Finite Group Databasebeta

← #29835334

Subgroups — #29835334

Order 15616.

590 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456331C211yesmaximal, characteristic, complemented
2#21456243C211yesmaximal, characteristic
3#21448063C212yesmaximal, complemented
4#21448059C212yesmaximal, complemented
5#21448057C212yesmaximal, complemented
6#21448054C212yesmaximal, complemented
7#21447527C211yesmaximal, characteristic, complemented
8#21447526C211yesmaximal, characteristic, complemented
9#21447524C211yesmaximal, characteristic, complemented
10C2×#16839537C211yesmaximal, characteristic, complemented
11C61×#2138C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C2×#14519797—21no—
14#16840315—21no—
15#16840308—21no—
16C22×#11758185C2211yescharacteristic
17#16840166C2212yes—
18#16840163C2212yes—
19#16840162—21no—
20#16840159—21no—
21#16840147C2211yescharacteristic
22#16840145C2211yescharacteristic
23#16840143—21no—
24#16840140C2211yescharacteristic
25#16840137C2211yescharacteristic
26#16840133—21no—
27#16839800C2211yescharacteristic
28#16839798—21no—
29C2×C8×Dic61C2211yescharacteristic
30#16839567C2212yes—
31#16839566C2212yes—
32#16839565C2212yes—
33#16839564C2212yes—
34#16839563C2212yes—
35#16839562C2212yes—
36#16839537C2212yes—
37#16839537C2212yes—
38C2×#135×C61C2211yescharacteristic
39C22×C16×C61C2211yescharacteristic
40C61×#427C2211yescharacteristic
41C61×#358C2212yes—
42C61×#357C2212yes—
43C2×#11758222—21no—
44#14519799—22no—
45#14519799—22no—
46#14519797—44no—
47C2×Dic244C2311yescharacteristic
48C2×#11758202—21no—
49C2×#11758190—21no—
50C2×#11758200—21no—
51C2×#11758201—21no—
52C2×D488C2311yescharacteristic
53C2×#11758199—21no—
54C22×#92418D411yescharacteristic
55#14519776—22no—
56#14519775—21no—
57#14519773—21no—
58#14519768—22no—
59#14519766—21no—
60#14519764—21no—
61#14519754—42no—
62#14519749—42no—
63C4×Dic122—42no—
64C4×C22⋊D61—42no—
65#14519736—21no—
66C2×C4×Dic61D411yescharacteristic
67C2×#11758185—22no—
68C2×#11758185C2312yes—
69#14519711C2312yes—
70#14519710—22no—
71#14519709—21no—
72#14519708—22no—
73C8×Dic61D411yescharacteristic
74C8×Dic61D411yescharacteristic
75C8×Dic61C2311yescharacteristic
76C8×Dic61C2311yescharacteristic
77C2×#41×C61—21no—
78#135×C61—22no—
79#135×C61—22no—
80C2×Q16×C61C2311yescharacteristic
81C2×SD16×C61—21no—
82C2×D8×C61C2311yescharacteristic
83C22×C8×C61C2311yescharacteristic
84#117×C61—21no—
85C2×C16×C61—22no—
86C2×C16×C61C2312yes—
87#107×C61C2312yes—
88#106×C61—21no—
89#41×C61—42no—
90#41×C61—42no—
91C2×Q8×C61—21no—
92C2×D4×C61—21no—
93C2×D4×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61C2×D411yescharacteristic
96C2×C22⋊D61—41no—
97C2×C22⋊D61—42no—
98C22×Dic61D812yes—
99#11758222—42no—
100#11758222—42no—
101C2×D244—21no—
102C2×C4×D61—41no—
103C2×Dic122—21no—
104C2×Dic122—42no—
105Q16×C61—21no—
106Q16×C61—21no—
107SD16×C61—42no—
108D8×C61—21no—
109D8×C61—21no—
110C2×C8×C61C2×D411yescharacteristic
111C2×C8×C61#4112yes—
112C2×C8×C61#4112yes—
113C2×C8×C61C2411yescharacteristic
114C4⋊C4×C61—22no—
115C22⋊C4×C61—41no—
116C42×C61—41no—
117C22⋊Dic61—41no—
118C22⋊Dic61—42no—
119#11758202—42no—
120#11758201—42no—
121#11758200—42no—
122#11758199—42no—
123#11758198—42no—
124C4⋊Dic61—22no—
125C4⋊Dic61—42no—
126#11758196—42no—
127#11758196—42no—
128#11758196—42no—
129C4×Dic61—41no—
130C4×Dic61D812yes—
131C4×Dic61C2×D411yescharacteristic
132C4×Dic61C2×D411yescharacteristic
133C2×#92418SD1614yes—
134C2×#92418C2×D411yescharacteristic
135C2×#92418C2×D411yescharacteristic
136Dic244—21no—
137Dic244—21no—
138D488—21no—
139D488—21no—
140#11758190—42no—
141C16×C61—22no—
142#11758185—22no—
143C23×C61—41no—
144C23×C61C2×D811yescharacteristic
145C22×D61—41no—
146Q8×C61—41no—
147Q8×C61—41no—
148D4×C61—41no—
149D4×C61—41no—
150D4×C61—82no—
151C2×C4×C61—42no—
152C2×C4×C61—41no—
153C2×C4×C61—82no—
154C2×C4×C61#12412yes—
155C2×C4×C61C2×D811yescharacteristic
156C2×C4×C61C2×SD1612yes—
157C2×C4×C61C22×D411yescharacteristic
158C22⋊D61—84no—
159C22⋊D61—82no—
160C2×Dic61—42no—
161C2×Dic61—42no—
162C2×Dic61—41no—
163C2×Dic61C2×D812yes—
164C2×Dic61C2×D812yes—
165D244—41no—
166D244—41no—
167C4×D61—82no—
168Dic122—41no—
169Dic122—41no—
170Dic122—84no—
171C8×C61#12412yes—
172C8×C61C2×#4111yescharacteristic
173C8×C61C2×#4111yescharacteristic
174#92418C2×SD1612yes—
175#92418C2×SD1612yes—
176#33701—611nomaximal, Sylow
177C22×C61—22no—
178C22×C61—81no—
179C22×C61—82no—
180C22×C61#48612yes—
181C22×C61C22×D811yescharacteristic
182C2×D61—81no—
183C2×D61—82no—
184C4×C61—81no—
185C4×C61—82no—
186C4×C61#48612yes—
187C4×C61C2×#12411yescharacteristic
188C4×C61C22×SD1611yescharacteristic
189Dic61—22no—
190Dic61—22no—
191Dic61—81no—
192Dic61—82no—
193#3160—611no—
194#3147—611no—
195#2245—612no—
196#2244—612no—
197#2243—612no—
198#2243—612no—
199#2138—611no—
200#2138—611no—
201#2138—611no—
202#2138D6111yescharacteristic, complemented
203C2×#346—611no—
204C2×C61—22no—
205C2×C61—161no—
206C2×C61C2×#48611yescharacteristic
207C2×C61#341412yes—
208D61—161no—
209C2×#135—611no—
210C2×#135—1221no—
211C2×#135C2×D6111yescharacteristic
212#533—1221no—
213#532—1221no—
214C22×C16—611no—
215C22×C16C2×D6111yescharacteristic
216#500—612no—
217#499—612no—
218#499—1221no—
219#498—1221no—
220#495—611no—
221#495—611no—
222#495—1221no—
223#494—611no—
224#493—611no—
225#492—1221no—
226#427—611no—
227#427—1221no—
228#427C2×D6111yescharacteristic
229C2×C4×C8—611no—
230#358—612no—
231#358—612no—
232#358—612no—
233#358C2×D6112yes—
234#357—612no—
235#357—612no—
236#357—612no—
237#357C2×D6112yes—
238#346—612no—
239#346—612no—
240C61#3370111yescharacteristic, complemented, Sylow
241C2×#41—21no—
242C2×#41—1221no—
243#135—22no—
244#135—22no—
245#135—1222no—
246#135—1222no—
247#135—2444no—
248C2×Q16—611no—
249C2×Q16—1221no—
250C2×Q16C22×D6111yescharacteristic
251C2×SD16—21no—
252C2×SD16—1221no—
253C2×SD16—1221no—
254C2×SD16—1221no—
255C2×D8—611no—
256C2×D8—1221no—
257C2×D8C22×D6111yescharacteristic
258C22×C8—611no—
259C22×C8C22×D6111yescharacteristic
260C4⋊Q8—1222no—
261C4⋊Q8—1221no—
262#127—1221no—
263#125—1222no—
264#124—1221no—
265#124—1221no—
266C22⋊Q8—2442no—
267#121—2442no—
268C4×Q8—2442no—
269C4×D4—2442no—
270#117—21no—
271#117—1221no—
272C2×C42—611no—
273C2×C16—22no—
274C2×C16—612no—
275C2×C16—1222no—
276C2×C16C22×D6112yes—
277#107—612no—
278#107—1222no—
279#107C22×D6112yes—
280#106—21no—
281#106—1222no—
282#106—1221no—
283C4×C8—611no—
284C4×C8—611no—
285C4×C8—611no—
286C4×C8—611no—
287#41—42no—
288#41—42no—
289#41—2442no—
290#41—2442no—
291C2×Q8—21no—
292C2×Q8—1221no—
293C2×Q8—2442no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—1221no—
297C2×D4—2442no—
298C2×D4—2441no—
299C22×C4—41no—
300C22×C4—612no—
301C22×C4—2441no—
302C22×C4D4×D6111yescharacteristic
303Q16—21no—
304Q16—21no—
305Q16—1221no—
306Q16—1221no—
307Q16—2442no—
308SD16—42no—
309SD16—2442no—
310SD16—2442no—
311SD16—2442no—
312D8—21no—
313D8—21no—
314D8—1221no—
315D8—1221no—
316D8—2442no—
317C2×C8—611no—
318C2×C8—611no—
319C2×C8—614no—
320C2×C8#1175822212yes—
321C2×C8D4×D6111yescharacteristic
322C2×C8#1175822412yes—
323C2×C8C23×D6111yescharacteristic
324C4⋊C4—22no—
325C4⋊C4—1222no—
326C4⋊C4—2442no—
327C4⋊C4—2442no—
328C4⋊C4—2442no—
329C4⋊C4—2442no—
330C22⋊C4—41no—
331C22⋊C4—2442no—
332C22⋊C4—2442no—
333C22⋊C4—2441no—
334C42—41no—
335C42—611no—
336C42—611no—
337C42—612no—
338C42—2441no—
339C16—22no—
340C16—1222no—
341C23—41no—
342C23—2441no—
343C23D8×D6111yescharacteristic
344Q8—41no—
345Q8—41no—
346Q8—4884no—
347Q8—2441no—
348Q8—2441no—
349D4—41no—
350D4—41no—
351D4—82no—
352D4—4884no—
353D4—4882no—
354D4—2441no—
355D4—2441no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—82no—
359C2×C4—612no—
360C2×C4—612no—
361C2×C4—4882no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4—2441no—
365C2×C4#1451976312yes—
366C2×C4D8×D6111yescharacteristic
367C2×C4SD16×D6112yes—
368C2×C4C2×D4×D6111yescharacteristic
369C8—612no—
370C8—612no—
371C8#1451976312yes—
372C8C2×#1175822411yescharacteristic
373C8C2×#1175822211yescharacteristic
374C22—22no—
375C22—82no—
376C22—81no—
377C22—4882no—
378C22—4881no—
379C22#1684010712yes—
380C22C2×D8×D6111yescharacteristic
381C4—82no—
382C4—81no—
383C4—1222no—
384C4—1222no—
385C4—4882no—
386C4—4881no—
387C4#1684010712yes—
388C4C2×#1451976311yescharacteristic
389C4C2×SD16×D6111yescharacteristic
390C2—22no—
391C2—161no—
392C2—9761no—
393C2C2×#1684010711yescharacteristic
394C2#2145895212yes—
subgroupG/H
centerC2×C4D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2138C2
socleC22×C61C22×D8
prime radical#21456243C2
hypercenter#2138D61
hypocenterC61#33701
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. #21456331C2semidirect
3. #21448063C2semidirect
4. #21448059C2semidirect
5. #21448057C2semidirect
6. #21448054C2semidirect
7. #21447527C2semidirect
8. #21447526C2semidirect
9. #21447524C2semidirect
10. C2×#16839537C2semidirect
11. C61×#2138C2semidirect
202. #2138D61semidirect
240. C61#33701semidirect

Maximal subgroups

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

subgroupindex
#214563312
#214562432
#214480632
#214480592
#214480572
#214480542
#214475272
#214475262
#214475242
C2×#168395372
C61×#21382
#3370161