Finite Group Databasebeta

← #29668019

Subgroups — #29668019

Order 15616.

754 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456744C211yesmaximal, characteristic, complemented
2#21456741C211yesmaximal, characteristic, complemented
3#21456394C211yesmaximal, characteristic
4C2×#16840165C211yesmaximal, characteristic, complemented
5#21456213C211yesmaximal, characteristic, complemented
6#21445473C212yesmaximal, complemented
7#21445469C212yesmaximal, complemented
8#21445427C212yesmaximal, complemented
9#21445410C212yesmaximal, complemented
10M16×#92418C211yesmaximal, characteristic, complemented
11C61×#3141C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13C2×#14519777C2211yescharacteristic
14#16840165—22no—
15#16840165C2211yescharacteristic
16#16840165C2212yes—
17#16840165C2212yes—
18#16840138C2211yescharacteristic
19#16840136C2212yes—
20#16840134C2212yes—
21#16840122—22no—
22#16840114C2212yes—
23#16840113C2212yes—
24#16840112C2212yes—
25#16840110C2211yescharacteristic
26#16840109C2212yes—
27#16840108C2211yescharacteristic
28#16839996—22no—
29#16839967—22no—
30#16839964—22no—
31#16839901—22no—
32#16839900—22no—
33C4×#11758199—22no—
34#16839815C2211yescharacteristic
35C2×#14519710—22no—
36#16839764—22no—
37C2×C4×#92418C2211yescharacteristic
38#16839483C2212yes—
39C8×#92418C2212yes—
40C61×#532C2211yescharacteristic
41C2×C4⋊Q8×C61C2211yescharacteristic
42C61×#489—22no—
43C61×#487C2212yes—
44C61×#486C2212yes—
45C61×#417—22no—
46C4×M16×C61C2211yescharacteristic
47C2×#11758222—21no—
48C22×Dic122—21no—
49C2×#11758200—24no—
50C2×#11758201—24no—
51C2×#11758199—22no—
52C22×#92418D412yes—
53#14519777—21no—
54#14519777—21no—
55#14519777—21no—
56#14519777C2312yes—
57#14519777C2311yescharacteristic
58#14519774C2311yescharacteristic
59#14519765—21no—
60#14519751—21no—
61#14519751—21no—
62C4×D244—21no—
63C4×D244—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66#14519710—22no—
67#14519710—24no—
68#14519710—24no—
69C4⋊#92418D412yes—
70#14519702—24no—
71#14519701—24no—
72#14519700—24no—
73#14519699—24no—
74#14519698—22no—
75#14519697—22no—
76C22⋊#92418D412yes—
77C4×#92418D412yes—
78C4×#92418C2311yescharacteristic
79C4×#92418C2311yescharacteristic
80C4×#92418C2312yes—
81C2×#41×C61—21no—
82C22×Q8×C61—21no—
83C2×M16×C61D412yes—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61C2312yes—
88C4⋊Q8×C61C2311yescharacteristic
89#127×C61C2311yescharacteristic
90#124×C61—21no—
91#121×C61—21no—
92#121×C61—21no—
93C4×D4×C61—21no—
94C4×D4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C42×C61C2311yescharacteristic
98#103×C61—24no—
99#102×C61—24no—
100#97×C61D412yes—
101C4×C8×C61C2312yes—
102#41×C61—44no—
103C2×Q8×C61—21no—
104C2×Q8×C61—22no—
105C2×Q8×C61—41no—
106C2×Q8×C61—82no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—41no—
111C22×C4×C61—41no—
112C22×C4×C61—41no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C2×C22⋊D61—41no—
117C22×Dic61—41no—
118#11758222—44no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244—21no—
122C2×C4×D61—41no—
123C2×Dic122—22no—
124C2×Dic122—21no—
125C2×Dic122—41no—
126C2×Dic122—82no—
127M16×C61—24no—
128C2×C8×C61C2×D414yes—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—41no—
134C4⋊C4×C61—41no—
135C22⋊C4×C61—41no—
136C22⋊C4×C61—41no—
137C42×C61C2×D411yescharacteristic
138C42×C61C2×D411yescharacteristic
139C42×C61C2×D411yescharacteristic
140C42×C61C2411yescharacteristic
141#11758201—84no—
142#11758200—84no—
143#11758199—44no—
144#11758198—41no—
145#11758198—41no—
146C4⋊Dic61—21no—
147C4⋊Dic61—22no—
148C4⋊Dic61—21no—
149C4⋊Dic61—22no—
150C4⋊Dic61—41no—
151C4⋊Dic61—41no—
152C2×#92418—24no—
153C2×#92418C2×D412yes—
154C2×#92418C2×D414yes—
155C2×#92418C2×D412yes—
156C23×C61—41no—
157C23×C61#12711yescharacteristic
158C22×D61—41no—
159Q8×C61—42no—
160Q8×C61—44no—
161Q8×C61—82no—
162D4×C61—42no—
163D4×C61—42no—
164D4×C61—42no—
165D4×C61—82no—
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61—41no—
169C2×C4×C61—42no—
170C2×C4×C61—82no—
171C2×C4×C61—82no—
172C2×C4×C61#12711yescharacteristic
173C2×C4×C61#12711yescharacteristic
174C2×C4×C61#12711yescharacteristic
175C2×C4×C61#13514yes—
176C2×C4×C61C22×D411yescharacteristic
177C2×C4×C61C22×D411yescharacteristic
178C2×C4×C61C22×D411yescharacteristic
179C22⋊D61—82no—
180C2×Dic61—42no—
181C2×Dic61—41no—
182C2×Dic61—82no—
183D244—42no—
184D244—42no—
185D244—42no—
186C4×D61—82no—
187Dic122—42no—
188Dic122—44no—
189Dic122—82no—
190C8×C61—24no—
191#92418—22no—
192#92418—24no—
193#92418#13714yes—
194#19638—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—81no—
198C22×C61—81no—
199C22×C61—81no—
200C22×C61#49512yes—
201C22×C61C2×#12711yescharacteristic
202C2×D61—81no—
203C2×D61—81no—
204C2×D61—81no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—81no—
208C4×C61—82no—
209C4×C61#49512yes—
210C4×C61C2×#13512yes—
211C4×C61C2×#13712yes—
212C4×C61#57612yes—
213Dic61—81no—
214Dic61—82no—
215#3206—611no—
216#3206—611no—
217#3169—611no—
218C2×#500—611no—
219#3141—611no—
220#3141D6111yescharacteristic, complemented
221#1719—612no—
222#1719—612no—
223#1710—612no—
224#1706—612no—
225C8×M16—611no—
226C2×C61—22no—
227C2×C61—161no—
228C2×C61C2×#49511yescharacteristic
229C2×C61#315211yescharacteristic
230C2×C61#315511yescharacteristic
231D61—161no—
232#532—611no—
233#532C2×D6111yescharacteristic
234C2×C4⋊Q8—611no—
235C2×C4⋊Q8C2×D6111yescharacteristic
236#500—612no—
237#500—612no—
238#500—611no—
239#500—1222no—
240#493—611no—
241#492—612no—
242#492—612no—
243#489—22no—
244#489—1222no—
245#487—612no—
246#487—612no—
247#487—612no—
248#487C2×D6112yes—
249#486—611no—
250#486—611no—
251#486—612no—
252#486C2×D6112yes—
253#466—1222no—
254#461—1222no—
255#461—1222no—
256#448—1222no—
257#448—1222no—
258C4×D8—1222no—
259#431—611no—
260C2×#107—1222no—
261#417—22no—
262#417—1222no—
263C4×M16C2×D6111yescharacteristic
264C2×C4×C8—611no—
265#322—612no—
266C82—612no—
267C61#1963811yescharacteristic, complemented, Sylow
268C2×#41—21no—
269C2×#41—1221no—
270C22×Q8—21no—
271C22×Q8—1221no—
272C2×SD16—1224no—
273C2×SD16—1224no—
274C2×D8—1222no—
275C2×M16C22⋊D6112yes—
276C22×C8—612no—
277C4⋊Q8—21no—
278C4⋊Q8—21no—
279C4⋊Q8—21no—
280C4⋊Q8—612no—
281C4⋊Q8—611no—
282C4⋊Q8—1221no—
283C4⋊Q8—1221no—
284C4⋊Q8—1221no—
285C4⋊Q8C22×D6112yes—
286C4⋊Q8C22×D6111yescharacteristic
287#127—611no—
288#127C22×D6111yescharacteristic
289#124—21no—
290#124—1221no—
291#121—21no—
292#121—21no—
293#121—1221no—
294#121—1221no—
295C4×D4—21no—
296C4×D4—21no—
297C4×D4—1221no—
298C4×D4—1221no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—1221no—
302C2×C4⋊C4—1221no—
303C2×C42C22×D6111yescharacteristic
304#107—1224no—
305#107—1224no—
306#107—1222no—
307C4⋊C8—612no—
308#103—24no—
309#103—1224no—
310#103—1224no—
311#103—1224no—
312#102—24no—
313#102—1222no—
314#102—1222no—
315#102—1224no—
316C22⋊C8—612no—
317#97C22⋊D6112yes—
318C4×C8—612no—
319C4×C8—611no—
320C4×C8—612no—
321C4×C8—611no—
322C4×C8C22×D6112yes—
323#41—44no—
324#41—2444no—
325C2×Q8—22no—
326C2×Q8—21no—
327C2×Q8—41no—
328C2×Q8—82no—
329C2×Q8—1222no—
330C2×Q8—1221no—
331C2×Q8—4882no—
332C2×Q8—2441no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—41no—
337C2×D4—1221no—
338C2×D4—1221no—
339C2×D4—1221no—
340C2×D4—2441no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—2441no—
344C22×C4—2441no—
345C22×C4D4×D6111yescharacteristic
346C22×C4D4×D6111yescharacteristic
347C22×C4C2×C22⋊D6111yescharacteristic
348SD16—4884no—
349SD16—4884no—
350D8—2444no—
351M16—24no—
352C2×C8—612no—
353C2×C8—612no—
354C2×C8—614no—
355C2×C8—1224no—
356C2×C8C2×C22⋊D6114yes—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—1221no—
364C4⋊C4—1221no—
365C4⋊C4—1222no—
366C4⋊C4—1222no—
367C4⋊C4—2441no—
368C4⋊C4—2441no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—2441no—
372C22⋊C4—2441no—
373C42D4×D6111yescharacteristic
374C42D4×D6111yescharacteristic
375C42C2×C22⋊D6111yescharacteristic
376C42C23×D6111yescharacteristic
377C23—41no—
378C23—2441no—
379C23#1451977311yescharacteristic
380Q8—42no—
381Q8—44no—
382Q8—82no—
383Q8—4882no—
384Q8—2442no—
385Q8—2444no—
386D4—42no—
387D4—42no—
388D4—42no—
389D4—82no—
390D4—4882no—
391D4—2442no—
392D4—2442no—
393D4—2442no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—4882no—
401C2×C4—4882no—
402C2×C4—2441no—
403C2×C4—2442no—
404C2×C4#1451977311yescharacteristic
405C2×C4#1451977311yescharacteristic
406C2×C4#1451977311yescharacteristic
407C2×C4#1451979714yes—
408C2×C4C2×D4×D6111yescharacteristic
409C2×C4C2×D4×D6111yescharacteristic
410C2×C4C22×C22⋊D6111yescharacteristic
411C8—24no—
412C8—614no—
413C8—1222no—
414C8—1224no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—4881no—
421C22—4881no—
422C22—4881no—
423C22#1684014412yes—
424C22C2×#1451977311yescharacteristic
425C4—21no—
426C4—21no—
427C4—81no—
428C4—82no—
429C4—4881no—
430C4—4882no—
431C4#1684014412yes—
432C4C2×#1451979712yes—
433C4#137×D6112yes—
434C4#1684056612yes—
435C2—22no—
436C2—161no—
437C2—9761no—
438C2C2×#1684014411yescharacteristic
439C2#2145627411yescharacteristic
440C2#2145630411yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3141C2
socleC22×C61C2×#127
prime radical#21456394C2
hypercenter#3141D61
hypocenterC61#19638
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3141

Complements

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

NH ≅ G/Ntype
1. #21456744C2semidirect
2. #21456741C2semidirect
4. C2×#16840165C2semidirect
5. #21456213C2semidirect
6. #21445473C2semidirect
7. #21445469C2semidirect
8. #21445427C2semidirect
9. #21445410C2semidirect
10. M16×#92418C2semidirect
11. C61×#3141C2semidirect
220. #3141D61semidirect
267. C61#19638semidirect

Maximal subgroups

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

subgroupindex
#214567442
#214567412
#214563942
C2×#168401652
#214562132
#214454732
#214454692
#214454272
#214454102
M16×#924182
C61×#31412
#1963861