Finite Group Databasebeta

← #29665325

Subgroups — #29665325

Order 15616.

610 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456777C211yesmaximal, characteristic
2#21455949C211yesmaximal, characteristic, complemented
3#21455850C211yesmaximal, characteristic
4#21445263C212yesmaximal, complemented
5#21445249C212yesmaximal, complemented
6#21445189C212yesmaximal, complemented
7#21445167C212yesmaximal, complemented
8#21444599C211yesmaximal, characteristic, complemented
9#21444522C211yesmaximal, characteristic
10#21444483C211yesmaximal, characteristic, complemented
11C61×#2982C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13#16840312C2211yescharacteristic
14#16840142C2211yescharacteristic
15#16840135C2211yescharacteristic
16#16840113C2211yescharacteristic
17#16840109C2211yescharacteristic
18#16840092—21no—
19#16840080—21no—
20#16840047C2212yes—
21#16840047C2212yes—
22#16839985—22no—
23#16839972C2212yes—
24#16839962C2212yes—
25#16839926—22no—
26#16839918—21no—
27#16839901—21no—
28#16839867—22no—
29C4×#11758201—22no—
30#16839819C2211yescharacteristic
31#16839815C2211yescharacteristic
32#16839515C2212yes—
33#16839513C2212yes—
34#16839509C2212yes—
35#16839509C2212yes—
36#16839507C2212yes—
37#16839505C2212yes—
38C2×C4×Q8×C61C2211yescharacteristic
39C8×Q8×C61—22no—
40C8×Q8×C61C2212yes—
41C8×Q8×C61C2212yes—
42C61×#431—22no—
43C61×#431C2211yescharacteristic
44C61×#431C2211yescharacteristic
45C2×#11758222—21no—
46C2×Dic244—22no—
47C2×#11758202—22no—
48C2×#11758190—22no—
49C2×#11758201—22no—
50#14519777C2311yescharacteristic
51#14519777C2311yescharacteristic
52#14519777C2311yescharacteristic
53#14519774C2311yescharacteristic
54#14519769—21no—
55#14519765—21no—
56#14519756—21no—
57#14519756—21no—
58#14519751—21no—
59#14519751—21no—
60C4×Dic122—21no—
61C4×Dic122—21no—
62C4×D244—21no—
63C4×D244—21no—
64#14519736—21no—
65#14519708—22no—
66#14519708—22no—
67C4⋊#92418C2312yes—
68C4⋊#92418C2312yes—
69#14519702—22no—
70#14519701—22no—
71#14519701—22no—
72#14519700—22no—
73#14519699—22no—
74#14519697—22no—
75C22⋊#92418—21no—
76C22⋊#92418—21no—
77#14519688—21no—
78C4×#92418—21no—
79C22×Q8×C61D411yescharacteristic
80C4×Q8×C61—21no—
81C4×Q8×C61—21no—
82C4×Q8×C61D412yes—
83C4×Q8×C61C2312yes—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61D411yescharacteristic
86C2×C42×C61—21no—
87C2×C42×C61C2311yescharacteristic
88C4⋊C8×C61—22no—
89C4⋊C8×C61—22no—
90C4⋊C8×C61D412yes—
91C4⋊C8×C61C2312yes—
92C22⋊C8×C61—21no—
93C22⋊C8×C61D412yes—
94C4×C8×C61—22no—
95C4×C8×C61—22no—
96C4×C8×C61—21no—
97C4×C8×C61C2311yescharacteristic
98C4×C8×C61C2311yescharacteristic
99C2×Q8×C61—24no—
100C2×Q8×C61—21no—
101C2×Q8×C61C2×D412yes—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61#4111yescharacteristic
107C2×C22⋊D61—41no—
108#11758222—82no—
109C2×D244—21no—
110C2×D244—21no—
111C2×D244—21no—
112C2×C4×D61—41no—
113C2×Dic122—21no—
114C2×Dic122—21no—
115C2×Dic122—21no—
116C2×C8×C61—22no—
117C2×C8×C61C2×D412yes—
118C2×C8×C61C2×D412yes—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—21no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61C2×D412yes—
123C42×C61—22no—
124C42×C61—22no—
125C42×C61C2×D411yescharacteristic
126C42×C61C2×D411yescharacteristic
127C42×C61#4111yescharacteristic
128C42×C61C2411yescharacteristic
129C22⋊Dic61—41no—
130#11758202—44no—
131#11758201—44no—
132#11758198—41no—
133#11758198—41no—
134C4⋊Dic61—21no—
135C4⋊Dic61—21no—
136C4⋊Dic61—21no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139C4⋊Dic61—21no—
140#11758196—41no—
141#11758196—41no—
142C4×Dic61—41no—
143C2×#92418—22no—
144C2×#92418—22no—
145Dic244—82no—
146#11758190—82no—
147C23×C61#12111yescharacteristic
148C22×D61—41no—
149Q8×C61—24no—
150Q8×C61—22no—
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—81no—
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C2×#4111yescharacteristic
165C22⋊D61—82no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169D244—42no—
170D244—42no—
171D244—81no—
172C4×D61—82no—
173Dic122—42no—
174Dic122—42no—
175Dic122—42no—
176C8×C61—22no—
177C8×C61—22no—
178C8×C61—42no—
179#92418—42no—
180#92418—42no—
181#19400—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61C2×#12111yescharacteristic
186C2×D61—81no—
187C2×D61—81no—
188C2×D61—81no—
189C4×C61—21no—
190C4×C61—24no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—42no—
195C4×C61C2×#13512yes—
196C4×C61#57712yes—
197Dic61—81no—
198Dic61—81no—
199Dic61—81no—
200#3210—611no—
201#3113—611no—
202#3103—611no—
203#2982D6111yescharacteristic, complemented
204#1682—612no—
205#1680—612no—
206#1670—612no—
207#1666—612no—
208#1572—611no—
209#1553—611no—
210#1544—611no—
211C2×C61—41no—
212C2×C61#304311yescharacteristic
213C2×C61#305011yescharacteristic
214C2×C61#306911yescharacteristic
215D61—161no—
216#533—611no—
217#532—611no—
218C2×C4×Q8C2×D6111yescharacteristic
219#494—611no—
220#492—611no—
221#487—611no—
222#486—611no—
223#483—1221no—
224#481—1221no—
225#475—612no—
226#475—612no—
227#464—1222no—
228#462—612no—
229#460—612no—
230#452—1222no—
231#451—1221no—
232#448—1221no—
233C8×Q8—22no—
234C8×Q8C2×D6112yes—
235C8×Q8C2×D6112yes—
236#441—1222no—
237C4×SD16—1222no—
238#432—611no—
239#431—22no—
240#431—611no—
241#431C2×D6111yescharacteristic
242#431C2×D6111yescharacteristic
243#335—612no—
244#334—612no—
245#332—612no—
246#332—612no—
247#332—612no—
248#331—612no—
249C61#1940011yescharacteristic, complemented, Sylow
250C2×#41—1221no—
251C22×Q8D24411yescharacteristic
252C2×Q16—1222no—
253C2×Q16—1222no—
254C2×SD16—1222no—
255C2×SD16—1222no—
256C4⋊Q8—611no—
257C4⋊Q8—611no—
258C4⋊Q8—611no—
259#127—611no—
260#125—1221no—
261#124—1221no—
262C22⋊Q8—1221no—
263C22⋊Q8—1221no—
264#121—1221no—
265#121—1221no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—1221no—
269C4×Q8—1221no—
270C4×Q8D24412yes—
271C4×Q8C22×D6112yes—
272C4×D4—1221no—
273C4×D4—1221no—
274#117—1221no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4D24411yescharacteristic
277C2×C42—21no—
278C2×C42C22×D6111yescharacteristic
279#106—1222no—
280#106—1222no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8—612no—
284C4⋊C8—612no—
285C4⋊C8C22⋊D6112yes—
286C4⋊C8C22×D6112yes—
287#103—1222no—
288#103—1222no—
289#103—1222no—
290#103—1222no—
291#102—1222no—
292#102—1222no—
293C22⋊C8—21no—
294C22⋊C8—1221no—
295C22⋊C8—1221no—
296C22⋊C8C22⋊D6112yes—
297#97—1221no—
298C4×C8—22no—
299C4×C8—22no—
300C4×C8—21no—
301C4×C8—1221no—
302C4×C8C22×D6111yescharacteristic
303C4×C8C22×D6111yescharacteristic
304#41—4882no—
305C2×Q8—24no—
306C2×Q8—21no—
307C2×Q8—1221no—
308C2×Q8—1221no—
309C2×Q8—1221no—
310C2×Q8C2×D24412yes—
311C2×D4—1221no—
312C2×D4—1221no—
313C2×D4—1221no—
314C2×D4—2441no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—2441no—
318C22×C4C2×D24411yescharacteristic
319C22×C4#1175822211yescharacteristic
320C22×C4C2×C22⋊D6111yescharacteristic
321Q16—4882no—
322Q16—2444no—
323SD16—4882no—
324SD16—2444no—
325C2×C8—22no—
326C2×C8—1222no—
327C2×C8—1222no—
328C2×C8C2×C22⋊D6112yes—
329C2×C8C2×C22⋊D6112yes—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—1221no—
334C4⋊C4—1221no—
335C4⋊C4—1221no—
336C4⋊C4—1221no—
337C4⋊C4—1221no—
338C4⋊C4—1221no—
339C4⋊C4—2441no—
340C4⋊C4—2441no—
341C4⋊C4C2×D24412yes—
342C22⋊C4—2441no—
343C22⋊C4—2441no—
344C22⋊C4—2441no—
345C42—22no—
346C42—22no—
347C42—2441no—
348C42C2×D24411yescharacteristic
349C42#1175822211yescharacteristic
350C42C2×C22⋊D6111yescharacteristic
351C42C23×D6111yescharacteristic
352C23—2441no—
353C23#1451975111yescharacteristic
354Q8—24no—
355Q8—22no—
356Q8—2442no—
357Q8—2442no—
358Q8—2442no—
359Q8#1451979914yes—
360D4—4881no—
361D4—4882no—
362D4—2442no—
363D4—2442no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—4882no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4#1451975111yescharacteristic
376C2×C4#1451975111yescharacteristic
377C2×C4#1451975111yescharacteristic
378C2×C4C22×C22⋊D6111yescharacteristic
379C2×C4C22×D24411yescharacteristic
380C2×C4C2×#1175822211yescharacteristic
381C8—22no—
382C8—22no—
383C8—42no—
384C8—2442no—
385C8—2442no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—4881no—
390C22—4881no—
391C22—4881no—
392C22C2×#1451975111yescharacteristic
393C4—24no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—4881no—
400C4—4881no—
401C4—4881no—
402C4C2×#1451979912yes—
403C4#1684057012yes—
404C2—41no—
405C2—9761no—
406C2#2145534011yescharacteristic
407C2#2145540011yescharacteristic
408C2#2145555611yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2982C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#2982D61
hypocenterC61#19400
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #21455949C2semidirect
4. #21445263C2semidirect
5. #21445249C2semidirect
6. #21445189C2semidirect
7. #21445167C2semidirect
8. #21444599C2semidirect
10. #21444483C2semidirect
11. C61×#2982C2semidirect
203. #2982D61semidirect
249. C61#19400semidirect

Maximal subgroups

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

subgroupindex
#214567772
#214559492
#214558502
#214452632
#214452492
#214451892
#214451672
#214445992
#214445222
#214444832
C61×#29822
#1940061