Finite Group Databasebeta

← #29663290

Subgroups — #29663290

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456972C211yesmaximal, characteristic, complemented
2#21456762C211yesmaximal, characteristic
3C2×#16840109C211yesmaximal, characteristic
4#21454753C211yesmaximal, characteristic, complemented
5#21445360C214yesmaximal, complemented
6#21445236C214yesmaximal, complemented
7#21444499C211yesmaximal, characteristic, complemented
8#21444486C211yesmaximal, characteristic
9C61×#1564C211yesmaximal, characteristic, complemented
10#16840319C2211yescharacteristic
11C2×#14519777C2211yescharacteristic
12C2×#14519774C2211yescharacteristic
13#16840167—21no—
14#16840135—21no—
15#16840113—21no—
16#16840109—21no—
17#16840109—21no—
18#16840109—21no—
19#16840109C2212yes—
20#16840109C2212yes—
21#16840098—22no—
22#16840058C2214yescomplemented
23#16839963C2214yes—
24#16839915—22no—
25#16839892—21no—
26#16839892C2212yescomplemented
27C8×Dic122—21no—
28C8×Dic122C2212yescomplemented
29#16839788—21no—
30#16839786C2211yescharacteristic
31C2×#14519699—21no—
32C2×#14519699—21no—
33#16839717—21no—
34#16839510C2214yescomplemented
35#16839505C2214yes—
36C61×#431C2211yescharacteristic
37C61×#431C2211yescharacteristic
38C2×C4×C8×C61C2211yescharacteristic
39C61×#335C2212yescomplemented
40C61×#334C2212yescomplemented
41C22×Dic122—21no—
42C22×D244—21no—
43C22×D244—21no—
44C22×D244—21no—
45C2×#11758190—24no—
46C2×#11758194—21no—
47#14519777—21no—
48#14519777—21no—
49#14519777—21no—
50#14519777D411yescharacteristic
51#14519777D411yescharacteristic
52#14519777C2312yes—
53#14519774—21no—
54#14519774—21no—
55#14519774—21no—
56#14519774C2312yes—
57#14519769—21no—
58#14519756—21no—
59#14519756D412yes—
60C4×Dic122—21no—
61C4×Dic122C2312yes—
62#14519736—21no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—21no—
65#14519711—24no—
66C4⋊#92418—21no—
67C4⋊#92418C2312yes—
68#14519706—22no—
69#14519706—22no—
70#14519702—24no—
71#14519699—22no—
72#14519699—22no—
73#14519699—22no—
74#14519699—22no—
75#14519699—24no—
76#14519687—22no—
77C8×Dic61—22no—
78C22×C8×C61D411yescharacteristic
79C22×C8×C61D411yescharacteristic
80C2×C42×C61C2311yescharacteristic
81C4⋊C8×C61C2312yes—
82C4⋊C8×C61C2312yes—
83C22⋊C8×C61—21no—
84C22⋊C8×C61—21no—
85C4×C8×C61—21no—
86C4×C8×C61—21no—
87C4×C8×C61D411yescharacteristic
88C4×C8×C61D411yescharacteristic
89C4×C8×C61C2311yescharacteristic
90C4×C8×C61C2311yescharacteristic
91C23×D61—41no—
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61#4111yescharacteristic
95C22×Dic61—41no—
96C2×D244—22no—
97C2×D244—22no—
98C2×D244—22no—
99C2×D244—42no—
100C2×D244—42no—
101C2×D244—41no—
102C2×D244—41no—
103C2×D244—41no—
104C2×D244—82no—
105C2×D244—82no—
106C2×Dic122—21no—
107C2×Dic122—22no—
108C2×Dic122—41no—
109C2×Dic122—82no—
110C2×Dic122C2×D412yes—
111C2×C8×C61—22no—
112C2×C8×C61—22no—
113C2×C8×C61—22no—
114C2×C8×C61—22no—
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—22no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127C4⋊Dic61—22no—
128C4⋊Dic61—22no—
129C4⋊Dic61—41no—
130C4⋊Dic61—41no—
131C4⋊Dic61C2×D412yes—
132#11758196—22no—
133#11758196—22no—
134C4×Dic61—22no—
135#11758194—42no—
136C2×#92418—22no—
137#11758190—84no—
138C23×C61#12111yescharacteristic
139C22×D61—42no—
140C22×D61—82no—
141C22×D61—82no—
142C22×D61—82no—
143C2×C4×C61—22no—
144C2×C4×C61—22no—
145C2×C4×C61#12111yescharacteristic
146C2×C4×C61#12111yescharacteristic
147C2×C4×C61#12111yescharacteristic
148C2×C4×C61#13512yes—
149C2×C4×C61#13712yes—
150C2×C4×C61C22×D411yescharacteristic
151C2×C4×C61C22×D411yescharacteristic
152C2×C4×C61C2×#4111yescharacteristic
153C2×Dic61—22no—
154C2×Dic61—22no—
155C2×Dic61—42no—
156C2×Dic61—82no—
157D244—42no—
158D244—42no—
159D244—44no—
160D244—82no—
161D244—82no—
162D244—82no—
163D244—82no—
164Dic122—24no—
165Dic122—42no—
166Dic122—82no—
167Dic122—82no—
168C8×C61—21no—
169C8×C61—21no—
170C8×C61—21no—
171C8×C61—21no—
172C8×C61—42no—
173C8×C61—42no—
174#92418—42no—
175#19212—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61#46412yes—
179C22×C61C2×#12111yescharacteristic
180C2×D61—82no—
181C2×D61—82no—
182C2×D61—82no—
183C2×D61—164no—
184C4×C61—21no—
185C4×C61—21no—
186C4×C61#46412yes—
187C4×C61#47212yes—
188C4×C61C2×#13511yescharacteristic
189C4×C61C2×#13711yescharacteristic
190C4×C61#57712yes—
191Dic61—42no—
192Dic61—42no—
193Dic61—82no—
194#3230—611no—
195#3208—611no—
196C2×#486—611no—
197#2960—611no—
198#1698—614no—
199#1678—614no—
200#1564D6111yescharacteristic, complemented
201#1548—611no—
202#1545—611no—
203C2×C61—22no—
204C2×C61C2×#46411yescharacteristic
205C2×C61#305711yescharacteristic
206C2×C61C2×#47211yescharacteristic
207D61—162no—
208#533—611no—
209C2×C4⋊Q8—611no—
210C2×#127—611no—
211#500—1221no—
212#492—1221no—
213#487—1221no—
214#486—612no—
215#486—612no—
216#486—1221no—
217#486—1221no—
218#486—1221no—
219#484—1222no—
220#477—614no—
221#460—614no—
222#450—1222no—
223#446—612no—
224#446—1221no—
225C8×Q8—612no—
226C8×Q8—1221no—
227#431C2×D6111yescharacteristic
228#431C2×D6111yescharacteristic
229#424—1221no—
230#423—611no—
231C2×#102—1221no—
232C2×#102—1221no—
233#405—1221no—
234C2×C4×C8C2×D6111yescharacteristic
235#335C2×D6112yescomplemented
236#334C2×D6112yescomplemented
237#333—614no—
238#331—614no—
239C61#1921211yescharacteristic, complemented, Sylow
240C22×Q8—1221no—
241C22×D4—1221no—
242C22×D4—1221no—
243C22×D4—1221no—
244C2×SD16—1224no—
245C2×M16—1221no—
246C22×C8D24411yescharacteristic
247C22×C8D24411yescharacteristic
248C4⋊Q8—611no—
249C4⋊Q8—611no—
250C4⋊Q8—612no—
251C4⋊Q8—1221no—
252C4⋊Q8—1221no—
253C4⋊Q8—1221no—
254#127—612no—
255#127—1221no—
256#127—1221no—
257#127—1221no—
258#125—1221no—
259C22⋊Q8—612no—
260C22⋊Q8—1221no—
261C4×Q8—612no—
262C4×Q8—1221no—
263#117—1221no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4—1221no—
266C2×C42C22×D6111yescharacteristic
267#107—1224no—
268C4⋊C8—612no—
269C4⋊C8—1222no—
270C4⋊C8—1222no—
271C4⋊C8—1221no—
272C4⋊C8C22×D6112yes—
273C4⋊C8C22×D6112yes—
274#103—1224no—
275#102—1224no—
276#102—1222no—
277#102—1222no—
278#102—1222no—
279#102—1222no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282#97—1222no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—1222no—
286C4×C8D24411yescharacteristic
287C4×C8D24411yescharacteristic
288C4×C8C22×D6111yescharacteristic
289C4×C8C22×D6111yescharacteristic
290C24—2441no—
291C2×Q8—612no—
292C2×Q8—1221no—
293C2×Q8—1222no—
294C2×Q8—4882no—
295C2×Q8—2441no—
296C2×D4—1222no—
297C2×D4—1222no—
298C2×D4—1222no—
299C2×D4—4882no—
300C2×D4—4882no—
301C2×D4—2442no—
302C2×D4—2441no—
303C2×D4—2441no—
304C2×D4—2441no—
305C2×D4—2442no—
306C22×C4—2441no—
307C22×C4C2×D24411yescharacteristic
308C22×C4D4×D6111yescharacteristic
309C22×C4#1175822611yescharacteristic
310SD16—4884no—
311M16—2442no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—1222no—
317C2×C8C2×D24411yescharacteristic
318C2×C8C2×D24411yescharacteristic
319C2×C8C2×D24411yescharacteristic
320C2×C8C2×D24411yescharacteristic
321C4⋊C4—612no—
322C4⋊C4—1222no—
323C4⋊C4—1222no—
324C4⋊C4—1221no—
325C4⋊C4—1221no—
326C4⋊C4—1221no—
327C4⋊C4—1222no—
328C4⋊C4—1222no—
329C4⋊C4—2441no—
330C4⋊C4—2441no—
331C22⋊C4—1222no—
332C42—1222no—
333C42C2×D24411yescharacteristic
334C42D4×D6111yescharacteristic
335C42#1175822611yescharacteristic
336C42C23×D6111yescharacteristic
337C23—4882no—
338C23—4882no—
339C23—4882no—
340C23—2442no—
341C23#1451975211yescharacteristic
342Q8—1224no—
343Q8—4882no—
344Q8—4882no—
345Q8—2442no—
346D4—4882no—
347D4—4882no—
348D4—4882no—
349D4—4882no—
350D4—2442no—
351D4—2442no—
352D4—2444no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—1222no—
356C2×C4—1222no—
357C2×C4—4882no—
358C2×C4—2442no—
359C2×C4#1451975211yescharacteristic
360C2×C4#1451975211yescharacteristic
361C2×C4#1451975211yescharacteristic
362C2×C4#1451979912yes—
363C2×C4#1451981012yes—
364C2×C4C2×D4×D6111yescharacteristic
365C2×C4C22×D24411yescharacteristic
366C2×C4C2×#1175822611yescharacteristic
367C8—21no—
368C8—21no—
369C8—21no—
370C8—21no—
371C8—42no—
372C8—42no—
373C8—2442no—
374C22—21no—
375C22—21no—
376C22—9764no—
377C22—4882no—
378C22—4882no—
379C22—4882no—
380C22#1683998512yes—
381C22C2×#1451975211yescharacteristic
382C4—21no—
383C4—21no—
384C4—4882no—
385C4—2442no—
386C4—2442no—
387C4#1683998512yes—
388C4#1684003312yes—
389C4C2×#1451979911yescharacteristic
390C4C2×#1451981011yescharacteristic
391C4#1684056912yes—
392C2—22no—
393C2—9762no—
394C2C2×#1683998511yescharacteristic
395C2#2145545911yescharacteristic
396C2C2×#1684003311yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1564C2
socleC22×C61C2×#121
prime radicalC2×#14519774C22
hypercenter#1564D61
hypocenterC61#19212
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
1. #21456972C2semidirect
4. #21454753C2semidirect
5. #21445360C2semidirect
6. #21445236C2semidirect
7. #21444499C2semidirect
9. C61×#1564C2semidirect
22. #16840058C22semidirect
26. #16839892C22semidirect
28. C8×Dic122C22semidirect
34. #16839510C22semidirect
39. C61×#335C22semidirect
40. C61×#334C22semidirect
200. #1564D61semidirect
235. #335C2×D61semidirect
236. #334C2×D61semidirect
239. C61#19212semidirect

Maximal subgroups

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

subgroupindex
#214569722
#214567622
C2×#168401092
#214547532
#214453602
#214452362
#214444992
#214444862
C61×#15642
#1921261