Finite Group Databasebeta

← #29668008

Subgroups — #29668008

Order 15616.

750 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456960C211yesmaximal, characteristic, complemented
2#21456760C211yesmaximal, characteristic
3#21456401C211yesmaximal, characteristic, complemented
4#21456200C211yesmaximal, characteristic, complemented
5C2×#16840108C211yesmaximal, characteristic
6#21445480C212yesmaximal, complemented
7#21445469C212yesmaximal, complemented
8#21445430C212yesmaximal, complemented
9#21445419C212yesmaximal, complemented
10M16×#92418C211yesmaximal, characteristic, complemented
11C61×#3138C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13C2×#14519777C2211yescharacteristic
14#16840165C2211yescharacteristic, complemented
15#16840158C2212yescomplemented
16#16840141C2211yescharacteristic, complemented
17#16840134C2212yes—
18#16840131—22no—
19#16840114C2211yescharacteristic, complemented
20#16840113C2212yescomplemented
21#16840113C2212yescomplemented
22#16840112C2211yescharacteristic, complemented
23#16840112C2212yescomplemented
24#16840108—22no—
25#16840108C2212yes—
26#16840108C2212yes—
27#16840108C2212yes—
28#16840090—22no—
29#16840047—22no—
30#16840001—22no—
31#16839959—22no—
32#16839913—22no—
33C4×#11758202—22no—
34#16839815C2211yescharacteristic
35#16839772—22no—
36C2×#14519698—22no—
37C2×C4×#92418C2211yescharacteristic
38#16839483C2212yescomplemented
39C8×#92418C2212yescomplemented
40C61×#533C2211yescharacteristic
41C2×#127×C61C2211yescharacteristic
42C61×#490—22no—
43C61×#486C2212yes—
44C61×#486C2212yes—
45C61×#418—22no—
46C4×M16×C61C2211yescharacteristic
47C22×Dic122—21no—
48C2×#11758202—22no—
49C2×#11758200—24no—
50C22×#92418D412yes—
51#14519777—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777C2312yes—
55#14519777C2311yescharacteristic
56#14519777C2311yescharacteristic
57#14519769—21no—
58#14519756—21no—
59#14519756—21no—
60C4×Dic122—21no—
61C4×Dic122—21no—
62#14519736—21no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—21no—
65#14519710—22no—
66#14519708—24no—
67C4⋊#92418D412yes—
68#14519702—24no—
69#14519702—24no—
70#14519701—22no—
71#14519701—24no—
72#14519700—22no—
73#14519698—24no—
74#14519698—24no—
75#14519698—24no—
76C22⋊#92418D412yes—
77C4×#92418D412yes—
78C4×#92418C2311yescharacteristic
79C4×#92418C2311yescharacteristic
80C4×#92418C2312yes—
81C22×D4×C61—21no—
82C22×D4×C61—21no—
83C22×D4×C61—21no—
84C2×M16×C61D412yes—
85C4⋊Q8×C61C2311yescharacteristic
86C4⋊Q8×C61C2311yescharacteristic
87#127×C61—21no—
88#127×C61—21no—
89#127×C61—21no—
90#127×C61C2312yes—
91#125×C61—21no—
92C22⋊Q8×C61—21no—
93C22⋊Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96#117×C61—21no—
97C2×C42×C61C2311yescharacteristic
98#102×C61—24no—
99#102×C61—24no—
100#97×C61D412yes—
101C4×C8×C61C2312yes—
102C24×C61—41no—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—44no—
109C2×D4×C61—41no—
110C2×D4×C61—41no—
111C2×D4×C61—41no—
112C2×D4×C61—44no—
113C2×D4×C61—82no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×Dic61—41no—
118C2×Dic122—22no—
119C2×Dic122—21no—
120C2×Dic122—21no—
121C2×Dic122—44no—
122C2×Dic122—41no—
123M16×C61—24no—
124C2×C8×C61C2×D414yes—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—41no—
130C4⋊C4×C61—41no—
131C22⋊C4×C61—41no—
132C42×C61—41no—
133C42×C61C2×D411yescharacteristic
134C42×C61C2×D411yescharacteristic
135C42×C61C2×D411yescharacteristic
136C42×C61C2411yescharacteristic
137C22⋊Dic61—41no—
138#11758202—44no—
139#11758200—84no—
140C4⋊Dic61—22no—
141C4⋊Dic61—21no—
142C4⋊Dic61—22no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—41no—
147C4⋊Dic61—41no—
148#11758196—41no—
149#11758196—41no—
150C4×Dic61—41no—
151C2×#92418—42no—
152C2×#92418C2×D412yes—
153C2×#92418C2×D412yes—
154C2×#92418C2×D414yes—
155C23×C61—42no—
156C23×C61—82no—
157C23×C61—82no—
158C23×C61—82no—
159C23×C61#12711yescharacteristic
160Q8×C61—42no—
161Q8×C61—42no—
162D4×C61—44no—
163D4×C61—44no—
164D4×C61—44no—
165D4×C61—82no—
166D4×C61—82no—
167D4×C61—82no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
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
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—42no—
182C2×Dic61—82no—
183Dic122—44no—
184Dic122—42no—
185Dic122—42no—
186Dic122—82no—
187C8×C61—24no—
188#92418—22no—
189#92418—22no—
190#92418—24no—
191#19637—611nomaximal, Sylow
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—82no—
195C22×C61—82no—
196C22×C61—82no—
197C22×C61—164no—
198C22×C61#49512yes—
199C22×C61C2×#12711yescharacteristic
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—81no—
203C4×C61—81no—
204C4×C61#49512yes—
205C4×C61C2×#13512yes—
206C4×C61#57512yes—
207C4×C61#57712yes—
208Dic61—82no—
209Dic61—81no—
210Dic61—81no—
211#3229—611no—
212#3208—611no—
213#3170—611no—
214#3139—611no—
215#3138D6111yescharacteristic
216C2×#486—611no—
217#1720—612no—
218#1719—612no—
219#1711—612no—
220#1708—612no—
221C8×M16—611no—
222C2×C61—22no—
223C2×C61—162no—
224C2×C61C2×#49511yescharacteristic
225C2×C61#315311yescharacteristic
226C2×C61#315411yescharacteristic
227#533—611no—
228#533C2×D6111yescharacteristic
229C2×C4⋊Q8—611no—
230C2×#127C2×D6111yescharacteristic
231#500—611no—
232#498—612no—
233#494—611no—
234#492—612no—
235#491—1222no—
236#490—22no—
237#487—611no—
238#487—611no—
239#487—612no—
240#487—612no—
241#487—612no—
242#486—612no—
243#486—612no—
244#486—612no—
245#486—1222no—
246#486C2×D6112yes—
247#486C2×D6112yes—
248#483—1222no—
249#475—1222no—
250#467—1222no—
251#460—1222no—
252#450—1222no—
253C4×Q16—1222no—
254#431—611no—
255#419—1222no—
256#418—22no—
257C2×#102—1222no—
258C4×M16C2×D6111yescharacteristic
259C2×C4×C8—611no—
260#322—612no—
261C82—612no—
262C61#1963711yescharacteristic, complemented, Sylow
263C22×Q8—1221no—
264C22×D4—21no—
265C22×D4—21no—
266C22×D4—21no—
267C2×Q16—1222no—
268C2×SD16—1224no—
269C2×M16C22⋊D6112yes—
270C22×C8—612no—
271C4⋊Q8—612no—
272C4⋊Q8—611no—
273C4⋊Q8—611no—
274C4⋊Q8—1221no—
275C4⋊Q8—1221no—
276C4⋊Q8—1221no—
277C4⋊Q8C22×D6111yescharacteristic
278C4⋊Q8C22×D6111yescharacteristic
279#127—21no—
280#127—21no—
281#127—21no—
282#127C22×D6112yes—
283#125—21no—
284#125—1221no—
285C22⋊Q8—21no—
286C22⋊Q8—21no—
287C22⋊Q8—1221no—
288C22⋊Q8—1221no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×Q8—1221no—
292C4×Q8—1221no—
293#117—21no—
294#117—1221no—
295C2×C4⋊C4—1221no—
296C2×C4⋊C4—1221no—
297C2×C42C22×D6111yescharacteristic
298#107—1222no—
299#106—1224no—
300C4⋊C8—612no—
301#103—1222no—
302#103—1222no—
303#103—1224no—
304#103—1224no—
305#103—1224no—
306#102—24no—
307#102—24no—
308#102—1224no—
309#102—1224no—
310#102—1224no—
311C22⋊C8—612no—
312#97C22⋊D6112yes—
313C4×C8—612no—
314C4×C8—611no—
315C4×C8—612no—
316C4×C8—611no—
317C4×C8C22×D6112yes—
318C24—41no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—1221no—
322C2×Q8—1221no—
323C2×Q8—1222no—
324C2×Q8—2444no—
325C2×Q8—2441no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—44no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—44no—
334C2×D4—82no—
335C22×C4—2441no—
336C22×C4D4×D6111yescharacteristic
337C22×C4D4×D6111yescharacteristic
338C22×C4C2×C22⋊D6111yescharacteristic
339Q16—2444no—
340SD16—4884no—
341M16—24no—
342C2×C8—612no—
343C2×C8—612no—
344C2×C8—614no—
345C2×C8—2442no—
346C2×C8C2×C22⋊D6114yes—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—1222no—
354C4⋊C4—1221no—
355C4⋊C4—1222no—
356C4⋊C4—1221no—
357C4⋊C4—1221no—
358C4⋊C4—1221no—
359C4⋊C4—2441no—
360C4⋊C4—2441no—
361C4⋊C4—2441no—
362C4⋊C4—2441no—
363C22⋊C4—41no—
364C22⋊C4—2441no—
365C42—41no—
366C42—2441no—
367C42D4×D6111yescharacteristic
368C42D4×D6111yescharacteristic
369C42C2×C22⋊D6111yescharacteristic
370C42C23×D6111yescharacteristic
371C23—42no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23#1451977311yescharacteristic
376Q8—42no—
377Q8—42no—
378Q8—4882no—
379Q8—2444no—
380Q8—2442no—
381Q8—2442no—
382D4—44no—
383D4—44no—
384D4—44no—
385D4—82no—
386D4—82no—
387D4—82no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—4882no—
393C2×C4—2441no—
394C2×C4—2441no—
395C2×C4—2442no—
396C2×C4#1451977311yescharacteristic
397C2×C4#1451977311yescharacteristic
398C2×C4#1451977311yescharacteristic
399C2×C4#1451979714yes—
400C2×C4C2×D4×D6111yescharacteristic
401C2×C4C2×D4×D6111yescharacteristic
402C2×C4C22×C22⋊D6111yescharacteristic
403C8—24no—
404C8—1222no—
405C8—1222no—
406C8—1224no—
407C22—21no—
408C22—21no—
409C22—82no—
410C22—82no—
411C22—82no—
412C22—164no—
413C22#1684014412yes—
414C22C2×#1451977311yescharacteristic
415C4—21no—
416C4—21no—
417C4—81no—
418C4—81no—
419C4—4882no—
420C4—4881no—
421C4—4881no—
422C4#1684014412yes—
423C4C2×#1451979712yes—
424C4#1684055812yes—
425C4#1684057212yes—
426C2—22no—
427C2—162no—
428C2C2×#1684014411yescharacteristic
429C2#2145628111yescharacteristic
430C2#2145629311yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3138C2
socleC22×C61C2×#127
prime radicalC2×#127×C61C22
hypercenter#3138D61
hypocenterC61#19637
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
1. #21456960C2semidirect
3. #21456401C2semidirect
4. #21456200C2semidirect
6. #21445480C2semidirect
7. #21445469C2semidirect
8. #21445430C2semidirect
9. #21445419C2semidirect
10. M16×#92418C2semidirect
14. #16840165C22semidirect
15. #16840158C22semidirect
16. #16840141C22semidirect
19. #16840114C22semidirect
20. #16840113C22semidirect
21. #16840113C22semidirect
22. #16840112C22semidirect
23. #16840112C22semidirect
38. #16839483C22semidirect
39. C8×#92418C22semidirect
262. C61#19637semidirect

Maximal subgroups

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

subgroupindex
#214569602
#214567602
#214564012
#214562002
C2×#168401082
#214454802
#214454692
#214454302
#214454192
M16×#924182
C61×#31382
#1963761