Finite Group Databasebeta

← #29667898

Subgroups — #29667898

Order 15616.

836 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456972C211yesmaximal, characteristic, complemented
2#21456374C211yesmaximal, characteristic, complemented
3C2×#16840134C211yesmaximal, characteristic, complemented
4#21456212C211yesmaximal, characteristic, complemented
5#21456193C211yesmaximal, characteristic, complemented
6#21445471C212yesmaximal, complemented
7#21445457C212yesmaximal, complemented
8#21445419C212yesmaximal, complemented
9#21445408C212yesmaximal, complemented
10#21444100C211yesmaximal, characteristic, complemented
11C61×#3205C211yesmaximal, characteristic
12C22×#11758200—22no—
13#16840319C2211yescharacteristic
14C2×#14519777C2211yescharacteristic
15#16840167C2212yescomplemented
16#16840158C2211yescharacteristic, complemented
17#16840134—22no—
18#16840134C2212yes—
19#16840134C2212yes—
20#16840134C2211yescharacteristic, complemented
21#16840126—22no—
22#16840121—22no—
23#16840114C2211yescharacteristic, complemented
24#16840113C2212yescomplemented
25#16840112C2212yescomplemented
26#16840108C2212yes—
27#16840108C2212yes—
28#16840108C2212yescomplemented
29#16840108C2211yescharacteristic, complemented
30#16840098—22no—
31#16840058—22no—
32#16839989—22no—
33C4×#11758200—22no—
34#16839767—22no—
35#16839763—22no—
36C4×#11758194C2211yescharacteristic
37C2×C4×#92418C2211yescharacteristic
38#16839484C2212yescomplemented
39C8×#92418C2212yescomplemented
40C61×#532C2211yescharacteristic
41C2×#127×C61C2211yescharacteristic
42C61×#493C2212yes—
43C61×#486C2212yes—
44C61×#459—22no—
45C61×#447—22no—
46C61×#431C2211yescharacteristic
47C22×Dic122—21no—
48C2×#11758200—22no—
49C2×#11758200—24no—
50C2×#11758200—24no—
51C2×#11758200—48no—
52C2×#11758194D412yes—
53C22×#92418D412yes—
54#14519777—21no—
55#14519777—21no—
56#14519777—21no—
57#14519777C2312yes—
58#14519777C2311yescharacteristic
59#14519777C2311yescharacteristic
60#14519769—21no—
61#14519756—21no—
62#14519756—21no—
63C4×Dic122—21no—
64C4×Dic122—21no—
65#14519736—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68#14519711—24no—
69#14519708—22no—
70#14519702—24no—
71#14519701—24no—
72#14519700—22no—
73#14519698—22no—
74#14519698—24no—
75#14519698—24no—
76#14519698—24no—
77#14519688D412yes—
78C4×#92418D412yes—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C4×#92418C2312yes—
82C2×#41×C61—21no—
83C22×D4×C61—21no—
84C22×D4×C61—21no—
85C22×D4×C61—21no—
86C2×D8×C61—24no—
87C4⋊Q8×C61C2311yescharacteristic
88#127×C61—21no—
89#127×C61—21no—
90#127×C61—21no—
91#127×C61C2312yes—
92#127×C61C2311yescharacteristic
93#124×C61—21no—
94#121×C61—21no—
95#121×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61D412yes—
100#102×C61—24no—
101C22⋊C8×C61D412yes—
102C4×C8×C61C2312yes—
103C24×C61—41no—
104#41×C61—44no—
105C2×Q8×C61—21no—
106C2×D4×C61—22no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—44no—
113C2×D4×C61—41no—
114C2×D4×C61—41no—
115C2×D4×C61—41no—
116C2×D4×C61—41no—
117C2×D4×C61—44no—
118C2×D4×C61—82no—
119C22×C4×C61—41no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×Dic61—41no—
124C2×Dic122—22no—
125C2×Dic122—21no—
126C2×Dic122—21no—
127C2×Dic122—44no—
128C2×Dic122—41no—
129D8×C61—84no—
130C2×C8×C61C2×D414yes—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C22⋊C4×C61—41no—
134C22⋊C4×C61—41no—
135C42×C61C2×D411yescharacteristic
136C42×C61C2×D411yescharacteristic
137C42×C61C2×D411yescharacteristic
138C42×C61C2411yescharacteristic
139C22⋊Dic61—41no—
140#11758200—48no—
141#11758200—44no—
142#11758200—48no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—22no—
146C4⋊Dic61—21no—
147C4⋊Dic61—22no—
148C4⋊Dic61—21no—
149C4⋊Dic61—41no—
150C4⋊Dic61—41no—
151#11758196—41no—
152#11758196—41no—
153C4×Dic61—41no—
154#11758194—24no—
155C2×#92418—24no—
156C2×#92418C2×D412yes—
157C2×#92418C2×D414yes—
158C2×#92418C2×D412yes—
159C23×C61—41no—
160C23×C61—42no—
161C23×C61—82no—
162C23×C61—82no—
163C23×C61—82no—
164C23×C61#12711yescharacteristic
165Q8×C61—42no—
166D4×C61—44no—
167D4×C61—42no—
168D4×C61—44no—
169D4×C61—42no—
170D4×C61—42no—
171D4×C61—44no—
172D4×C61—82no—
173D4×C61—82no—
174D4×C61—82no—
175D4×C61—82no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—41no—
179C2×C4×C61—82no—
180C2×C4×C61#12711yescharacteristic
181C2×C4×C61#12711yescharacteristic
182C2×C4×C61#12711yescharacteristic
183C2×C4×C61#13514yes—
184C2×C4×C61C22×D411yescharacteristic
185C2×C4×C61C22×D411yescharacteristic
186C2×C4×C61C22×D411yescharacteristic
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—42no—
190C2×Dic61—82no—
191Dic122—44no—
192Dic122—42no—
193Dic122—42no—
194Dic122—82no—
195C8×C61—24no—
196#92418—24no—
197#92418—22no—
198#92418AGL(1,ℤ/8)14yes—
199#19625—611nomaximal, Sylow
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—81no—
203C22×C61—82no—
204C22×C61—82no—
205C22×C61—81no—
206C22×C61—82no—
207C22×C61—81no—
208C22×C61—164no—
209C22×C61#49512yes—
210C22×C61C2×#12711yescharacteristic
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—81no—
214C4×C61#49512yes—
215C4×C61C2×#13512yes—
216C4×C61C2×AGL(1,ℤ/8)12yes—
217C4×C61#57712yes—
218Dic61—82no—
219Dic61—81no—
220Dic61—81no—
221#3230—611no—
222#3205D6111yescharacteristic
223#3167—611no—
224C2×#492—611no—
225#3141—611no—
226#3138—611no—
227#1719—612no—
228#1717—612no—
229#1708—612no—
230#1706—612no—
231C8×M16—611no—
232C2×C61—22no—
233C2×C61—161no—
234C2×C61—162no—
235C2×C61C2×#49511yescharacteristic
236C2×C61#315111yescharacteristic
237C2×C61#315411yescharacteristic
238C22×SD16—1222no—
239#533—611no—
240#532C2×D6111yescharacteristic
241C2×C4⋊Q8—611no—
242C2×#127C2×D6111yescharacteristic
243#500—612no—
244#498—611no—
245#493C2×D6112yes—
246#492—612no—
247#492—612no—
248#492—611no—
249#492—1222no—
250#490—1222no—
251#489—1222no—
252#487—612no—
253#487—611no—
254#487—612no—
255#486—612no—
256#486—612no—
257#486—612no—
258#486—611no—
259#486C2×D6112yes—
260#484—1222no—
261#477—1222no—
262#465—1222no—
263#459—22no—
264#447—22no—
265C4×SD16—1222no—
266#431C2×D6111yescharacteristic
267#418—1222no—
268#417—1222no—
269C4×M16—611no—
270C2×C4×C8—611no—
271#322—612no—
272C82—612no—
273C61#1962511yescharacteristic, complemented, Sylow
274C2×#41—21no—
275C22×Q8—1221no—
276C22×D4—21no—
277C22×D4—21no—
278C22×D4—21no—
279C2×SD16—1224no—
280C2×SD16—1224no—
281C2×SD16—1222no—
282C2×SD16—2448no—
283C2×D8—24no—
284C2×M16—612no—
285C22×C8—612no—
286C4⋊Q8—612no—
287C4⋊Q8—611no—
288C4⋊Q8—611no—
289C4⋊Q8—1221no—
290C4⋊Q8—1221no—
291C4⋊Q8—1221no—
292C4⋊Q8C22×D6111yescharacteristic
293#127—21no—
294#127—21no—
295#127—21no—
296#127C22×D6112yes—
297#127C22×D6111yescharacteristic
298#125—1221no—
299#124—21no—
300C22⋊Q8—1221no—
301C22⋊Q8—1221no—
302#121—21no—
303#121—21no—
304C4×Q8—1221no—
305C4×Q8—1221no—
306C4×D4—21no—
307C4×D4—21no—
308#117—1221no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1221no—
311C2×C42C22×D6111yescharacteristic
312#107—1224no—
313#106—1222no—
314C4⋊C8C22⋊D6112yes—
315#103—1222no—
316#103—1224no—
317#103—1224no—
318#102—24no—
319#102—1224no—
320#102—1224no—
321#102—1224no—
322#102—1222no—
323C22⋊C8C22⋊D6112yes—
324#97—612no—
325C4×C8—612no—
326C4×C8—611no—
327C4×C8—612no—
328C4×C8—611no—
329C4×C8C22×D6112yes—
330C24—41no—
331#41—44no—
332C2×Q8—21no—
333C2×Q8—1221no—
334C2×Q8—1222no—
335C2×Q8—1221no—
336C2×Q8—2444no—
337C2×Q8—2441no—
338C2×D4—21no—
339C2×D4—22no—
340C2×D4—21no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—21no—
344C2×D4—44no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—44no—
350C2×D4—82no—
351C22×C4—41no—
352C22×C4—2441no—
353C22×C4D4×D6111yescharacteristic
354C22×C4D4×D6111yescharacteristic
355C22×C4C2×C22⋊D6111yescharacteristic
356SD16—2444no—
357SD16—2448no—
358SD16—2448no—
359D8—84no—
360M16—1224no—
361C2×C8—612no—
362C2×C8—612no—
363C2×C8—614no—
364C2×C8—1224no—
365C2×C8C2×C22⋊D6114yes—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—1221no—
369C4⋊C4—1221no—
370C4⋊C4—1222no—
371C4⋊C4—1221no—
372C4⋊C4—1222no—
373C4⋊C4—1221no—
374C4⋊C4—2441no—
375C4⋊C4—2441no—
376C4⋊C4—2441no—
377C4⋊C4—2441no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—2441no—
381C42—2441no—
382C42D4×D6111yescharacteristic
383C42D4×D6111yescharacteristic
384C42C2×C22⋊D6111yescharacteristic
385C42C23×D6111yescharacteristic
386C23—41no—
387C23—42no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23#1451977311yescharacteristic
392Q8—42no—
393Q8—4882no—
394Q8—2444no—
395Q8—2442no—
396Q8—2442no—
397D4—44no—
398D4—42no—
399D4—44no—
400D4—42no—
401D4—42no—
402D4—44no—
403D4—82no—
404D4—82no—
405D4—82no—
406D4—82no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—82no—
411C2×C4—4882no—
412C2×C4—2441no—
413C2×C4—2441no—
414C2×C4—2442no—
415C2×C4#1451977311yescharacteristic
416C2×C4#1451977311yescharacteristic
417C2×C4#1451977311yescharacteristic
418C2×C4#1451979614yes—
419C2×C4C2×D4×D6111yescharacteristic
420C2×C4C2×D4×D6111yescharacteristic
421C2×C4C22×C22⋊D6111yescharacteristic
422C8—24no—
423C8—614no—
424C8—1224no—
425C8—1222no—
426C22—21no—
427C22—21no—
428C22—81no—
429C22—82no—
430C22—82no—
431C22—81no—
432C22—82no—
433C22—81no—
434C22—164no—
435C22#1684014512yes—
436C22C2×#1451977311yescharacteristic
437C4—21no—
438C4—21no—
439C4—81no—
440C4—4882no—
441C4—4881no—
442C4—4881no—
443C4#1684014512yes—
444C4C2×#1451979612yes—
445C4AGL(1,ℤ/8)×D6112yes—
446C4#1684057312yes—
447C2—22no—
448C2—162no—
449C2—161no—
450C2C2×#1684014511yescharacteristic
451C2#2145626711yescharacteristic
452C2#2145629111yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3205C2
socleC22×C61C2×#127
hypercenter#3205D61
hypocenterC61#19625
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #21456972C2semidirect
2. #21456374C2semidirect
3. C2×#16840134C2semidirect
4. #21456212C2semidirect
5. #21456193C2semidirect
6. #21445471C2semidirect
7. #21445457C2semidirect
8. #21445419C2semidirect
9. #21445408C2semidirect
10. #21444100C2semidirect
15. #16840167C22semidirect
16. #16840158C22semidirect
20. #16840134C22semidirect
23. #16840114C22semidirect
24. #16840113C22semidirect
25. #16840112C22semidirect
28. #16840108C22semidirect
29. #16840108C22semidirect
38. #16839484C22semidirect
39. C8×#92418C22semidirect
273. C61#19625semidirect

Maximal subgroups

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

subgroupindex
#214569722
#214563742
C2×#168401342
#214562122
#214561932
#214454712
#214454572
#214454192
#214454082
#214441002
C61×#32052
#1962561