Finite Group Databasebeta

← #29834538

Subgroups — #29834538

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840192C211yesmaximal, characteristic, complemented
2C2×#16840179C211yesmaximal, characteristic, complemented
3#21455489C211yesmaximal, characteristic, complemented
4#21447925C212yesmaximal, complemented
5#21447920C212yesmaximal, complemented
6#21447917C212yesmaximal, complemented
7#21447913C212yesmaximal, complemented
8C2×#16839589C211yesmaximal, characteristic, complemented
9#21447531C211yesmaximal, characteristic, complemented
10#21447529C211yesmaximal, characteristic, complemented
11C61×#3061C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13#16840340—21no—
14#16840192—24no—
15C2×#14519723C2211yescharacteristic
16C2×#14519721C2211yescharacteristic
17C2×#14519720C2211yescharacteristic
18C2×#14519718C2211yescharacteristic
19#16840179—24no—
20C2×#14519717C2211yescharacteristic
21C2×#14519717C2212yes—
22C22×#11758185C2212yes—
23C22×#11758185C2211yescharacteristic
24#16840008—22no—
25#16840003C2212yes—
26#16839998C2212yes—
27C2×#14519711C2211yescharacteristic
28#16839764—21no—
29#16839589C2212yes—
30#16839589C2212yes—
31#16839566C2212yes—
32#16839565C2212yes—
33#16839563C2212yes—
34#16839562C2212yes—
35C2×#135×C61C2211yescharacteristic
36C2×#131×C61C2211yescharacteristic
37C61×#537—21no—
38C61×#468—22no—
39C61×#467C2212yes—
40C61×#466C2212yes—
41C2×#107×C61C2211yescharacteristic
42C61×#417—21no—
43C2×#11758222—21no—
44#14519799—22no—
45#14519799—22no—
46C2×Dic244C2311yescharacteristic
47C2×#11758190—21no—
48C2×D488C2311yescharacteristic
49#14519756—22no—
50#14519753—41no—
51#14519751—22no—
52#14519749—42no—
53#14519748—41no—
54C2×C4⋊Dic61—21no—
55#14519723—22no—
56#14519721—22no—
57#14519720—22no—
58#14519718—22no—
59#14519717—22no—
60#14519717—24no—
61C2×#11758185—24no—
62C2×#11758185D414yescomplemented
63C2×#11758185C2312yes—
64C2×#11758185C2311yescharacteristic
65C2×#11758185C2311yescharacteristic
66#14519711—21no—
67#14519711C2312yes—
68#14519702—22no—
69#14519699—22no—
70C2×#41×C61—21no—
71C2×#41×C61D411yescharacteristic
72#135×C61—22no—
73#135×C61—22no—
74C2×Q16×C61C2311yescharacteristic
75C2×SD16×C61—21no—
76C2×D8×C61C2311yescharacteristic
77#131×C61—22no—
78#131×C61—22no—
79C2×M16×C61D412yes—
80C2×M16×C61D411yescharacteristic
81C22×C8×C61C2312yes—
82C22×C8×C61C2311yescharacteristic
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—41no—
85#121×C61—22no—
86#121×C61—41no—
87#121×C61—42no—
88C2×C4⋊C4×C61—21no—
89#107×C61—21no—
90#107×C61C2312yes—
91#103×C61—22no—
92#102×C61—22no—
93#41×C61—22no—
94#41×C61—22no—
95#41×C61—42no—
96#41×C61—42no—
97C2×Q8×C61—21no—
98C2×Q8×C61D811yescharacteristic
99C2×D4×C61—21no—
100C2×D4×C61—41no—
101C2×D4×C61—42no—
102C2×D4×C61D811yescharacteristic
103C2×D4×C61D812yes—
104C22×C4×C61—41no—
105C22×C4×C61—41no—
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D411yescharacteristic
108C2×C22⋊D61—41no—
109C2×C22⋊D61—42no—
110C22×Dic61—41no—
111#11758222—42no—
112#11758222—42no—
113C2×D244—21no—
114C2×C4×D61—41no—
115C2×Dic122—21no—
116Q16×C61—21no—
117Q16×C61—21no—
118SD16×C61—42no—
119D8×C61—21no—
120D8×C61—21no—
121M16×C61—22no—
122M16×C61—24no—
123C2×C8×C61—22no—
124C2×C8×C61—22no—
125C2×C8×C61C2×D412yes—
126C2×C8×C61C2×D411yescharacteristic
127C2×C8×C61C2×D412yes—
128C2×C8×C61#4112yes—
129C2×C8×C61C2411yescharacteristic
130C4⋊C4×C61—22no—
131C4⋊C4×C61—41no—
132C4⋊C4×C61—42no—
133C22⋊C4×C61—42no—
134C22⋊C4×C61—42no—
135C22⋊Dic61—42no—
136#11758198—42no—
137C4⋊Dic61—22no—
138C4⋊Dic61—41no—
139#11758196—42no—
140Dic244—21no—
141Dic244—21no—
142D488—21no—
143D488—21no—
144#11758190—42no—
145#11758185—22no—
146#11758185C2×D412yes—
147#11758185C2×D412yes—
148C23×C61—41no—
149C23×C61C2×D812yes—
150C23×C61C2×D811yescharacteristic
151C22×D61—41no—
152Q8×C61—22no—
153Q8×C61—41no—
154Q8×C61—41no—
155D4×C61—24no—
156D4×C61—22no—
157D4×C61—41no—
158D4×C61—41no—
159D4×C61—82no—
160D4×C61—162no—
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—42no—
164C2×C4×C61—41no—
165C2×C4×C61—82no—
166C2×C4×C61—161no—
167C2×C4×C61#12112yes—
168C2×C4×C61C2×D812yes—
169C2×C4×C61C2×D811yescharacteristic
170C2×C4×C61#13512yes—
171C2×C4×C61C22×D411yescharacteristic
172C22⋊D61—82no—
173C22⋊D61—162no—
174C2×Dic61—41no—
175C2×Dic61—42no—
176C2×Dic61—161no—
177D244—41no—
178D244—41no—
179C4×D61—82no—
180Dic122—41no—
181Dic122—41no—
182C8×C61—22no—
183C8×C61#12112yes—
184C8×C61C22×D411yescharacteristic
185C8×C61C2×#4111yescharacteristic
186#33611—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—22no—
190C22×C61—22no—
191C22×C61—22no—
192C22×C61—81no—
193C22×C61—81no—
194C22×C61—81no—
195C22×C61#46612yes—
196C22×C61C22×D811yescharacteristic
197C2×D61—81no—
198C2×D61—81no—
199C2×D61—81no—
200C4×C61—22no—
201C4×C61—82no—
202C4×C61—81no—
203C4×C61#46612yes—
204C4×C61C2×#12111yescharacteristic
205C4×C61C2×#13511yescharacteristic
206Dic61—82no—
207Dic61—81no—
208C2×#508—611no—
209C2×#504—611no—
210#3061—611no—
211#3061D6111yescharacteristic, complemented
212#2219—612no—
213#2218—612no—
214#2217—612no—
215#2217—612no—
216C2×#367—611no—
217#2139—611no—
218#2139—611no—
219C2×C61—22no—
220C2×C61—42no—
221C2×C61—161no—
222C2×C61C2×#46611yescharacteristic
223C2×C61#341611yescharacteristic
224C2×C61#341611yescharacteristic
225D61—161no—
226C2×#135—611no—
227C2×#135C2×D6111yescharacteristic
228C2×#131C2×D6111yescharacteristic
229#537—21no—
230#537—1221no—
231#508—1224no—
232C2×Q32—611no—
233C2×SD32—611no—
234C2×SD32—611no—
235C2×D16—611no—
236#504—1224no—
237C2×M32—611no—
238C2×M32—612no—
239C22×C16—611no—
240C22×C16—612no—
241#468—22no—
242#468—1222no—
243#467—612no—
244#467C2×D6112yes—
245#466—612no—
246#466C2×D6112yes—
247C2×#107—611no—
248C2×#107C2×D6111yescharacteristic
249#417—21no—
250#417—1221no—
251#367—612no—
252#367—612no—
253#358—612no—
254#358—612no—
255#357—612no—
256#357—612no—
257C61#3361111yescharacteristic, complemented, Sylow
258C2×#41—21no—
259C2×#41—1221no—
260C2×#41C22⋊D6111yescharacteristic
261#135—22no—
262#135—22no—
263#135—1222no—
264#135—1222no—
265C2×Q16—611no—
266C2×Q16C22×D6111yescharacteristic
267C2×SD16—21no—
268C2×SD16—1221no—
269C2×D8—611no—
270C2×D8C22×D6111yescharacteristic
271#131—22no—
272#131—22no—
273C2×M16C22⋊D6111yescharacteristic
274C2×M16C22⋊D6112yes—
275C22×C8C22×D6112yes—
276C22×C8C22×D6111yescharacteristic
277C22⋊Q8—22no—
278C22⋊Q8—41no—
279C22⋊Q8—1222no—
280C22⋊Q8—2441no—
281#121—22no—
282#121—41no—
283#121—42no—
284#121—1222no—
285#121—2441no—
286#121—2442no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—1221no—
289Q32—1222no—
290SD32—1222no—
291SD32—1222no—
292D16—1222no—
293M32—1224no—
294M32—1222no—
295C2×C16—614no—
296C2×C16—611no—
297C2×C16—611no—
298C2×C16—612no—
299C2×C16—1224no—
300#107—21no—
301#107—612no—
302#107—1221no—
303#107C22×D6112yes—
304#103—22no—
305#103—1222no—
306#102—22no—
307#102—1222no—
308#41—22no—
309#41—22no—
310#41—42no—
311#41—42no—
312#41—2442no—
313#41—2442no—
314C2×Q8—21no—
315C2×Q8—1221no—
316C2×Q8#1175819911yescharacteristic
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—42no—
320C2×D4—1221no—
321C2×D4—2442no—
322C2×D4—2441no—
323C2×D4#1175819912yes—
324C2×D4#1175819911yescharacteristic
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—2441no—
328C22×C4—2441no—
329C22×C4C2×C22⋊D6112yes—
330C22×C4C2×C22⋊D6111yescharacteristic
331Q16—21no—
332Q16—21no—
333Q16—1221no—
334Q16—1221no—
335SD16—42no—
336SD16—2442no—
337D8—21no—
338D8—21no—
339D8—1221no—
340D8—1221no—
341M16—22no—
342M16—24no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8D4×D6112yes—
346C2×C8#1175822412yes—
347C2×C8C2×C22⋊D6112yes—
348C2×C8C2×C22⋊D6111yescharacteristic
349C2×C8C23×D6111yescharacteristic
350C4⋊C4—22no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—1222no—
354C4⋊C4—2441no—
355C4⋊C4—2442no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—2442no—
359C22⋊C4—2442no—
360C16—612no—
361C16—612no—
362C16—1222no—
363C23—41no—
364C23—2441no—
365C23C2×#1175819912yes—
366C23C2×#1175819911yescharacteristic
367Q8—22no—
368Q8—41no—
369Q8—41no—
370Q8—2441no—
371Q8—2441no—
372D4—24no—
373D4—22no—
374D4—41no—
375D4—41no—
376D4—82no—
377D4—162no—
378D4—9762no—
379D4—4882no—
380D4—2441no—
381D4—2441no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—82no—
387C2×C4—161no—
388C2×C4—9761no—
389C2×C4—4882no—
390C2×C4—2442no—
391C2×C4—2441no—
392C2×C4#1451974912yes—
393C2×C4C2×#1175819912yes—
394C2×C4C2×#1175819911yescharacteristic
395C2×C4#1451979712yes—
396C2×C4C22×C22⋊D6111yescharacteristic
397C8—22no—
398C8#1451974912yes—
399C8C2×D4×D6111yescharacteristic
400C8C2×#1175822411yescharacteristic
401C22—21no—
402C22—21no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—4881no—
410C22—4881no—
411C22—4881no—
412C22#1683999612yes—
413C22C22×#1175819911yescharacteristic
414C4—22no—
415C4—82no—
416C4—81no—
417C4—4882no—
418C4—4881no—
419C4#1683999612yes—
420C4C2×#1451974911yescharacteristic
421C4C2×#1451979711yescharacteristic
422C2—22no—
423C2—42no—
424C2—161no—
425C2—9761no—
426C2C2×#1683999611yescharacteristic
427C2#2145896311yescharacteristic
428C2#2145896611yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3061C2
socleC22×C61C22×D8
hypercenter#3061D61
hypocenterC61#33611
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ #3061

Complements

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

NH ≅ G/Ntype
1. C2×#16840192C2semidirect
2. C2×#16840179C2semidirect
3. #21455489C2semidirect
4. #21447925C2semidirect
5. #21447920C2semidirect
6. #21447917C2semidirect
7. #21447913C2semidirect
8. C2×#16839589C2semidirect
9. #21447531C2semidirect
10. #21447529C2semidirect
11. C61×#3061C2semidirect
62. C2×#11758185D4semidirect
211. #3061D61semidirect
257. C61#33611semidirect

Maximal subgroups

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

subgroupindex
C2×#168401922
C2×#168401792
#214554892
#214479252
#214479202
#214479172
#214479132
C2×#168395892
#214475312
#214475292
C61×#30612
#3361161