Finite Group Databasebeta

← #29834685

Subgroups — #29834685

Order 15616.

574 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#500×D61C211yesmaximal, characteristic, complemented
2C4×#14519800C211yesmaximal, characteristic
3#21447971C211yesmaximal, characteristic, complemented
4#21447967C211yesmaximal, characteristic, complemented
5#21447963C211yesmaximal, characteristic, complemented
6#21447959C212yesmaximal, complemented
7#21447955C212yesmaximal, complemented
8#21447604C212yesmaximal, complemented
9#21447573C211yesmaximal, characteristic, complemented
10#21447530C212yesmaximal
11C61×#2227C211yesmaximal, characteristic, complemented
12C2×#14519800C2211yescharacteristic
13C4⋊Q8×D61—21no—
14C4×#11758226—21no—
15C2×#14519716C2212yes—
16#16840167C2211yescharacteristic
17#16840166—21no—
18#16840166C2212yes—
19#16840165—21no—
20Q16×Dic61C2211yescharacteristic
21C4×#11758201—21no—
22#16839847—21no—
23C4×D488C2211yescharacteristic
24#16839844C2211yescharacteristic
25#16839799C2211yescharacteristic
26#107×D61—21no—
27#107×D61C2212yes—
28#16839757—21no—
29C4×C8×D61C2211yescharacteristic
30#16839590C2212yes—
31#16839589C2212yes—
32#16839587C2212yes—
33C4⋊#11758185C2211yescharacteristic
34#16839580C2212yes—
35#16839566C2212yes—
36#16839564C2212yes—
37#16839562C2212yes—
38C61×#500C2211yescharacteristic
39C4×Q16×C61C2211yescharacteristic
40C61×#366C2212yes—
41C61×C4⋊C16C2211yescharacteristic
42C61×#358C2212yes—
43C2×#11758226—21no—
44C2×Q8×D61—41no—
45#14519800—22no—
46#14519800Q814yescomplemented
47C2×#11758201—21no—
48C2×D488C2311yescharacteristic
49C2×C8×D61D411yescharacteristic
50C2×C8×D61C2311yescharacteristic
51#14519777—21no—
52#14519776—22no—
53#14519776—41no—
54#14519775—41no—
55Q8×Dic61—21no—
56C4×D244—21no—
57C4×D244—41no—
58#14519739—21no—
59#14519739—41no—
60#14519735—21no—
61#14519735—41no—
62C4⋊C4×D61—22no—
63C4⋊C4×D61—41no—
64C42×D61—41no—
65C42×D61D411yescharacteristic
66#14519716—24no—
67C2×#11758185C2312yes—
68#14519711—21no—
69#14519711C2312yes—
70#14519711C2311yescharacteristic
71#14519710—22no—
72#14519710—22no—
73#14519708—21no—
74#14519701—21no—
75#14519699—21no—
76#14519697—21no—
77C4×#92418D411yescharacteristic
78C8×Dic61D411yescharacteristic
79C8×Dic61C2311yescharacteristic
80C2×Q16×C61C2311yescharacteristic
81C4⋊Q8×C61—21no—
82C4×Q8×C61—21no—
83C2×C16×C61C2312yes—
84#107×C61—21no—
85#107×C61C2312yes—
86#107×C61C2311yescharacteristic
87#103×C61—21no—
88C4×C8×C61C2311yescharacteristic
89C2×Q8×C61—21no—
90C2×Q8×C61—41no—
91#11758226—24no—
92#11758226—42no—
93Q8×D61—84no—
94C2×D244—21no—
95C2×D244—41no—
96C2×C4×D61—41no—
97C2×C4×D61—42no—
98C2×C4×D61—41no—
99C2×C4×D61—41no—
100C2×C4×D61D811yescharacteristic
101C2×C4×D61D811yescharacteristic
102C2×C4×D61C2×D411yescharacteristic
103C2×Dic122—41no—
104C2×Dic122—42no—
105Q16×C61—21no—
106Q16×C61C2×Q812yescomplemented
107C2×C8×C61C2×D411yescharacteristic
108C2×C8×C61C2411yescharacteristic
109C4⋊C4×C61—21no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—41no—
113C42×C61—41no—
114C42×C61C2×D411yescharacteristic
115#11758201—24no—
116#11758198—41no—
117#11758198—41no—
118#11758198—41no—
119C4⋊Dic61—21no—
120C4⋊Dic61—22no—
121C4⋊Dic61—41no—
122C4⋊Dic61—41no—
123#11758196—42no—
124#11758196—42no—
125C4×Dic61—41no—
126C4×Dic61—41no—
127C4×Dic61D811yescharacteristic
128C4×Dic61D811yescharacteristic
129C4×Dic61C2×D411yescharacteristic
130C2×#92418C2×D411yescharacteristic
131C2×#92418C2×D411yescharacteristic
132D488—21no—
133D488C2×Q812yescomplemented
134C8×D61—22no—
135C8×D61C2×Q812yes—
136C8×D61#4112yes—
137C16×C61—22no—
138#11758185—22no—
139C22×D61—41no—
140C22×D61C2×D811yescharacteristic
141Q8×C61—22no—
142Q8×C61—41no—
143Q8×C61—82no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—42no—
147C2×C4×C61C2×D811yescharacteristic
148C2×C4×C61C2×D811yescharacteristic
149C2×C4×C61C22×D411yescharacteristic
150C2×Dic61—41no—
151C2×Dic61—42no—
152C2×Dic61C2×D811yescharacteristic
153C2×Dic61C2×D811yescharacteristic
154C2×Dic61C2×D811yescharacteristic
155D244—22no—
156D244—44no—
157D244—41no—
158C4×D61—22no—
159C4×D61—22no—
160C4×D61—44no—
161C4×D61—84no—
162C4×D61—82no—
163C4×D61—82no—
164C4×D61C22⋊Q812yes—
165C4×D61#13712yes—
166Dic122—82no—
167Dic122—84no—
168C8×C61—21no—
169C8×C61C22×Q811yescharacteristic
170C8×C61C2×#4111yescharacteristic
171#92418—21no—
172#92418C22⋊Q812yes—
173#33629—611nomaximal, Sylow
174C22×C61C22×D811yescharacteristic
175C2×D61—21no—
176C2×D61—21no—
177C2×D61—42no—
178C2×D61—81no—
179C2×D61—81no—
180C2×D61#47412yes—
181C4×C61—21no—
182C4×C61—42no—
183C4×C61—82no—
184C4×C61—81no—
185C4×C61#51012yes—
186C4×C61C2×C22⋊Q811yescharacteristic
187C4×C61C2×#13711yescharacteristic
188Dic61—21no—
189Dic61—82no—
190Dic61—81no—
191Dic61#47412yes—
192Dic61#50912yes—
193C2×#500—611no—
194C4×#135—611no—
195#2227—611no—
196#2227D6111yescharacteristic, complemented
197#2226—611no—
198#2226—611no—
199#2225—612no—
200#2224—612no—
201#2159—612no—
202#2150—611no—
203#2139—612no—
204C2×C61C2×#47411yescharacteristic
205C2×C61C2×#50911yescharacteristic
206C2×C61C2×#51011yescharacteristic
207D61—22no—
208D61—82no—
209C2×#135—611no—
210C2×C4⋊Q8—1221no—
211C4×#41—1221no—
212C2×M32—612no—
213#500—612no—
214#500—611no—
215#500—1221no—
216#500—1221no—
217#500C2×D6111yescharacteristic
218C4×Q16—611no—
219C4×Q16C2×D6111yescharacteristic
220C4×SD16—1221no—
221C4×SD16—1221no—
222C4×D8—611no—
223C4×D8—611no—
224#427—611no—
225C2×#107—612no—
226C2×#107—1221no—
227#416—1221no—
228C2×C4×C8—611no—
229#367—612no—
230#367—612no—
231#366—612no—
232#366C2×D6112yes—
233C4⋊C16—612no—
234C4⋊C16—611no—
235C4⋊C16C2×D6111yescharacteristic
236#358—612no—
237#358C2×D6112yes—
238#357—612no—
239#357—612no—
240C61#3362911yescharacteristic, complemented, Sylow
241C2×#41—1221no—
242C22×Q8—2441no—
243#135—614no—
244#135—1222no—
245C2×Q16C22×D6111yescharacteristic
246C2×SD16—1221no—
247C2×D8—611no—
248C22×C8—611no—
249C22×C8—611no—
250C4⋊Q8—21no—
251C4⋊Q8—1222no—
252C4⋊Q8—1221no—
253C4⋊Q8—2441no—
254C4⋊Q8—2441no—
255C4×Q8—21no—
256C4×Q8—1221no—
257C4×D4—1221no—
258C4×D4—1221no—
259C4×D4—2441no—
260C4×D4—2441no—
261#117—1221no—
262#117—2441no—
263C2×C4⋊C4—1222no—
264C2×C4⋊C4—2441no—
265C2×C42—611no—
266C2×C42—2441no—
267M32—1224no—
268C2×C16—612no—
269C2×C16C22×D6112yes—
270#107—21no—
271#107—612no—
272#107—611no—
273#107—1222no—
274#107—1222no—
275#107—1221no—
276#107C22×D6112yes—
277#107C22×D6111yescharacteristic
278#106—1221no—
279#103—21no—
280#103—1221no—
281#102—1221no—
282#102—1221no—
283C4×C8—611no—
284C4×C8—611no—
285C4×C8—611no—
286C4×C8C22×D6111yescharacteristic
287#41—1224no—
288#41—2442no—
289C2×Q8—21no—
290C2×Q8—41no—
291C2×Q8—4884no—
292C2×Q8—2442no—
293C2×Q8—2441no—
294C2×D4—1221no—
295C2×D4—2441no—
296C22×C4—611no—
297C22×C4—611no—
298C22×C4—611no—
299C22×C4—2441no—
300C22×C4—2441no—
301C22×C4—2442no—
302C22×C4—2441no—
303Q16—21no—
304Q16Q8×D6112yescomplemented
305SD16—1224no—
306D8—612no—
307D8—1221no—
308C2×C8—611no—
309C2×C8—612no—
310C2×C8—612no—
311C2×C8—611no—
312C2×C8—1222no—
313C2×C8D4×D6111yescharacteristic
314C2×C8C23×D6111yescharacteristic
315C4⋊C4—21no—
316C4⋊C4—22no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—1221no—
320C4⋊C4—1222no—
321C4⋊C4—2442no—
322C4⋊C4—2442no—
323C4⋊C4—2441no—
324C4⋊C4—2441no—
325C22⋊C4—2441no—
326C22⋊C4—2441no—
327C22⋊C4—2441no—
328C42—41no—
329C42—611no—
330C42—611no—
331C42—611no—
332C42—2441no—
333C42—2441no—
334C42D4×D6111yescharacteristic
335C16—22no—
336C16—1222no—
337C23—611no—
338C23—2441no—
339Q8—22no—
340Q8—41no—
341Q8—82no—
342Q8—4882no—
343Q8—4884no—
344D4—1222no—
345D4—2444no—
346D4—2441no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—611no—
351C2×C4—611no—
352C2×C4—612no—
353C2×C4—612no—
354C2×C4—611no—
355C2×C4—1222no—
356C2×C4—1222no—
357C2×C4—4882no—
358C2×C4—4884no—
359C2×C4—4882no—
360C2×C4—2444no—
361C2×C4—2441no—
362C2×C4—2442no—
363C2×C4D8×D6111yescharacteristic
364C2×C4D8×D6111yescharacteristic
365C2×C4C2×D4×D6111yescharacteristic
366C8—21no—
367C8—612no—
368C8—1221no—
369C8C2×Q8×D6111yescharacteristic
370C8#41×D6111yescharacteristic
371C22—612no—
372C22—1221no—
373C22—1221no—
374C22—4881no—
375C22—4881no—
376C22—2442no—
377C22C2×D8×D6111yescharacteristic
378C4—21no—
379C4—42no—
380C4—81no—
381C4—82no—
382C4—612no—
383C4—612no—
384C4—1221no—
385C4—4881no—
386C4—4882no—
387C4#1684020412yes—
388C4C22⋊Q8×D6111yescharacteristic
389C4#137×D6111yescharacteristic
390C2—1222no—
391C2—4882no—
392C2#474×D6111yescharacteristic
393C2#509×D6111yescharacteristic
394C2C2×#1684020411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2227C2
socleC22×C61C22×D8
prime radicalC2×#14519800C22
hypercenter#2227D61
hypocenterC61#33629
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2227

Complements

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

NH ≅ G/Ntype
1. #500×D61C2semidirect
3. #21447971C2semidirect
4. #21447967C2semidirect
5. #21447963C2semidirect
6. #21447959C2semidirect
7. #21447955C2semidirect
8. #21447604C2semidirect
9. #21447573C2semidirect
11. C61×#2227C2semidirect
46. #14519800Q8semidirect
106. Q16×C61C2×Q8semidirect
133. D488C2×Q8semidirect
196. #2227D61semidirect
240. C61#33629semidirect
304. Q16Q8×D61semidirect

Maximal subgroups

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

subgroupindex
#500×D612
C4×#145198002
#214479712
#214479672
#214479632
#214479592
#214479552
#214476042
#214475732
#214475302
C61×#22272
#3362961