Finite Group Databasebeta

← #29836377

Subgroups — #29836377

Order 15616.

528 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456402C211yesmaximal, characteristic, complemented
2#21456399C211yesmaximal, characteristic
3#21447968C211yesmaximal, characteristic, complemented
4#21447955C212yesmaximal, complemented
5#21447935C212yesmaximal, complemented
6#21447933C212yesmaximal, complemented
7#21447863C212yesmaximal, complemented
8#21447860C211yesmaximal, characteristic, complemented
9#21447859C211yesmaximal, characteristic, complemented
10#21447573C211yesmaximal, characteristic, complemented
11C61×#2227C211yesmaximal, characteristic, complemented
12#16840316—21no—
13#16840313—21no—
14C2×#14519722C2212yes—
15C2×#14519720C2212yes—
16C2×#14519716C2212yes—
17#16840166C2211yescharacteristic
18#16840140C2211yescharacteristic
19#16840139C2211yescharacteristic
20#16840111—21no—
21#16840110—21no—
22#16840004C2211yescharacteristic
23#16840002—21no—
24#16840000C2211yescharacteristic
25#16839999—21no—
26#16839999C2212yes—
27C4×Dic244C2211yescharacteristic
28C4×#11758202—21no—
29#16839844—21no—
30#16839844C2212yes—
31C4×C8×D61C2211yescharacteristic
32#16839587C2212yes—
33C4⋊#11758185C2211yescharacteristic
34#16839580C2212yes—
35#16839565C2212yes—
36#16839564C2212yes—
37#16839562C2212yes—
38C61×#500C2211yescharacteristic
39C4×Q16×C61C2211yescharacteristic
40C61×#366C2212yes—
41C61×C4⋊C16C2211yescharacteristic
42C61×#358C2212yes—
43C2×#11758226—41no—
44C2×Dic244C2311yescharacteristic
45C2×#11758202—21no—
46C2×D488—21no—
47C2×D488C2312yes—
48C2×C8×D61D411yescharacteristic
49C2×C8×D61C2311yescharacteristic
50#14519776—21no—
51#14519775—21no—
52#14519774—21no—
53#14519767—41no—
54#14519766—41no—
55#14519757—21no—
56#14519755—41no—
57#14519753—21no—
58#14519752—22no—
59#14519752—41no—
60C4×Dic122—21no—
61C4×Dic122—41no—
62#14519739—22no—
63#14519739—41no—
64#14519734—41no—
65C42×D61D411yescharacteristic
66#14519722—42no—
67#14519720—42no—
68#14519716—42no—
69C2×#11758185C2312yes—
70#14519711C2311yescharacteristic
71#14519710—21no—
72#14519702—21no—
73#14519701—21no—
74#14519700—21no—
75#14519697—22no—
76#14519697—22no—
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—84no—
92C2×D244—22no—
93C2×D244—42no—
94C2×D244—41no—
95C2×D244—41no—
96C2×C4×D61—42no—
97C2×C4×D61D811yescharacteristic
98C2×C4×D61D811yescharacteristic
99C2×C4×D61C2×D411yescharacteristic
100C2×Dic122—21no—
101C2×Dic122—41no—
102Q16×C61—21no—
103Q16×C61—21no—
104C2×C8×C61C2×D411yescharacteristic
105C2×C8×C61C2411yescharacteristic
106C4⋊C4×C61—21no—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—41no—
109C4⋊C4×C61—41no—
110C42×C61—41no—
111C42×C61C2×D411yescharacteristic
112#11758202—42no—
113#11758198—41no—
114#11758198—41no—
115#11758198—42no—
116#11758198—42no—
117C4⋊Dic61—21no—
118C4⋊Dic61—41no—
119#11758196—41no—
120#11758196—41no—
121#11758196—41no—
122#11758196—41no—
123#11758196—41no—
124C4×Dic61—41no—
125C4×Dic61D811yescharacteristic
126C4×Dic61D811yescharacteristic
127C4×Dic61C2×D411yescharacteristic
128C2×#92418C2×D411yescharacteristic
129C2×#92418C2×D411yescharacteristic
130Dic244—21no—
131Dic244—21no—
132D488—22no—
133D488—22no—
134D488—42no—
135C8×D61—21no—
136C8×D61—21no—
137C8×D61—41no—
138C16×C61—22no—
139#11758185—22no—
140C22×D61—42no—
141C22×D61C2×D811yescharacteristic
142Q8×C61—41no—
143Q8×C61—41no—
144Q8×C61—82no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—42no—
148C2×C4×C61C2×D811yescharacteristic
149C2×C4×C61C2×D811yescharacteristic
150C2×C4×C61C22×D411yescharacteristic
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61C2×D811yescharacteristic
154C2×Dic61C2×D811yescharacteristic
155C2×Dic61C2×D811yescharacteristic
156D244—42no—
157D244—42no—
158D244—82no—
159D244—84no—
160D244—82no—
161C4×D61—21no—
162C4×D61—21no—
163C4×D61—22no—
164C4×D61—22no—
165C4×D61—84no—
166Dic122—41no—
167Dic122—41no—
168Dic122—82no—
169C8×C61—21no—
170C8×C61H(5,𝔽2)11yescharacteristic
171C8×C612−1+411yescharacteristic
172#92418—21no—
173#92418—21no—
174#33820—611nomaximal, Sylow
175C22×C61C22×D811yescharacteristic
176C2×D61—21no—
177C2×D61—21no—
178C2×D61—21no—
179C2×D61—82no—
180C2×D61—82no—
181C2×D61—82no—
182C4×C61—21no—
183C4×C61—81no—
184C4×C61—82no—
185C4×C61—81no—
186C4×C61#51012yes—
187C4×C61#53711yescharacteristic
188C4×C61#57511yescharacteristic
189Dic61—21no—
190Dic61—21no—
191Dic61—81no—
192Dic61—81no—
193Dic61#50912yes—
194#3170—611no—
195#3169—611no—
196#2227—611no—
197#2227D6111yescharacteristic, complemented
198#2224—612no—
199#2220—612no—
200#2220—612no—
201#2208—612no—
202#2207—611no—
203#2207—611no—
204#2150—611no—
205C2×C61#324511yescharacteristic
206C2×C61C2×#50911yescharacteristic
207C2×C61C2×#51011yescharacteristic
208D61—41no—
209D61—162no—
210#533—1221no—
211#532—1221no—
212C2×SD32—612no—
213C2×SD32—612no—
214C2×M32—612no—
215#500—611no—
216#500C2×D6111yescharacteristic
217#494—611no—
218#493—611no—
219#487—1221no—
220#486—1221no—
221#467—611no—
222#467—611no—
223#467—1221no—
224#466—612no—
225#466—1221no—
226C4×Q16—611no—
227C4×Q16—1221no—
228C4×Q16C2×D6111yescharacteristic
229C4×D8—612no—
230C4×D8—1221no—
231C2×C4×C8—611no—
232#366—612no—
233#366C2×D6112yes—
234C4⋊C16—612no—
235C4⋊C16—611no—
236C4⋊C16C2×D6111yescharacteristic
237#358—612no—
238#358C2×D6112yes—
239#357—612no—
240#357—612no—
241C61#3382011yescharacteristic, complemented, Sylow
242C2×#41—2441no—
243C2×Q16—611no—
244C2×Q16—1221no—
245C2×Q16C22×D6111yescharacteristic
246C2×D8—612no—
247C2×D8—1221no—
248C22×C8—611no—
249C22×C8—611no—
250C4⋊Q8—21no—
251C4⋊Q8—1221no—
252C4⋊Q8—1221no—
253#127—1221no—
254#125—2441no—
255#124—2441no—
256C22⋊Q8—1221no—
257C22⋊Q8—1221no—
258C22⋊Q8—2441no—
259#121—1222no—
260#121—2441no—
261C4×Q8—21no—
262C4×Q8—1221no—
263C4×Q8—2441no—
264C4×D4—1222no—
265C4×D4—2441no—
266#117—2441no—
267C2×C42—611no—
268SD32—2442no—
269SD32—2442no—
270M32—2442no—
271C2×C16—612no—
272C2×C16C22×D6112yes—
273#107—21no—
274#107—611no—
275#107—1221no—
276#107C22×D6111yescharacteristic
277#107C22×D6112yes—
278#103—21no—
279#103—1221no—
280#103—1221no—
281#103—1221no—
282#102—1222no—
283#102—1222no—
284C4×C8—611no—
285C4×C8—611no—
286C4×C8—611no—
287C4×C8C22×D6111yescharacteristic
288#41—4884no—
289C2×Q8—21no—
290C2×Q8—41no—
291C2×Q8—1221no—
292C2×Q8—2441no—
293C2×D4—1222no—
294C2×D4—2442no—
295C2×D4—2441no—
296C2×D4—2441no—
297C22×C4—611no—
298C22×C4—611no—
299C22×C4—611no—
300C22×C4—2442no—
301Q16—21no—
302Q16—21no—
303Q16—1221no—
304Q16—1221no—
305Q16—2442no—
306D8—1222no—
307D8—1222no—
308D8—2442no—
309C2×C8—611no—
310C2×C8—611no—
311C2×C8—1221no—
312C2×C8—1221no—
313C2×C8—2441no—
314C2×C8D4×D6111yescharacteristic
315C2×C8C23×D6111yescharacteristic
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—1221no—
321C4⋊C4—2441no—
322C4⋊C4—2441no—
323C4⋊C4—2441no—
324C4⋊C4—2441no—
325C4⋊C4—2441no—
326C4⋊C4—2441no—
327C22⋊C4—2442no—
328C22⋊C4—2442no—
329C22⋊C4—2441no—
330C22⋊C4—2441no—
331C42—41no—
332C42—611no—
333C42—611no—
334C42—611no—
335C42—2441no—
336C42D4×D6111yescharacteristic
337C16—22no—
338C16—1222no—
339C23—611no—
340C23—2442no—
341Q8—41no—
342Q8—41no—
343Q8—82no—
344Q8—4882no—
345Q8—2441no—
346Q8—2441no—
347D4—4882no—
348D4—4884no—
349D4—4882no—
350D4—2442no—
351D4—2442no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—611no—
356C2×C4—611no—
357C2×C4—611no—
358C2×C4—1222no—
359C2×C4—1222no—
360C2×C4—1221no—
361C2×C4—1221no—
362C2×C4—4884no—
363C2×C4—2441no—
364C2×C4—2441no—
365C2×C4D8×D6111yescharacteristic
366C2×C4D8×D6111yescharacteristic
367C2×C4C2×D4×D6111yescharacteristic
368C8—21no—
369C8—1221no—
370C8—1221no—
371C8#1451982311yescharacteristic
372C8#1451982511yescharacteristic
373C22—1221no—
374C22—1221no—
375C22—1221no—
376C22—4882no—
377C22—4882no—
378C22—4882no—
379C22C2×D8×D6111yescharacteristic
380C4—21no—
381C4—81no—
382C4—82no—
383C4—81no—
384C4—612no—
385C4—1221no—
386C4—1221no—
387C4—4881no—
388C4—4881no—
389C4#1684020412yes—
390C4#1684034211yescharacteristic
391C4#1684056111yescharacteristic
392C2—9762no—
393C2—2441no—
394C2#2145712811yescharacteristic
395C2#509×D6111yescharacteristic
396C2C2×#1684020411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2227C2
socleC22×C61C22×D8
prime radical#21456399C2
hypercenter#2227D61
hypocenterC61#33820
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. #21456402C2semidirect
3. #21447968C2semidirect
4. #21447955C2semidirect
5. #21447935C2semidirect
6. #21447933C2semidirect
7. #21447863C2semidirect
8. #21447860C2semidirect
9. #21447859C2semidirect
10. #21447573C2semidirect
11. C61×#2227C2semidirect
197. #2227D61semidirect
241. C61#33820semidirect

Maximal subgroups

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

subgroupindex
#214564022
#214563992
#214479682
#214479552
#214479352
#214479332
#214478632
#214478602
#214478592
#214475732
C61×#22272
#3382061