Finite Group Databasebeta

← #29667950

Subgroups — #29667950

Order 15616.

858 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456942C211yesmaximal, characteristic, complemented
2#21456394C211yesmaximal, characteristic
3C2×#16840165C211yesmaximal, characteristic, complemented
4#21456279C211yesmaximal, characteristic, complemented
5#21456276C211yesmaximal, characteristic, complemented
6#21445492C212yesmaximal, complemented
7#21445489C212yesmaximal, complemented
8#21445436C212yesmaximal, complemented
9#21445410C212yesmaximal, complemented
10#21444100C211yesmaximal, characteristic, complemented
11C61×#3226C211yesmaximal, characteristic, complemented
12C2×#14519808—22no—
13C2×#14519807—22no—
14#16840312C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16#16840167C2212yes—
17#16840165—22no—
18#16840165C2211yescharacteristic
19#16840165C2212yes—
20#16840165C2212yes—
21#16840157—22no—
22#16840154—22no—
23#16840141C2212yes—
24#16840141C2212yes—
25#16840138C2211yescharacteristic
26#16840136C2212yes—
27#16840134C2212yes—
28#16840110C2211yescharacteristic
29#16840109C2212yes—
30#16840108C2211yescharacteristic
31#16840076—22no—
32#16840043—22no—
33#16839996—22no—
34C4×#11758199—22no—
35C2×#14519710—22no—
36C4×#11758194C2211yescharacteristic
37C2×C4×#92418C2211yescharacteristic
38#16839484C2212yes—
39C8×#92418C2212yes—
40C61×#532C2211yescharacteristic
41C2×C4⋊Q8×C61C2211yescharacteristic
42C61×#500C2212yes—
43C61×#486C2212yes—
44C61×#480—22no—
45C61×#474—22no—
46C61×#431C2211yescharacteristic
47C2×#11758222—21no—
48C22×Dic122—21no—
49#14519808—48no—
50#14519807—48no—
51C2×#11758202—24no—
52C2×#11758202—24no—
53C2×#11758200—24no—
54C2×#11758201—24no—
55C2×#11758199—22no—
56C2×#11758194D412yes—
57C22×#92418D412yes—
58#14519777—21no—
59#14519777—21no—
60#14519777—21no—
61#14519777C2312yes—
62#14519777C2311yescharacteristic
63#14519774C2311yescharacteristic
64#14519765—21no—
65#14519751—21no—
66#14519751—21no—
67C4×D244—21no—
68C4×D244—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71#14519711—24no—
72#14519710—22no—
73#14519710—24no—
74#14519710—24no—
75#14519699—24no—
76#14519698—22no—
77#14519697—22no—
78#14519688D412yes—
79C4×#92418D412yes—
80C4×#92418C2311yescharacteristic
81C4×#92418C2311yescharacteristic
82C4×#92418C2312yes—
83C2×#41×C61—21no—
84C22×Q8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61—21no—
88C4⋊Q8×C61C2312yes—
89C4⋊Q8×C61C2311yescharacteristic
90#127×C61C2311yescharacteristic
91#124×C61—21no—
92#121×C61—21no—
93#121×C61—21no—
94C4×D4×C61—21no—
95C4×D4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C42×C61C2311yescharacteristic
99#107×C61—24no—
100C4⋊C8×C61D412yes—
101#102×C61—24no—
102C22⋊C8×C61D412yes—
103C4×C8×C61C2312yes—
104#41×C61—44no—
105C2×Q8×C61—21no—
106C2×Q8×C61—22no—
107C2×Q8×C61—41no—
108C2×Q8×C61—44no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—41no—
113C22×C4×C61—41no—
114C22×C4×C61—41no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C2×C22⋊D61—41no—
119C22×Dic61—41no—
120#11758222—44no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×C4×D61—41no—
125C2×Dic122—22no—
126C2×Dic122—21no—
127C2×Dic122—44no—
128C2×Dic122—41no—
129C2×C8×C61C2×D414yes—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—41no—
135C4⋊C4×C61—41no—
136C22⋊C4×C61—41no—
137C22⋊C4×C61—41no—
138C42×C61C2×D411yescharacteristic
139C42×C61C2×D411yescharacteristic
140C42×C61C2×D411yescharacteristic
141C42×C61C2411yescharacteristic
142#11758202—48no—
143#11758202—48no—
144#11758201—48no—
145#11758200—48no—
146#11758199—44no—
147#11758198—41no—
148#11758198—41no—
149C4⋊Dic61—21no—
150C4⋊Dic61—22no—
151C4⋊Dic61—22no—
152C4⋊Dic61—21no—
153C4⋊Dic61—41no—
154C4⋊Dic61—41no—
155#11758194D818yescomplemented
156C2×#92418—24no—
157C2×#92418C2×D412yes—
158C2×#92418C2×D414yes—
159C2×#92418C2×D412yes—
160C23×C61—41no—
161C23×C61#12711yescharacteristic
162C22×D61—41no—
163Q8×C61—44no—
164Q8×C61—42no—
165Q8×C61—82no—
166D4×C61—42no—
167D4×C61—42no—
168D4×C61—42no—
169D4×C61—82no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—41no—
173C2×C4×C61—42no—
174C2×C4×C61—82no—
175C2×C4×C61—82no—
176C2×C4×C61#12711yescharacteristic
177C2×C4×C61#12711yescharacteristic
178C2×C4×C61#12711yescharacteristic
179C2×C4×C61C2×D814yes—
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C22×D411yescharacteristic
183C22⋊D61—82no—
184C2×Dic61—42no—
185C2×Dic61—41no—
186C2×Dic61—82no—
187D244—42no—
188D244—42no—
189D244—42no—
190C4×D61—82no—
191Dic122—44no—
192Dic122—42no—
193Dic122—82no—
194C8×C61—24no—
195#92418—22no—
196#92418C2×D818yes—
197#92418#13714yescomplemented
198#19630—611nomaximal, Sylow
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—81no—
202C22×C61—81no—
203C22×C61—81no—
204C22×C61#49312yes—
205C22×C61C2×#12711yescharacteristic
206C2×D61—81no—
207C2×D61—81no—
208C2×D61—81no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—81no—
212C4×C61—82no—
213C4×C61#49312yes—
214C4×C61C22×D812yes—
215C4×C61C2×#13712yes—
216C4×C61#57812yes—
217Dic61—82no—
218Dic61—81no—
219#3226—611no—
220#3226D6111yescharacteristic, complemented
221#3169—611no—
222C2×#500—611no—
223#3152—611no—
224#3152—611no—
225#1722—612no—
226#1722—612no—
227#1713—612no—
228#1706—612no—
229C8×M16—611no—
230C2×C61—22no—
231C2×C61—161no—
232C2×C61C2×#49311yescharacteristic
233C2×C61#315211yescharacteristic
234C2×C61#315611yescharacteristic
235D61—161no—
236C2×#137—1222no—
237C2×#137—1222no—
238#532—611no—
239#532C2×D6111yescharacteristic
240C2×C4⋊Q8—611no—
241C2×C4⋊Q8C2×D6111yescharacteristic
242#500—612no—
243#500—612no—
244#500—611no—
245#500—612no—
246#500—1222no—
247#500C2×D6112yes—
248#497—1222no—
249#497—1222no—
250#494—612no—
251#494—612no—
252#493—611no—
253#492—612no—
254#492—612no—
255#486—611no—
256#486—611no—
257#486—612no—
258#486C2×D6112yes—
259#480—22no—
260#480—1222no—
261#474—22no—
262#474—1222no—
263#466—1222no—
264C4×D8—1222no—
265#431C2×D6111yescharacteristic
266C2×#107—1222no—
267C4×M16—611no—
268C2×C4×C8—611no—
269#322—612no—
270C82—612no—
271C61#1963011yescharacteristic, complemented, Sylow
272C2×#41—21no—
273C2×#41—1221no—
274C22×Q8—21no—
275C22×Q8—1221no—
276#137—2448no—
277#137—2448no—
278C2×Q16—1224no—
279C2×Q16—1224no—
280C2×SD16—1224no—
281C2×SD16—1224no—
282C2×D8—1222no—
283C2×M16—612no—
284C22×C8—612no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—612no—
289C4⋊Q8—611no—
290C4⋊Q8—1221no—
291C4⋊Q8—1221no—
292C4⋊Q8—1221no—
293C4⋊Q8C22×D6112yes—
294C4⋊Q8C22×D6111yescharacteristic
295#127—611no—
296#127C22×D6111yescharacteristic
297#124—21no—
298#124—1221no—
299#121—21no—
300#121—21no—
301#121—1221no—
302#121—1221no—
303C4×D4—21no—
304C4×D4—21no—
305C4×D4—1221no—
306C4×D4—1221no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1221no—
311C2×C42C22×D6111yescharacteristic
312#107—24no—
313#107—1224no—
314#107—1224no—
315#107—1222no—
316#107—1224no—
317C4⋊C8C22⋊D6112yes—
318#102—24no—
319#102—1222no—
320#102—1222no—
321#102—1224no—
322C22⋊C8C22⋊D6112yes—
323#97—612no—
324C4×C8—612no—
325C4×C8—611no—
326C4×C8—612no—
327C4×C8—611no—
328C4×C8C22×D6112yes—
329#41—44no—
330#41—2444no—
331C2×Q8—21no—
332C2×Q8—22no—
333C2×Q8—41no—
334C2×Q8—44no—
335C2×Q8—1221no—
336C2×Q8—1222no—
337C2×Q8—2444no—
338C2×Q8—2441no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—41no—
343C2×D4—1221no—
344C2×D4—1221no—
345C2×D4—1221no—
346C2×D4—2441no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—2441no—
350C22×C4—2441no—
351C22×C4D4×D6111yescharacteristic
352C22×C4D4×D6111yescharacteristic
353C22×C4C2×C22⋊D6111yescharacteristic
354Q16—2448no—
355Q16—2448no—
356SD16—2448no—
357SD16—2448no—
358D8—2444no—
359M16—618no—
360C2×C8—612no—
361C2×C8—614no—
362C2×C8—612no—
363C2×C8—1224no—
364C2×C8C2×C22⋊D6114yes—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—21no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—1221no—
372C4⋊C4—1222no—
373C4⋊C4—1222no—
374C4⋊C4—1221no—
375C4⋊C4—2441no—
376C4⋊C4—2441no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—2441no—
380C22⋊C4—2441no—
381C42D4×D6111yescharacteristic
382C42D4×D6111yescharacteristic
383C42C2×C22⋊D6111yescharacteristic
384C42C23×D6111yescharacteristic
385C23—41no—
386C23—2441no—
387C23#1451977311yescharacteristic
388Q8—44no—
389Q8—42no—
390Q8—82no—
391Q8—4882no—
392Q8—2444no—
393Q8—2442no—
394D4—42no—
395D4—42no—
396D4—42no—
397D4—82no—
398D4—4882no—
399D4—2442no—
400D4—2442no—
401D4—2442no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—4882no—
409C2×C4—4882no—
410C2×C4—2441no—
411C2×C4—2442no—
412C2×C4#1451977311yescharacteristic
413C2×C4#1451977311yescharacteristic
414C2×C4#1451977311yescharacteristic
415C2×C4D8×D6114yes—
416C2×C4C2×D4×D6111yescharacteristic
417C2×C4C2×D4×D6111yescharacteristic
418C2×C4C22×C22⋊D6111yescharacteristic
419C8—24no—
420C8—618no—
421C8—614no—
422C8—1222no—
423C22—21no—
424C22—21no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—4881no—
429C22—4881no—
430C22—4881no—
431C22#1684013712yes—
432C22C2×#1451977311yescharacteristic
433C4—21no—
434C4—21no—
435C4—81no—
436C4—82no—
437C4—4881no—
438C4—4882no—
439C4#1684013712yes—
440C4C2×D8×D6112yes—
441C4#137×D6112yes—
442C4#1684057712yes—
443C2—22no—
444C2—161no—
445C2—9761no—
446C2C2×#1684013711yescharacteristic
447C2#2145627411yescharacteristic
448C2#2145631011yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3226C2
socleC22×C61C2×#127
prime radical#21456394C2
hypercenter#3226D61
hypocenterC61#19630
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3226

Complements

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

NH ≅ G/Ntype
1. #21456942C2semidirect
3. C2×#16840165C2semidirect
4. #21456279C2semidirect
5. #21456276C2semidirect
6. #21445492C2semidirect
7. #21445489C2semidirect
8. #21445436C2semidirect
9. #21445410C2semidirect
10. #21444100C2semidirect
11. C61×#3226C2semidirect
155. #11758194D8semidirect
197. #92418#137semidirect
220. #3226D61semidirect
271. C61#19630semidirect

Maximal subgroups

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

subgroupindex
#214569422
#214563942
C2×#168401652
#214562792
#214562762
#214454922
#214454892
#214454362
#214454102
#214441002
C61×#32262
#1963061