Finite Group Databasebeta

← #29663450

Subgroups — #29663450

Order 15616.

562 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457000C212yesmaximal
2#21456405C211yesmaximal, characteristic, complemented
3#21454794C211yesmaximal, characteristic
4#21445348C211yesmaximal, characteristic, complemented
5#21445343C211yesmaximal, characteristic, complemented
6#21445314C211yesmaximal, characteristic, complemented
7#21445309C211yesmaximal, characteristic, complemented
8#21445278C212yesmaximal, complemented
9#21445271C212yesmaximal, complemented
10#21444527C212yesmaximal, complemented
11C61×#1564C211yesmaximal, characteristic, complemented
12#16840319C2212yes—
13#16840296C2211yescharacteristic
14#16840167C2211yescharacteristic
15#16840164—22no—
16#16840142C2211yescharacteristic
17#16840117—22no—
18#16840113C2212yes—
19#16840104—22no—
20#16840087—22no—
21#16840058C2212yes—
22#16840053C2212yes—
23#16840048C2212yes—
24#16840043C2212yes—
25#16840003—21no—
26#16840003—21no—
27#16839892C2211yescharacteristic
28C8×Dic122C2211yescharacteristic
29C4×Dic244—21no—
30C4×Dic244—21no—
31#16839841C2211yescharacteristic
32#16839833—21no—
33C8×D244C2211yescharacteristic
34C8×C22⋊D61—21no—
35#16839821—21no—
36#16839819C2211yescharacteristic
37#16839716—21no—
38#16839510C2212yes—
39#16839510C2212yes—
40#16839507C2212yes—
41#16839507C2212yes—
42C61×#431C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C61×#335C2212yes—
45C61×#334C2211yescharacteristic
46C61×#334C2211yescharacteristic
47C2×Dic244—21no—
48C2×Dic244—21no—
49C2×#11758189—21no—
50C2×C8×D61—21no—
51#14519777C2312yes—
52#14519777C2312yes—
53#14519771—21no—
54#14519769—22no—
55#14519769D411yescharacteristic
56#14519765D411yescharacteristic
57#14519760—21no—
58#14519756—22no—
59#14519756—22no—
60#14519756D411yescharacteristic
61#14519751D411yescharacteristic
62C4×Dic122—22no—
63C4×Dic122—22no—
64C4×Dic122C2311yescharacteristic
65C4×D244C2311yescharacteristic
66C4×C22⋊D61—21no—
67#14519736—22no—
68#14519734—21no—
69#14519711—21no—
70#14519711—21no—
71#14519711—22no—
72#14519711—22no—
73#14519709—22no—
74#14519709—22no—
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2311yescharacteristic
77#14519706—21no—
78#14519706—21no—
79#14519702—22no—
80#14519702—22no—
81#14519702—22no—
82#14519702—22no—
83#14519699—22no—
84#14519699—22no—
85C22⋊#92418—21no—
86#14519689—21no—
87#14519689—21no—
88#14519688—21no—
89#14519687—21no—
90C8×Dic61—21no—
91C22×C8×C61D411yescharacteristic
92C22×C8×C61D411yescharacteristic
93C2×C42×C61C2311yescharacteristic
94C4⋊C8×C61C2312yes—
95C4⋊C8×C61C2312yes—
96C22⋊C8×C61—22no—
97C4×C8×C61—22no—
98C4×C8×C61D411yescharacteristic
99C4×C8×C61D411yescharacteristic
100C4×C8×C61C2311yescharacteristic
101C4×C8×C61C2311yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61#4111yescharacteristic
105C2×C22⋊D61—21no—
106C2×D244C2×D411yescharacteristic
107C2×C4×D61—21no—
108C2×Dic122—22no—
109C2×Dic122—22no—
110C2×Dic122C2×D411yescharacteristic
111C2×C8×C61—22no—
112C2×C8×C61—22no—
113C2×C8×C61—22no—
114C2×C8×C61—41no—
115C2×C8×C61C2×D411yescharacteristic
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61C2×D411yescharacteristic
118C2×C8×C61C2×D411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61C2×D411yescharacteristic
121C42×C61#4111yescharacteristic
122C42×C61C2411yescharacteristic
123C22⋊Dic61—21no—
124C22⋊Dic61—42no—
125#11758198—21no—
126#11758198—21no—
127#11758198—21no—
128#11758198—21no—
129C4⋊Dic61—22no—
130C4⋊Dic61—22no—
131C4⋊Dic61—22no—
132C4⋊Dic61—22no—
133C4⋊Dic61C2×D411yescharacteristic
134C4⋊Dic61C2×D411yescharacteristic
135#11758196—21no—
136#11758196—21no—
137#11758196—21no—
138#11758196—21no—
139#11758196—42no—
140#11758196—42no—
141C4×Dic61—21no—
142C4×Dic61—42no—
143C2×#92418—21no—
144C2×#92418—21no—
145Dic244—42no—
146Dic244—42no—
147#11758189—42no—
148C8×D61—42no—
149C23×C61#12111yescharacteristic
150C22×D61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—22no—
154C2×C4×C61—41no—
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#12111yescharacteristic
158C2×C4×C61C22×D411yescharacteristic
159C2×C4×C61C22×D411yescharacteristic
160C2×C4×C61C2×#4111yescharacteristic
161C22⋊D61—42no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—42no—
166C2×Dic61—42no—
167D244—22no—
168C4×D61—41no—
169C4×D61—41no—
170Dic122—22no—
171Dic122—42no—
172Dic122—42no—
173Dic122—82no—
174C8×C61—21no—
175C8×C61—21no—
176C8×C61—21no—
177C8×C61—42no—
178C8×C61—42no—
179C8×C61#13512yes—
180#92418—41no—
181#92418—41no—
182#19226—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61C2×#12111yescharacteristic
187C2×D61—41no—
188C2×D61—41no—
189C2×D61—41no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61C2×#13511yescharacteristic
195C4×C61#57511yescharacteristic
196C4×C61#57812yes—
197Dic61—41no—
198Dic61—41no—
199Dic61—41no—
200Dic61—82no—
201Dic61—82no—
202#3233—612no—
203#3170—611no—
204#2966—611no—
205#1696—611no—
206#1695—611no—
207#1690—611no—
208#1689—611no—
209#1684—612no—
210#1683—612no—
211#1564D6111yescharacteristic, complemented
212#1554—612no—
213C2×C61—41no—
214C2×C61#303511yescharacteristic
215C2×C61#305911yescharacteristic
216C2×C61#306711yescharacteristic
217D61—81no—
218#533—612no—
219#529—611no—
220#500—611no—
221#499—1222no—
222#494—611no—
223#488—1222no—
224#487—612no—
225#485—1222no—
226#482—1222no—
227#477—612no—
228#476—612no—
229#475—612no—
230#474—612no—
231#467—1221no—
232#467—1221no—
233#446—611no—
234C8×Q8—611no—
235C4×Q16—1221no—
236C4×Q16—1221no—
237#436—611no—
238#435—1221no—
239C8×D4—611no—
240C8×D4—1221no—
241#433—1221no—
242#432—611no—
243#431C2×D6112yes—
244#405—1221no—
245C2×C4×C8C2×D6111yescharacteristic
246#335C2×D6112yes—
247#334C2×D6111yescharacteristic
248#334C2×D6111yescharacteristic
249#333—612no—
250#333—612no—
251#332—612no—
252#332—612no—
253C61#1922611yescharacteristic, complemented, Sylow
254C2×Q16—1221no—
255C2×Q16—1221no—
256C2×M16—1221no—
257C22×C8—1221no—
258C22×C8D24411yescharacteristic
259C22×C8D24411yescharacteristic
260C4⋊Q8—612no—
261C4⋊Q8—612no—
262#126—1221no—
263#125—611no—
264#125—1222no—
265#124—611no—
266#123—1221no—
267C22⋊Q8—611no—
268C22⋊Q8—1222no—
269C22⋊Q8—1222no—
270#121—611no—
271C4×Q8—611no—
272C4×Q8—1222no—
273C4×Q8—1222no—
274C4×D4—611no—
275C4×D4—1221no—
276#117—1221no—
277#117—1222no—
278C2×C42C22×D6111yescharacteristic
279#107—1222no—
280#107—1222no—
281#107—1221no—
282#107—1221no—
283#106—1222no—
284#106—1222no—
285C4⋊C8—611no—
286C4⋊C8—611no—
287C4⋊C8—1221no—
288C4⋊C8—1221no—
289C4⋊C8C22×D6112yes—
290C4⋊C8C22×D6112yes—
291#103—1222no—
292#103—1222no—
293#103—1222no—
294#103—1222no—
295#102—1222no—
296#102—1222no—
297C22⋊C8—22no—
298C22⋊C8—1221no—
299C22⋊C8—1221no—
300C22⋊C8—1221no—
301#97—1221no—
302#97—1221no—
303C4×C8—22no—
304C4×C8—1221no—
305C4×C8D24411yescharacteristic
306C4×C8D24411yescharacteristic
307C4×C8C22×D6111yescharacteristic
308C4×C8C22×D6111yescharacteristic
309C2×Q8—611no—
310C2×Q8—1222no—
311C2×Q8—1222no—
312C2×D4—611no—
313C2×D4—1221no—
314C22×C4—1221no—
315C22×C4C2×D24411yescharacteristic
316C22×C4D4×D6111yescharacteristic
317C22×C4#1175822611yescharacteristic
318Q16—2442no—
319Q16—2442no—
320M16—2442no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—41no—
325C2×C8—1221no—
326C2×C8—1221no—
327C2×C8—2442no—
328C2×C8C2×D24411yescharacteristic
329C2×C8C2×D24411yescharacteristic
330C2×C8C2×D24411yescharacteristic
331C2×C8C2×D24411yescharacteristic
332C4⋊C4—611no—
333C4⋊C4—611no—
334C4⋊C4—1221no—
335C4⋊C4—1221no—
336C4⋊C4—1221no—
337C4⋊C4—1221no—
338C4⋊C4—1222no—
339C4⋊C4—1222no—
340C4⋊C4—1222no—
341C4⋊C4—1222no—
342C4⋊C4—2442no—
343C4⋊C4—2442no—
344C22⋊C4—1221no—
345C22⋊C4—1221no—
346C22⋊C4—1221no—
347C22⋊C4—1221no—
348C22⋊C4—1221no—
349C22⋊C4—2442no—
350C42—1221no—
351C42—2442no—
352C42C2×D24411yescharacteristic
353C42D4×D6111yescharacteristic
354C42#1175822611yescharacteristic
355C42C23×D6111yescharacteristic
356C23—1221no—
357C23#1451975211yescharacteristic
358Q8—1222no—
359Q8—4882no—
360Q8—2442no—
361Q8—2442no—
362D4—1222no—
363D4—2442no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—1221no—
369C2×C4—1221no—
370C2×C4—1221no—
371C2×C4—2442no—
372C2×C4—2442no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4#1451975211yescharacteristic
376C2×C4#1451975211yescharacteristic
377C2×C4#1451975211yescharacteristic
378C2×C4C2×D4×D6111yescharacteristic
379C2×C4C22×D24411yescharacteristic
380C2×C4C2×#1175822611yescharacteristic
381C8—21no—
382C8—21no—
383C8—21no—
384C8—42no—
385C8—42no—
386C8—2441no—
387C8—2441no—
388C8#1451979912yes—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—2441no—
393C22—2441no—
394C22—2441no—
395C22C2×#1451975211yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—4882no—
401C4—4882no—
402C4—2441no—
403C4—2441no—
404C4—2441no—
405C4C2×#1451979911yescharacteristic
406C4#1684056211yescharacteristic
407C4#1684057612yes—
408C2—41no—
409C2—4881no—
410C2#2145527611yescharacteristic
411C2#2145547511yescharacteristic
412C2#2145554011yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1564C2
socleC22×C61C2×#121
prime radical#14519751D4
hypercenter#1564D61
hypocenterC61#19226
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1564

Complements

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

NH ≅ G/Ntype
2. #21456405C2semidirect
4. #21445348C2semidirect
5. #21445343C2semidirect
6. #21445314C2semidirect
7. #21445309C2semidirect
8. #21445278C2semidirect
9. #21445271C2semidirect
10. #21444527C2semidirect
11. C61×#1564C2semidirect
211. #1564D61semidirect
253. C61#19226semidirect

Maximal subgroups

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

subgroupindex
#214570002
#214564052
#214547942
#214453482
#214453432
#214453142
#214453092
#214452782
#214452712
#214445272
C61×#15642
#1922661