Finite Group Databasebeta

← #29834679

Subgroups — #29834679

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×#14519796C211yesmaximal, characteristic
3#21447970C212yesmaximal, complemented
4#21447964C212yesmaximal, complemented
5#21447962C211yesmaximal, characteristic, complemented
6#21447958C211yesmaximal, characteristic, complemented
7#21447956C211yesmaximal, characteristic, complemented
8#21447604C212yesmaximal, complemented
9#21447573C211yesmaximal, characteristic, complemented
10#21447528C212yesmaximal
11C61×#2224C211yesmaximal, characteristic, complemented
12C2×#14519796C2211yescharacteristic
13C4⋊Q8×D61—21no—
14C4×#11758224—21no—
15C2×#14519716C2212yes—
16#16840167C2211yescharacteristic
17#16840166—21no—
18#16840166C2212yes—
19#16840165—21no—
20C4×Dic244C2211yescharacteristic
21#16839856C2211yescharacteristic
22#16839853—21no—
23C4×#11758200—21no—
24D8×Dic61C2211yescharacteristic
25#16839799C2211yescharacteristic
26#107×D61—21no—
27#107×D61C2212yes—
28#16839755—21no—
29C4×C8×D61C2211yescharacteristic
30#16839590C2212yes—
31#16839589C2212yes—
32#16839587C2212yes—
33C4⋊#11758185C2211yescharacteristic
34#16839580C2212yes—
35#16839567C2212yes—
36#16839565C2212yes—
37#16839563C2212yes—
38C61×#500C2211yescharacteristic
39C4×D8×C61C2211yescharacteristic
40C61×#366C2212yes—
41C61×C4⋊C16C2211yescharacteristic
42C61×#357C2212yes—
43C2×#11758224—21no—
44C2×Q8×D61—41no—
45#14519796—22no—
46#14519796Q814yescomplemented
47C2×Dic244C2311yescharacteristic
48C2×#11758200—21no—
49C2×C8×D61D411yescharacteristic
50C2×C8×D61C2311yescharacteristic
51#14519777—21no—
52#14519776—22no—
53#14519776—41no—
54#14519775—41no—
55C4×Dic122—21no—
56#14519742—21no—
57D4×Dic61—21no—
58C4×C22⋊D61—41no—
59#14519737—41no—
60#14519735—21no—
61#14519733—41no—
62C4⋊C4×D61—22no—
63C4⋊C4×D61—41no—
64C2×C4×Dic61—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#14519702—21no—
75#14519700—21no—
76#14519698—21no—
77C4×#92418D411yescharacteristic
78C8×Dic61D411yescharacteristic
79C8×Dic61C2311yescharacteristic
80C2×D8×C61C2311yescharacteristic
81C4⋊Q8×C61—21no—
82C4×D4×C61—21no—
83C2×C16×C61C2312yes—
84#107×C61—21no—
85#107×C61C2312yes—
86#107×C61C2311yescharacteristic
87#102×C61—21no—
88C4×C8×C61C2311yescharacteristic
89C2×Q8×C61—41no—
90C2×D4×C61—21no—
91C22×C4×C61—41no—
92C2×C22⋊D61—41no—
93C22×Dic61—41no—
94C22×Dic61—41no—
95Q8×D61—84no—
96#11758224—24no—
97#11758224—42no—
98C2×C4×D61—42no—
99C2×C4×D61D811yescharacteristic
100C2×C4×D61D811yescharacteristic
101C2×C4×D61C2×D411yescharacteristic
102C2×Dic122—21no—
103C2×Dic122—41no—
104C2×Dic122—42no—
105D8×C61—21no—
106D8×C61C2×Q812yescomplemented
107C2×C8×C61C2×D411yescharacteristic
108C2×C8×C61C2411yescharacteristic
109C4⋊C4×C61—21no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113C42×C61C2×D411yescharacteristic
114C22⋊Dic61—41no—
115#11758200—24no—
116#11758198—41no—
117C4⋊Dic61—21no—
118C4⋊Dic61—22no—
119C4⋊Dic61—41no—
120#11758196—41no—
121#11758196—42no—
122#11758196—42no—
123#11758196—41no—
124C4×Dic61—41no—
125C4×Dic61—41no—
126C4×Dic61—41no—
127C4×Dic61D811yescharacteristic
128C4×Dic61D811yescharacteristic
129C4×Dic61C2×D411yescharacteristic
130C2×#92418C2×D411yescharacteristic
131C2×#92418C2×D411yescharacteristic
132Dic244—21no—
133Dic244C2×Q812yescomplemented
134C8×D61—22no—
135C8×D61C2×Q812yes—
136C8×D61#4112yes—
137C16×C61—22no—
138#11758185—22no—
139C23×C61—41no—
140C22×D61C2×D811yescharacteristic
141Q8×C61—82no—
142D4×C61—22no—
143D4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—42no—
146C2×C4×C61—82no—
147C2×C4×C61C2×D811yescharacteristic
148C2×C4×C61C2×D811yescharacteristic
149C2×C4×C61C22×D411yescharacteristic
150C22⋊D61—44no—
151C2×Dic61—41no—
152C2×Dic61—44no—
153C2×Dic61—41no—
154C2×Dic61—42no—
155C2×Dic61—82no—
156C2×Dic61C2×D811yescharacteristic
157C2×Dic61C2×D811yescharacteristic
158C2×Dic61C2×D811yescharacteristic
159C4×D61—22no—
160C4×D61—22no—
161C4×D61—84no—
162C4×D61C22⋊Q812yes—
163C4×D61#13712yes—
164Dic122—22no—
165Dic122—41no—
166Dic122—84no—
167Dic122—82no—
168C8×C61—21no—
169C8×C61C22×Q811yescharacteristic
170C8×C61C2×#4111yescharacteristic
171#92418—21no—
172#92418C22⋊Q812yes—
173#33629—611nomaximal, Sylow
174C22×C61—42no—
175C22×C61—81no—
176C22×C61—81no—
177C22×C61C22×D811yescharacteristic
178C2×D61—21no—
179C2×D61—21no—
180C2×D61#47412yes—
181C4×C61—21no—
182C4×C61—82no—
183C4×C61—81no—
184C4×C61#50912yes—
185C4×C61C2×C22⋊Q811yescharacteristic
186C4×C61C2×#13711yescharacteristic
187Dic61—21no—
188Dic61—42no—
189Dic61—82no—
190Dic61—81no—
191Dic61#47412yes—
192Dic61#51012yes—
193C2×#500—611no—
194C4×#135—611no—
195#2227—612no—
196#2226—612no—
197#2225—611no—
198#2225—611no—
199#2224—611no—
200#2224D6111yescharacteristic, complemented
201#2159—612no—
202#2150—611no—
203#2139—612no—
204C2×C61—82no—
205C2×C61C2×#47411yescharacteristic
206C2×C61C2×#50911yescharacteristic
207C2×C61C2×#51011yescharacteristic
208D61—22no—
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×Q16—611no—
220C4×SD16—1221no—
221C4×SD16—1221no—
222C4×D8—611no—
223C4×D8C2×D6111yescharacteristic
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#358—612no—
238#357—612no—
239#357C2×D6112yes—
240C61#3362911yescharacteristic, complemented, Sylow
241C2×#41—1221no—
242C22×Q8—2441no—
243#135—614no—
244#135—1222no—
245C2×Q16—611no—
246C2×SD16—1221no—
247C2×D8C22×D6111yescharacteristic
248C22×C8—611no—
249C22×C8—611no—
250C4⋊Q8—21no—
251C4⋊Q8—1222no—
252C4⋊Q8—1221no—
253C4⋊Q8—2441no—
254C4⋊Q8—2441no—
255C4×Q8—1221no—
256C4×Q8—1221no—
257C4×D4—21no—
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—1221no—
280#103—1221no—
281#102—21no—
282#102—1221no—
283C4×C8—611no—
284C4×C8—611no—
285C4×C8—611no—
286C4×C8C22×D6111yescharacteristic
287#41—1224no—
288#41—2442no—
289C2×Q8—41no—
290C2×Q8—1221no—
291C2×Q8—4884no—
292C2×Q8—2442no—
293C2×Q8—2441no—
294C2×D4—21no—
295C2×D4—2441no—
296C22×C4—41no—
297C22×C4—611no—
298C22×C4—611no—
299C22×C4—611no—
300C22×C4—2442no—
301C22×C4—2441no—
302C22×C4—2441no—
303Q16—612no—
304Q16—1221no—
305SD16—1224no—
306D8—21no—
307D8Q8×D6112yescomplemented
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—1221no—
319C4⋊C4—1222no—
320C4⋊C4—2442no—
321C4⋊C4—2442no—
322C4⋊C4—2441no—
323C4⋊C4—2441no—
324C4⋊C4—2441no—
325C22⋊C4—41no—
326C22⋊C4—2441no—
327C22⋊C4—2441no—
328C42—611no—
329C42—611no—
330C42—611no—
331C42—2441no—
332C42—2441no—
333C42—2441no—
334C42D4×D6111yescharacteristic
335C16—22no—
336C16—1222no—
337C23—41no—
338C23—611no—
339Q8—82no—
340Q8—1222no—
341Q8—4884no—
342Q8—4882no—
343Q8—2441no—
344D4—22no—
345D4—41no—
346D4—2444no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—82no—
350C2×C4—611no—
351C2×C4—611no—
352C2×C4—612no—
353C2×C4—612no—
354C2×C4—611no—
355C2×C4—1222no—
356C2×C4—1222no—
357C2×C4—4884no—
358C2×C4—4882no—
359C2×C4—2444no—
360C2×C4—2441no—
361C2×C4—2442no—
362C2×C4—2441no—
363C2×C4D8×D6111yescharacteristic
364C2×C4D8×D6111yescharacteristic
365C2×C4C2×D4×D6111yescharacteristic
366C8—21no—
367C8—612no—
368C8—1221no—
369C8C2×Q8×D6111yescharacteristic
370C8#41×D6111yescharacteristic
371C22—42no—
372C22—81no—
373C22—81no—
374C22—612no—
375C22—1221no—
376C22—1221no—
377C22C2×D8×D6111yescharacteristic
378C4—21no—
379C4—81no—
380C4—82no—
381C4—612no—
382C4—612no—
383C4—1221no—
384C4—4881no—
385C4—4882no—
386C4—2442no—
387C4#1684019612yes—
388C4C22⋊Q8×D6111yescharacteristic
389C4#137×D6111yescharacteristic
390C2—82no—
391C2—1222no—
392C2#474×D6111yescharacteristic
393C2C2×#1684019611yescharacteristic
394C2#510×D6111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2224C2
socleC22×C61C22×D8
prime radicalC2×#14519796C22
hypercenter#2224D61
hypocenterC61#33629
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2224

Complements

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

NH ≅ G/Ntype
1. #500×D61C2semidirect
3. #21447970C2semidirect
4. #21447964C2semidirect
5. #21447962C2semidirect
6. #21447958C2semidirect
7. #21447956C2semidirect
8. #21447604C2semidirect
9. #21447573C2semidirect
11. C61×#2224C2semidirect
46. #14519796Q8semidirect
106. D8×C61C2×Q8semidirect
133. Dic244C2×Q8semidirect
200. #2224D61semidirect
240. C61#33629semidirect
307. D8Q8×D61semidirect

Maximal subgroups

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

subgroupindex
#500×D612
C4×#145197962
#214479702
#214479642
#214479622
#214479582
#214479562
#214476042
#214475732
#214475282
C61×#22242
#3362961