—
204#2234—612no—
205#2227—611no—
206#2226—611no—
207#2207—611no—
208#2206—611no—
209#2186—612no—
210C4×SD32—612no—
211C4×D16—612no—
212#2162D6111yescharacteristic, complemented
213C2×C61—41no—
214C2×C61#310011yescharacteristic
215C2×C61C2×#50811yescharacteristic
216C2×C61C2×#50911yescharacteristic
217D61—82no—
218D61—162no—
219C2×#135—611no—
220#532—1221no—
221C4×#41—1221no—
222C2×SD32—612no—
223C2×D16—612no—
224#500—611no—
225#493—611no—
226#486—1221no—
227#466—612no—
228#466—1221no—
229C4×Q16—611no—
230C4×Q16—611no—
231C4×SD16—1221no—
232C4×SD16—1221no—
233C4×D8—612no—
234C4×D8—611no—
235C4×D8—611no—
236C4×D8—1221no—
237#427—611no—
238#416—1221no—
239C2×C4×C8C2×D6111yescharacteristic
240#367—612no—
241#366—612no—
242C4⋊C16C2×D6111yescharacteristic
243C4⋊C16C2×D6111yescharacteristic
244#358—612no—
245#357—612no—
246#357—612no—
247#357—612no—
248C22⋊C16C2×D6112yes—
249C4×C16C2×D6112yes—
250C61#3366611yescharacteristic, complemented, Sylow
251C2×#41—1221no—
252C2×#41—2441no—
253#135—1222no—
254#135—1222no—
255C2×Q16—611no—
256C2×SD16—1221no—
257C2×D8—611no—
258C2×D8—612no—
259C2×D8—1221no—
260C22×C8D24411yescharacteristic
261C22×C8C22×D6111yescharacteristic
262C4⋊Q8—1221no—
263#127—1221no—
264#124—2441no—
265#121—1222no—
266#121—2441no—
267C4×Q8—1221no—
268C4×Q8—1221no—
269C4×D4—1221no—
270C4×D4—1221no—
271C4×D4—1222no—
272C4×D4—2441no—
273C4×D4—2441no—
274C4×D4—2441no—
275#117—1221no—
276#117—2441no—
277C2×C42—2441no—
278C2×C42D24411yescharacteristic
279SD32—1224no—
280D16—1224no—
281C2×C16C22×D6112yes—
282C2×C16C22×D6112yes—
283#107—611no—
284#107—611no—
285#107—612no—
286#107—1221no—
287#106—1221no—
288#103—1221no—
289#103—1221no—
290#102—1221no—
291#102—1221no—
292#102—1222no—
293#102—1222no—
294C4×C8D24411yescharacteristic
295C4×C8D24411yescharacteristic
296C4×C8C22×D6111yescharacteristic
297C4×C8C22×D6111yescharacteristic
298#41—1224no—
299#41—4884no—
300#41—2442no—
301C2×Q8—1221no—
302C2×Q8—2441no—
303C2×D4—1221no—
304C2×D4—1222no—
305C2×D4—2442no—
306C2×D4—2441no—
307C2×D4—2441no—
308C2×D4—2441no—
309C22×C4—2441no—
310C22×C4—2441no—
311C22×C4—2442no—
312C22×C4—2441no—
313C22×C4D48811yescharacteristic
314C22×C4D48811yescharacteristic
315C22×C4C2×D24411yescharacteristic
316Q16—612no—
317Q16—1221no—
318SD16—1224no—
319D8—612no—
320D8—1221no—
321D8—1222no—
322D8—1222no—
323D8—2442no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—41no—
327C2×C8C2×D24411yescharacteristic
328C2×C8C2×D24411yescharacteristic
329C2×C8C2×D24411yescharacteristic
330C2×C8C23×D6111yescharacteristic
331C4⋊C4—1222no—
332C4⋊C4—1221no—
333C4⋊C4—1221no—
334C4⋊C4—2441no—
335C4⋊C4—2441no—
336C4⋊C4—2441no—
337C22⋊C4—2441no—
338C22⋊C4—2441no—
339C22⋊C4—2441no—
340C22⋊C4—2442no—
341C22⋊C4—2442no—
342C42—2441no—
343C42—2441no—
344C42—2441no—
345C42D48811yescharacteristic
346C42D48811yescharacteristic
347C42C2×D24411yescharacteristic
348C42C2×D24411yescharacteristic
349C16—22no—
350C16—22no—
351C23—2442no—
352C23—2441no—
353C23C2×D48811yescharacteristic
354Q8—1222no—
355Q8—4882no—
356Q8—2441no—
357D4—1222no—
358D4—4882no—
359D4—4884no—
360D4—4882no—
361D4—2444no—
362D4—2442no—
363D4—2442no—
364D4—2441no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—4882no—
370C2×C4—4882no—
371C2×C4—4884no—
372C2×C4—2444no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2442no—
376C2×C4—2441no—
377C2×C4C2×D48811yescharacteristic
378C2×C4C2×D48811yescharacteristic
379C2×C4C2×D48811yescharacteristic
380C2×C4C2×D48811yescharacteristic
381C2×C4C2×D48811yescharacteristic
382C2×C4C22×D24411yescharacteristic
383C8—21no—
384C8—21no—
385C8—21no—
386C8#41×D6111yescharacteristic
387C8#41×D6111yescharacteristic
388C22—21no—
389C22—21no—
390C22—21no—
391C22—4882no—
392C22—4882no—
393C22—4882no—
394C22—4881no—
395C22—4881no—
396C22—2442no—
397C22C22×D48811yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—4881no—
402C4—4882no—
403C4—4881no—
404C4—2442no—
405C4#1684019412yes—
406C4#1684020012yes—
407C4#1684027411yescharacteristic
408C4#1684056211yescharacteristic
409C2—41no—
410C2—9762no—
411C2—4882no—
412C2#2145582011yescharacteristic
413C2C2×#1684019411yescharacteristic
414C2C2×#1684020011yescharacteristic
subgroupG/H
centerC2×C4C2×D488
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2162C2
socleC22×C61C22×D8
hypercenter#2162D61
hypocenterC61#33666
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #2162

Complements

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

NH ≅ G/Ntype
1. #21456398C2semidirect
2. C4×#14519799C2semidirect
3. #21448011C2semidirect
4. #21447972C2semidirect
5. #21447967C2semidirect
6. #21447860C2semidirect
7. #21447855C2semidirect
8. #21447744C2semidirect
9. C4×#14519722C2semidirect
10. C4×D976C2semidirect
11. C61×#2162C2semidirect
134. Dic244#41semidirect
139. D488#41semidirect
212. #2162D61semidirect
250. C61#33666semidirect

Maximal subgroups

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

subgroupindex
#214563982
C4×#145197992
#214480112
#214479722
#214479672
#214478602
#214478552
#214477442
C4×#145197222
C4×D9762
C61×#21622
#3366661