Finite Group Databasebeta

← #29573256

Subgroups — #29573256

Order 15616.

430 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454530C211yesmaximal, characteristic
2#21445990C211yesmaximal, characteristic, complemented
3C2×#16839512C211yesmaximal, characteristic, complemented
4#21444497C211yesmaximal, characteristic
5#21444494C211yesmaximal, characteristic
6#21443693C211yesmaximal, characteristic, complemented
7C61×#1292C211yesmaximal, characteristic, complemented
8C2×#14519785—21no—
9#16840312C2211yescharacteristic
10#16839839—21no—
11#16839839C412yes—
12#16839827—21no—
13#16839827C412yes—
14C2×#14519712—21no—
15C2×#14519712—21no—
16#16839786C2211yescharacteristic
17C2×C4⋊#92418C2211yescharacteristic
18#16839730—21no—
19C2×C4×#92418C2211yescharacteristic
20#16839512—21no—
21#16839512—21no—
22#16839509—21no—
23#16839507—21no—
24#16839505—21no—
25#16839505—21no—
26C61×#423C2211yescharacteristic
27C61×#423C2211yescharacteristic
28C2×C4⋊C8×C61C2211yescharacteristic
29C2×#11758222—21no—
30#14519785—42no—
31#14519785—42no—
32C2×#11758194—21no—
33C2×#11758189—22no—
34C22×#92418—21no—
35C22×#92418D411yescharacteristic
36C22×#92418D411yescharacteristic
37C2×C8×D61—22no—
38#14519777D411yescharacteristic
39#14519774D411yescharacteristic
40#14519765—21no—
41#14519751—21no—
42#14519751Q812yes—
43C4×D244—21no—
44C4×D244C2×C412yes—
45#14519712—42no—
46#14519712—42no—
47C4⋊#92418—21no—
48C4⋊#92418—21no—
49C4⋊#92418—21no—
50C4⋊#92418—21no—
51#14519689—22no—
52#14519689—22no—
53C4×#92418C2×C411yescharacteristic
54C4×#92418C2×C411yescharacteristic
55C4×#92418D411yescharacteristic
56C4×#92418D411yescharacteristic
57C2×M16×C61—21no—
58C2×M16×C61—21no—
59C22×C8×C61—21no—
60C2×C42×C61C2311yescharacteristic
61C4⋊C8×C61—21no—
62C4⋊C8×C61—21no—
63C4⋊C8×C61—21no—
64C4⋊C8×C61—21no—
65C4⋊C8×C61—21no—
66C4⋊C8×C61C2×C412yes—
67C22×C4×C61C2×D411yescharacteristic
68C22×C4×C61C2×Q811yescharacteristic
69C22×C4×C61#4111yescharacteristic
70C2×C22⋊D61—22no—
71#11758222—41no—
72#11758222—41no—
73#11758222—41no—
74#11758222—41no—
75C2×D244—21no—
76C2×D244—21no—
77C2×D244C4⋊C412yes—
78C2×C4×D61—22no—
79C2×Dic122—21no—
80M16×C61—41no—
81M16×C61—41no—
82M16×C61—41no—
83M16×C61—41no—
84C2×C8×C61—22no—
85C2×C8×C61—41no—
86C2×C8×C61—41no—
87C2×C8×C61—81no—
88C42×C61C22×C411yescharacteristic
89C42×C61C2×D411yescharacteristic
90C42×C61C2×D411yescharacteristic
91C42×C61#4111yescharacteristic
92#11758198—22no—
93#11758198—22no—
94C4⋊Dic61—21no—
95C4⋊Dic61C4⋊C412yes—
96#11758194—41no—
97#11758194—41no—
98C2×#92418—21no—
99C2×#92418—21no—
100C2×#92418—41no—
101C2×#92418—41no—
102C2×#92418—81no—
103C2×#92418C22⋊C411yescharacteristic
104C2×#92418C22⋊C411yescharacteristic
105C2×#92418C22⋊C411yescharacteristic
106C2×#92418C22⋊C411yescharacteristic
107C2×#92418D812yes—
108C2×#92418SD1612yes—
109#11758189—44no—
110C8×D61—44no—
111C23×C61C22⋊Q811yescharacteristic
112C22×D61—22no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—41no—
116C2×C4×C61C2×C22⋊C411yescharacteristic
117C2×C4×C61C2×C4⋊C411yescharacteristic
118C2×C4×C61#11711yescharacteristic
119C2×C4×C61#12111yescharacteristic
120C2×C4×C61#12111yescharacteristic
121C2×C4×C61C22⋊Q811yescharacteristic
122C2×C4×C61C2×D811yescharacteristic
123C2×C4×C61C2×SD1611yescharacteristic
124C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C61#13711yescharacteristic
126C22⋊D61—42no—
127C22⋊D61—42no—
128C2×Dic61—22no—
129D244—42no—
130D244—41no—
131D244—41no—
132D244—41no—
133D244—41no—
134C4×D61—42no—
135C4×D61—42no—
136Dic122—41no—
137Dic122—41no—
138C8×C61—42no—
139C8×C61—81no—
140C8×C61—81no—
141#92418—21no—
142#92418—21no—
143#92418—81no—
144#92418—81no—
145#92418#10212yes—
146#92418#10212yes—
147#8682—611nomaximal, Sylow
148C22×C61—21no—
149C22×C61—21no—
150C22×C61#38011yescharacteristic
151C22×C61#47411yescharacteristic
152C22×C61#47611yescharacteristic
153C2×D61—42no—
154C2×D61—42no—
155C2×D61—42no—
156C4×C61—21no—
157C4×C61—21no—
158C4×C61—21no—
159C4×C61C2×#10211yescharacteristic
160C4×C61#41711yescharacteristic
161C4×C61#45911yescharacteristic
162C4×C61#46111yescharacteristic
163C4×C61#47111yescharacteristic
164C4×C61#47111yescharacteristic
165Dic61—42no—
166#2931—611no—
167#1848—611no—
168C2×#334—611no—
169#1547—611no—
170#1547—611no—
171#1292—611no—
172#1292D6111yescharacteristic, complemented
173C2×C61—21no—
174C2×C61—21no—
175C2×C61#180211yescharacteristic
176C2×C61#181311yescharacteristic
177C2×C61#181511yescharacteristic
178D61—82no—
179C2×#131—1221no—
180#532—611no—
181#436—612no—
182#436—1221no—
183C8×D4—612no—
184C8×D4—1221no—
185C2×#108—1221no—
186C2×#108—1221no—
187#423—611no—
188#423C2×D6111yescharacteristic
189#423C2×D6111yescharacteristic
190C2×C4⋊C8—611no—
191C2×C4⋊C8C2×D6111yescharacteristic
192#408—1221no—
193C2×C4×C8—611no—
194#334—1221no—
195#334—1221no—
196#332—1221no—
197#332—1221no—
198#331—1221no—
199#331—1221no—
200C61#868211yescharacteristic, complemented, Sylow
201C2×#41—1221no—
202#131—2442no—
203#131—2442no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—1222no—
207C2×M16—1221no—
208C22×C8—21no—
209C22×C8—611no—
210C22×C8—611no—
211C22×C8—1222no—
212C22×C8—1221no—
213C4⋊Q8—611no—
214#127—611no—
215#124—1221no—
216#121—612no—
217#121—1221no—
218C4×D4—612no—
219C4×D4—1221no—
220C2×C42C22×D6111yescharacteristic
221#108—2442no—
222#108—2442no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—1221no—
229C4⋊C8—1221no—
230C4⋊C8—1221no—
231C4⋊C8—1221no—
232C4⋊C8C4×D6112yes—
233C22⋊C8—1222no—
234C22⋊C8—1222no—
235C4×C8—611no—
236C4×C8—611no—
237C4×C8—611no—
238C4×C8—611no—
239#41—2441no—
240#41—2441no—
241#41—2441no—
242#41—2441no—
243C2×Q8—1221no—
244C2×D4—612no—
245C2×D4—1222no—
246C2×D4—1221no—
247C2×D4—1221no—
248C22×C4—1222no—
249C22×C4#1175822211yescharacteristic
250C22×C4D4×D6111yescharacteristic
251C22×C4Q8×D6111yescharacteristic
252M16—41no—
253M16—41no—
254M16—41no—
255M16—41no—
256M16—2444no—
257M16—2441no—
258M16—2441no—
259C2×C8—22no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—81no—
263C2×C8—611no—
264C2×C8—611no—
265C2×C8—612no—
266C2×C8—612no—
267C2×C8—611no—
268C2×C8—611no—
269C2×C8—1221no—
270C2×C8—1221no—
271C2×C8—4881no—
272C2×C8—2444no—
273C2×C8—2441no—
274C2×C8—2441no—
275C4⋊C4—612no—
276C4⋊C4—1221no—
277C22⋊C4—1222no—
278C22⋊C4—1222no—
279C42C2×C4×D6111yescharacteristic
280C42#1175822211yescharacteristic
281C42D4×D6111yescharacteristic
282C42D4×D6111yescharacteristic
283C23—1222no—
284C23#1451975511yescharacteristic
285Q8—2441no—
286Q8—2441no—
287D4—2442no—
288D4—2442no—
289D4—2442no—
290D4—2441no—
291D4—2441no—
292D4—2441no—
293D4—2441no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—1222no—
298C2×C4—2442no—
299C2×C4—2442no—
300C2×C4C22⋊C4×D6111yescharacteristic
301C2×C4C4⋊C4×D6111yescharacteristic
302C2×C4#1451973411yescharacteristic
303C2×C4#1451975011yescharacteristic
304C2×C4#1451975011yescharacteristic
305C2×C4#1451975511yescharacteristic
306C2×C4D8×D6111yescharacteristic
307C2×C4SD16×D6111yescharacteristic
308C2×C4#1451980411yescharacteristic
309C2×C4#1451980911yescharacteristic
310C8—42no—
311C8—81no—
312C8—81no—
313C8—612no—
314C8—612no—
315C8—1221no—
316C8—1221no—
317C8—4881no—
318C8—4881no—
319C22—21no—
320C22—21no—
321C22—2442no—
322C22—2442no—
323C22—2442no—
324C22#1683962111yescharacteristic
325C22#1684004111yescharacteristic
326C22#1684005111yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—2442no—
331C4#102×D6111yescharacteristic
332C4#1683976211yescharacteristic
333C4#1683995611yescharacteristic
334C4#1683996611yescharacteristic
335C4#1684002411yescharacteristic
336C4#1684002811yescharacteristic
337C2—21no—
338C2—21no—
339C2—4882no—
340C2#2144581811yescharacteristic
341C2#2144586811yescharacteristic
342C2#2144587711yescharacteristic
subgroupG/H
centerC22#16839621
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1292C2
socleC22×C61#380
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1292D61
hypocenterC61#8682
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1292

Complements

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

NH ≅ G/Ntype
2. #21445990C2semidirect
3. C2×#16839512C2semidirect
6. #21443693C2semidirect
7. C61×#1292C2semidirect
172. #1292D61semidirect
200. C61#8682semidirect

Maximal subgroups

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

subgroupindex
#214545302
#214459902
C2×#168395122
#214444972
#214444942
#214436932
C61×#12922
#868261