Finite Group Databasebeta

← #29834871

Subgroups — #29834871

Order 15616.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455475C211yesmaximal, characteristic
2#21454416C211yesmaximal, characteristic
3#21448030C212yesmaximal, complemented
4#21448029C212yesmaximal, complemented
5#21448023C212yesmaximal, complemented
6#21448018C212yesmaximal, complemented
7#21447608C211yesmaximal, characteristic, complemented
8#21447538C211yesmaximal, characteristic, complemented
9C2×#16839567C211yesmaximal, characteristic, complemented
10#21447432C211yesmaximal, characteristic
11C61×#2158C211yesmaximal, characteristic, complemented
12C22×Dic244C2211yescharacteristic
13C2×#14519783C2211yescharacteristic
14#16840338—21no—
15#16840242—21no—
16C2×#14519717C2211yescharacteristic
17#16840021—22no—
18#16840004C2212yes—
19#16840004C2212yes—
20#16839805—21no—
21#16839801—22no—
22#16839799C2212yes—
23C2×#14519711C2211yescharacteristic
24#107×D61C2212yes—
25#16839770—21no—
26#16839589C2212yes—
27#16839587C2212yes—
28#16839567C2212yes—
29#16839567C2212yes—
30#16839565C2212yes—
31#16839565C2212yes—
32#16839541C2212yes—
33#16839541C2212yes—
34C22×C16×C61C2211yescharacteristic
35C61×#427C2211yescharacteristic
36C61×#427C2211yescharacteristic
37C61×#367C2211yescharacteristic
38C61×#367C2211yescharacteristic
39C61×#366C2211yescharacteristic
40C61×#366C2211yescharacteristic
41C2×#11758222D411yescharacteristic
42C22×Dic122—21no—
43C2×Dic244—22no—
44C2×Dic244—22no—
45C2×Dic244—21no—
46C2×Dic244C2312yes—
47#14519783—22no—
48#14519783—22no—
49C2×#11758194D411yescharacteristic
50C2×#11758189D412yes—
51C2×C8×D61C2312yes—
52#14519777—41no—
53#14519765—41no—
54#14519761—42no—
55#14519757—22no—
56#14519757—22no—
57C4×Dic122—41no—
58C4×D244—41no—
59#14519737—42no—
60#14519735—22no—
61C2×C4⋊Dic61—21no—
62C4⋊C4×D61—22no—
63#14519717—22no—
64C2×#11758185C2312yes—
65#14519711—21no—
66#14519711C2312yes—
67#14519710—22no—
68#14519708—22no—
69#14519700—22no—
70#14519700—22no—
71C22×C8×C61C2311yescharacteristic
72#117×C61—21no—
73#117×C61—21no—
74C2×C16×C61—22no—
75C2×C16×C61C2311yescharacteristic
76C2×C16×C61C2311yescharacteristic
77#107×C61C2311yescharacteristic
78#107×C61C2311yescharacteristic
79#107×C61C2311yescharacteristic
80#107×C61C2311yescharacteristic
81#106×C61—21no—
82#106×C61—21no—
83C22×C4×C61C2×D411yescharacteristic
84C2×C22⋊D61D812yes—
85C22×Dic61—41no—
86C22×Dic61—41no—
87#11758222—22no—
88#11758222—22no—
89C2×D244D811yescharacteristic
90C2×C4×D61—42no—
91C2×C4×D61C2×D412yes—
92C2×Dic122—22no—
93C2×Dic122—42no—
94C2×Dic122—42no—
95C2×Dic122—41no—
96C2×Dic122D811yescharacteristic
97C2×C8×C61C2×D411yescharacteristic
98C2×C8×C61#4112yes—
99C2×C8×C61#4112yes—
100C2×C8×C61C2411yescharacteristic
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—21no—
105C22⋊C4×C61—41no—
106C22⋊C4×C61—41no—
107C42×C61—41no—
108C42×C61—41no—
109#11758198—42no—
110#11758198—42no—
111#11758198—42no—
112C4⋊Dic61—22no—
113C4⋊Dic61—41no—
114C4⋊Dic61—42no—
115C4⋊Dic61—42no—
116#11758196—42no—
117C4×Dic61—42no—
118#11758194—22no—
119C2×#92418C2×D412yes—
120Dic244—22no—
121Dic244—22no—
122Dic244—42no—
123#11758189—24no—
124C8×D61—22no—
125C8×D61—22no—
126C16×C61—21no—
127C16×C61—21no—
128#11758185—22no—
129C23×C61C2×D811yescharacteristic
130C22×D61C2×D812yes—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61#12312yes—
136C2×C4×C61C2×D811yescharacteristic
137C2×C4×C61#13712yes—
138C2×C4×C61C22×D411yescharacteristic
139C22⋊D61—24no—
140C2×Dic61—42no—
141C2×Dic61—42no—
142C2×Dic61—82no—
143C2×Dic61—161no—
144C2×Dic61C2×D812yes—
145D244—22no—
146C4×D61—22no—
147C4×D61—22no—
148C4×D61—162no—
149Dic122—22no—
150Dic122—42no—
151Dic122—42no—
152Dic122—82no—
153C8×C61#12312yes—
154C8×C61C2×#4111yescharacteristic
155C8×C61C2×#4111yescharacteristic
156#92418—22no—
157#33651—611nomaximal, Sylow
158C22×C61—21no—
159C22×C61—21no—
160C22×C61#48012yes—
161C22×C61C22×D811yescharacteristic
162C2×D61—22no—
163C2×D61—22no—
164C2×D61—22no—
165C4×C61—81no—
166C4×C61—81no—
167C4×C61—81no—
168C4×C61—81no—
169C4×C61#48012yes—
170C4×C61C2×#12311yescharacteristic
171C4×C61C2×#13711yescharacteristic
172Dic61—22no—
173Dic61—82no—
174Dic61—82no—
175#3059—611no—
176#2913—611no—
177#2237—612no—
178#2237—612no—
179#2236—612no—
180#2235—612no—
181#2159—611no—
182#2158D6111yescharacteristic, complemented
183#2141—611no—
184C2×#358—611no—
185#2113—611no—
186C2×C61—22no—
187C2×C61C2×#48011yescharacteristic
188C2×C61#341611yescharacteristic
189C2×C61#341711yescharacteristic
190D61—42no—
191C22×Q16—611no—
192C2×#131—611no—
193#536—1221no—
194#520—1221no—
195C2×M32—611no—
196C22×C16C2×D6111yescharacteristic
197#470—1222no—
198#467—612no—
199#467—612no—
200#428—1222no—
201#428—1221no—
202#427—612no—
203#427C2×D6111yescharacteristic
204#427C2×D6111yescharacteristic
205C2×#107—611no—
206C2×#107—612no—
207#419—1221no—
208#367—612no—
209#367C2×D6111yescharacteristic
210#367C2×D6111yescharacteristic
211#366—612no—
212#366C2×D6111yescharacteristic
213#366C2×D6111yescharacteristic
214#358—612no—
215#358—612no—
216#358—612no—
217#358—612no—
218C22⋊C16—612no—
219C22⋊C16—612no—
220C61#3365111yescharacteristic, complemented, Sylow
221C2×#41—611no—
222C22×Q8—1221no—
223C2×Q16—612no—
224C2×Q16—1222no—
225C2×Q16—1221no—
226C2×Q16—1222no—
227#131—1222no—
228#131—1222no—
229C2×M16—611no—
230C2×M16—612no—
231C22×C8—612no—
232C22×C8C22×D6111yescharacteristic
233C4⋊Q8—2441no—
234#124—2441no—
235#123—2442no—
236C22⋊Q8—1222no—
237C22⋊Q8—1222no—
238C4×Q8—2441no—
239C4×D4—2441no—
240C4×D4—2442no—
241#117—21no—
242#117—21no—
243#117—1222no—
244C2×C4⋊C4—1222no—
245C2×C4⋊C4—1221no—
246M32—1222no—
247C2×C16—22no—
248C2×C16—612no—
249C2×C16C22×D6111yescharacteristic
250C2×C16C22×D6111yescharacteristic
251#107—612no—
252#107—1222no—
253#107—1221no—
254#107C22×D6111yescharacteristic
255#107C22×D6111yescharacteristic
256#107C22×D6111yescharacteristic
257#107C22×D6111yescharacteristic
258#106—21no—
259#106—21no—
260#106—1222no—
261#103—1222no—
262#103—1222no—
263#41—1222no—
264#41—1222no—
265C2×Q8—611no—
266C2×Q8—1222no—
267C2×Q8—2442no—
268C2×Q8—2442no—
269C2×Q8—2441no—
270C2×D4—612no—
271C2×D4—611no—
272C22×C4—612no—
273C22×C4—2442no—
274C22×C4—2441no—
275C22×C4—2441no—
276C22×C4D4×D6111yescharacteristic
277Q16—1222no—
278Q16—1222no—
279Q16—2442no—
280M16—1222no—
281M16—1224no—
282C2×C8—612no—
283C2×C8—1222no—
284C2×C8—1222no—
285C2×C8#1175822212yes—
286C2×C8D4×D6111yescharacteristic
287C2×C8#1175822612yes—
288C2×C8C23×D6111yescharacteristic
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—1222no—
294C4⋊C4—2442no—
295C4⋊C4—2442no—
296C4⋊C4—2442no—
297C4⋊C4—2441no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—2442no—
301C22⋊C4—2442no—
302C22⋊C4—2442no—
303C42—41no—
304C42—41no—
305C42—2442no—
306C16—21no—
307C16—21no—
308C16—1222no—
309C23—612no—
310C23D8×D6111yescharacteristic
311Q8—1222no—
312Q8—4882no—
313Q8—2442no—
314Q8—2442no—
315D4—1224no—
316D4—1222no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—612no—
322C2×C4—1222no—
323C2×C4—1222no—
324C2×C4—9762no—
325C2×C4—9761no—
326C2×C4—4882no—
327C2×C4—2442no—
328C2×C4—2442no—
329C2×C4#1451976212yes—
330C2×C4D8×D6111yescharacteristic
331C2×C4#1451980912yes—
332C2×C4C2×D4×D6111yescharacteristic
333C8—1222no—
334C8#1451976212yes—
335C8C2×#1175822211yescharacteristic
336C8C2×#1175822611yescharacteristic
337C22—21no—
338C22—21no—
339C22—1222no—
340C22—1222no—
341C22—1222no—
342C22#1684007712yes—
343C22C2×D8×D6111yescharacteristic
344C4—81no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—1222no—
349C4—4882no—
350C4—4882no—
351C4#1684007712yes—
352C4C2×#1451976211yescharacteristic
353C4C2×#1451980911yescharacteristic
354C2—22no—
355C2—2442no—
356C2C2×#1684007711yescharacteristic
357C2#2145896211yescharacteristic
358C2#2145896911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2158C2
socleC22×C61C22×D8
prime radicalC2×#11758222D4
hypercenter#2158D61
hypocenterC61#33651
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2158

Complements

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

NH ≅ G/Ntype
3. #21448030C2semidirect
4. #21448029C2semidirect
5. #21448023C2semidirect
6. #21448018C2semidirect
7. #21447608C2semidirect
8. #21447538C2semidirect
9. C2×#16839567C2semidirect
11. C61×#2158C2semidirect
182. #2158D61semidirect
220. C61#33651semidirect

Maximal subgroups

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

subgroupindex
#214554752
#214544162
#214480302
#214480292
#214480232
#214480182
#214476082
#214475382
C2×#168395672
#214474322
C61×#21582
#3365161