Finite Group Databasebeta

← #29665380

Subgroups — #29665380

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456955C211yesmaximal, characteristic
2#21455941C211yesmaximal, characteristic
3#21454900C211yesmaximal, characteristic, complemented
4#21445265C212yesmaximal, complemented
5#21445237C212yesmaximal, complemented
6#21445221C212yesmaximal, complemented
7#21445199C212yesmaximal, complemented
8#21444597C211yesmaximal, characteristic, complemented
9#21444529C211yesmaximal, characteristic, complemented
10#21444479C211yesmaximal, characteristic
11C61×#3104C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13#16840165C2211yescharacteristic
14#16840158C2211yescharacteristic
15#16840110C2211yescharacteristic
16#16840110C2211yescharacteristic
17#16840086—22no—
18#16840075—21no—
19#16840059—22no—
20#16840057C2212yes—
21#16839962C2212yes—
22#16839901—21no—
23Q8×#92418—22no—
24Q8×#92418C2212yes—
25Q8×#92418C2212yes—
26#16839860—22no—
27#16839819C2211yescharacteristic
28#16839815—22no—
29#16839815C2211yescharacteristic
30#16839815C2211yescharacteristic
31#16839514C2212yes—
32#16839512C2212yes—
33#16839511C2212yes—
34#16839510C2212yes—
35#16839508C2212yes—
36#16839505C2212yes—
37C61×#533C2211yescharacteristic
38C2×C4×Q8×C61C2211yescharacteristic
39C61×#484—21no—
40C61×#477C2212yes—
41C61×#462C2212yes—
42C61×#451—21no—
43C4×Q16×C61—22no—
44C61×#431C2211yescharacteristic
45C2×#11758222—21no—
46C2×#11758201—22no—
47#14519777C2311yescharacteristic
48#14519774C2311yescharacteristic
49#14519765—21no—
50#14519751—21no—
51#14519751—21no—
52C4×D244—21no—
53C4×D244—21no—
54#14519710—22no—
55#14519710—22no—
56#14519708—22no—
57C4⋊#92418—22no—
58C4⋊#92418—22no—
59C4⋊#92418D412yes—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519701—22no—
63#14519697—22no—
64#14519697—22no—
65#14519697—22no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C22⋊#92418D412yes—
69#14519688—21no—
70C4×#92418—21no—
71C4×#92418—22no—
72C4×#92418—22no—
73C4×#92418C2311yescharacteristic
74C4×#92418C2311yescharacteristic
75C22×Q8×C61D411yescharacteristic
76C2×Q16×C61—22no—
77C4⋊Q8×C61C2311yescharacteristic
78C4⋊Q8×C61C2311yescharacteristic
79#125×C61—21no—
80C22⋊Q8×C61—21no—
81C22⋊Q8×C61—21no—
82C4×Q8×C61—21no—
83C4×Q8×C61—21no—
84C4×Q8×C61—21no—
85C4×Q8×C61—21no—
86C4×Q8×C61D412yes—
87C4×Q8×C61C2312yes—
88#117×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61D411yescharacteristic
91C2×C42×C61—21no—
92C2×C42×C61C2311yescharacteristic
93#107×C61—22no—
94C4⋊C8×C61C2312yes—
95#103×C61—22no—
96#103×C61—22no—
97C22⋊C8×C61—21no—
98C4×C8×C61—21no—
99C2×Q8×C61—24no—
100C2×Q8×C61—21no—
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61C2×D412yes—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61#4111yescharacteristic
109C2×C22⋊D61—41no—
110#11758222—82no—
111C2×D244—21no—
112C2×D244—21no—
113C2×D244—21no—
114C2×C4×D61—41no—
115C2×Dic122—21no—
116Q16×C61—44no—
117C2×C8×C61—22no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—21no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—41no—
126C4⋊C4×C61—41no—
127C4⋊C4×C61C2×D412yes—
128C22⋊C4×C61—41no—
129C42×C61—22no—
130C42×C61—22no—
131C42×C61—41no—
132C42×C61C2×D411yescharacteristic
133C42×C61C2×D411yescharacteristic
134C42×C61#4111yescharacteristic
135C42×C61C2411yescharacteristic
136#11758201—44no—
137#11758198—41no—
138#11758198—41no—
139C4⋊Dic61—21no—
140C4⋊Dic61—21no—
141C2×#92418—22no—
142C2×#92418—22no—
143C2×#92418C2×D412yes—
144C2×#92418C2×D412yes—
145C23×C61#12111yescharacteristic
146C22×D61—41no—
147Q8×C61—24no—
148Q8×C61—22no—
149Q8×C61—42no—
150Q8×C61—81no—
151Q8×C61#13514yes—
152C2×C4×C61—24no—
153C2×C4×C61—22no—
154C2×C4×C61—22no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—81no—
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C22×D411yescharacteristic
166C2×C4×C61C2×#4111yescharacteristic
167C22⋊D61—82no—
168C2×Dic61—41no—
169D244—42no—
170D244—42no—
171D244—81no—
172C4×D61—82no—
173Dic122—81no—
174C8×C61—42no—
175#92418—22no—
176#92418—22no—
177#92418—42no—
178#92418—42no—
179#19406—611nomaximal, Sylow
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61C2×#12111yescharacteristic
184C2×D61—81no—
185C2×D61—81no—
186C2×D61—81no—
187C4×C61—21no—
188C4×C61—24no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—42no—
193C4×C61—81no—
194C4×C61—81no—
195C4×C61C2×#13512yes—
196C4×C61#57812yes—
197Dic61—81no—
198#3228—611no—
199#3112—611no—
200#3104D6111yescharacteristic, complemented
201#2982—611no—
202#1682—612no—
203#1678—612no—
204#1676—612no—
205#1672—612no—
206#1572—611no—
207#1554—611no—
208#1543—611no—
209C2×C61—41no—
210C2×C61#304411yescharacteristic
211C2×C61#305011yescharacteristic
212C2×C61#306811yescharacteristic
213D61—161no—
214#533C2×D6111yescharacteristic
215#532—611no—
216C2×C4×Q8C2×D6111yescharacteristic
217#500—611no—
218#498—611no—
219#486—611no—
220#486—611no—
221#484—21no—
222#482—1222no—
223#480—1221no—
224#478—1222no—
225#477—612no—
226#477C2×D6112yes—
227#462C2×D6112yes—
228#460—612no—
229#451—21no—
230#448—1221no—
231C8×Q8—612no—
232C8×Q8—612no—
233C8×Q8—1222no—
234#440—1222no—
235C4×Q16—22no—
236#432—611no—
237#431—611no—
238#431—611no—
239#431—1222no—
240#431C2×D6111yescharacteristic
241#335—612no—
242#334—612no—
243#333—612no—
244#333—612no—
245#332—612no—
246#331—612no—
247C61#1940611yescharacteristic, complemented, Sylow
248C2×#41—1221no—
249C22×Q8C22⋊D6111yescharacteristic
250C2×Q16—22no—
251C2×SD16—1222no—
252C4⋊Q8—611no—
253C4⋊Q8C22×D6111yescharacteristic
254C4⋊Q8C22×D6111yescharacteristic
255#127—611no—
256#125—21no—
257#124—1221no—
258C22⋊Q8—21no—
259C22⋊Q8—21no—
260#121—1221no—
261#121—1221no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8C22⋊D6112yes—
267C4×Q8C22×D6112yes—
268C4×D4—1221no—
269C4×D4—1221no—
270#117—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4C22⋊D6111yescharacteristic
273C2×C42—21no—
274C2×C42C22×D6111yescharacteristic
275#107—22no—
276#107—1222no—
277#107—1222no—
278#106—1222no—
279C4⋊C8—612no—
280C4⋊C8—612no—
281C4⋊C8—612no—
282C4⋊C8—1222no—
283C4⋊C8—1222no—
284C4⋊C8C22×D6112yes—
285#103—22no—
286#103—22no—
287#103—1222no—
288#102—1222no—
289#102—1222no—
290#102—1222no—
291C22⋊C8—21no—
292C22⋊C8—612no—
293C22⋊C8—1221no—
294C22⋊C8—1221no—
295#97—1221no—
296C4×C8—21no—
297C4×C8—611no—
298C4×C8—611no—
299C4×C8—1222no—
300C4×C8—1222no—
301C4×C8—1221no—
302#41—4882no—
303C2×Q8—24no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—1221no—
308C2×Q8C2×C22⋊D6112yes—
309C2×D4—1221no—
310C2×D4—1221no—
311C2×D4—1221no—
312C2×D4—2441no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—2441no—
316C22×C4D4×D6111yescharacteristic
317C22×C4#1175822411yescharacteristic
318C22×C4C2×C22⋊D6111yescharacteristic
319Q16—44no—
320SD16—2444no—
321C2×C8—22no—
322C2×C8—612no—
323C2×C8—612no—
324C2×C8—1222no—
325C2×C8—1222no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—1221no—
336C4⋊C4—1221no—
337C4⋊C4C2×C22⋊D6112yes—
338C22⋊C4—41no—
339C22⋊C4—2441no—
340C22⋊C4—2441no—
341C42—22no—
342C42—22no—
343C42—41no—
344C42D4×D6111yescharacteristic
345C42#1175822411yescharacteristic
346C42C2×C22⋊D6111yescharacteristic
347C42C23×D6111yescharacteristic
348C23—2441no—
349C23#1451974911yescharacteristic
350Q8—24no—
351Q8—22no—
352Q8—42no—
353Q8—81no—
354Q8—4881no—
355Q8#1451979714yes—
356D4—4881no—
357D4—4882no—
358D4—2442no—
359D4—2442no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—81no—
369C2×C4—4882no—
370C2×C4—2441no—
371C2×C4#1451974911yescharacteristic
372C2×C4#1451974911yescharacteristic
373C2×C4#1451974911yescharacteristic
374C2×C4C2×D4×D6111yescharacteristic
375C2×C4C22×C22⋊D6111yescharacteristic
376C2×C4C2×#1175822411yescharacteristic
377C8—42no—
378C8—1222no—
379C8—1222no—
380C8—2442no—
381C8—2442no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—4881no—
386C22—4881no—
387C22—4881no—
388C22C2×#1451974911yescharacteristic
389C4—24no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—81no—
396C4—81no—
397C4—4881no—
398C4C2×#1451979712yes—
399C4#1684057512yes—
400C2—41no—
401C2—9761no—
402C2#2145534611yescharacteristic
403C2#2145539711yescharacteristic
404C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3104C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#3104D61
hypocenterC61#19406
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
3. #21454900C2semidirect
4. #21445265C2semidirect
5. #21445237C2semidirect
6. #21445221C2semidirect
7. #21445199C2semidirect
8. #21444597C2semidirect
9. #21444529C2semidirect
11. C61×#3104C2semidirect
200. #3104D61semidirect
247. C61#19406semidirect

Maximal subgroups

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

subgroupindex
#214569552
#214559412
#214549002
#214452652
#214452372
#214452212
#214451992
#214445972
#214445292
#214444792
C61×#31042
#1940661