Finite Group Databasebeta

← #29657819

Subgroups — #29657819

Order 15616.

1058 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457037C212yesmaximal, complemented
2C2×#16840134C211yesmaximal, characteristic, complemented
3#21444946C214yesmaximal, complemented
4#21444940C212yesmaximal, complemented
5#21444935C212yesmaximal, complemented
6#21444533C211yesmaximal, characteristic, complemented
7#21444279C212yesmaximal, complemented
8C61×#3543C211yesmaximal, characteristic
9C22×#11758200—22no—
10C2×#14519777C2211yescharacteristic, complemented
11#16840172—22no—
12#16840134—22no—
13#16840134C2212yes—
14#16840134C2212yescomplemented
15#16840121—22no—
16#16840090—22no—
17#16840073—22no—
18#16840057C2214yescomplemented
19#16840052C2214yescomplemented
20#16839819C2212yescomplemented
21C2×C4×#92418C2211yescharacteristic, complemented
22#16839506C2212yes—
23#16839506C2212yescomplemented
24Q8⋊#92418C2214yescomplemented
25D4⋊#92418C2214yescomplemented
26C61×#560—22no—
27C61×#559C2211yescharacteristic
28C61×#550C2212yes—
29C61×#546—22no—
30D42×C61—22no—
31D42×C61C2211yescharacteristic
32C61×#540—22no—
33C61×#535—22no—
34C61×#534—22no—
35C2×#127×C61C2211yescharacteristic
36C61×#529C2212yes—
37C22×Dic122—21no—
38C2×#11758200—24no—
39C2×#11758200—24no—
40C2×#11758200—48no—
41C22×#92418—21no—
42#14519777—22no—
43#14519777—21no—
44#14519777C2312yes—
45C2×C4⋊Dic61—22no—
46#14519710—24no—
47#14519708—24no—
48C4⋊#92418C2314yes—
49#14519701—24no—
50#14519698—24no—
51C22⋊#92418—22no—
52#14519688—22no—
53C4×#92418—21no—
54C4×#92418C2312yes—
55C2×#41×C61—22no—
56C22×D4×C61—22no—
57C22×D4×C61—22no—
58C22×D4×C61—22no—
59C22×D4×C61D412yes—
60#127×C61—22no—
61#127×C61—22no—
62#127×C61D412yes—
63#127×C61C2311yescharacteristic
64#127×C61C2311yescharacteristic
65#126×C61—22no—
66#125×C61D412yes—
67#124×C61—22no—
68#124×C61D412yes—
69#123×C61—22no—
70#123×C61—24no—
71#123×C61—22no—
72#123×C61—22no—
73C22⋊Q8×C61D412yes—
74#121×C61—24no—
75#121×C61—24no—
76#121×C61—24no—
77#121×C61—24no—
78#121×C61—22no—
79#121×C61—22no—
80#121×C61D412yes—
81#120×C61—22no—
82#120×C61—24no—
83#120×C61—24no—
84#120×C61—22no—
85C4×Q8×C61C2312yes—
86C4×D4×C61—22no—
87C4×D4×C61—22no—
88C4×D4×C61—22no—
89C4×D4×C61C2312yes—
90#117×C61—22no—
91C2×C22⋊C4×C61—22no—
92C2×C42×C61C2311yescharacteristic
93C24×C61—22no—
94C24×C61—22no—
95#41×C61—84no—
96C2×Q8×C61C2×D412yes—
97C2×D4×C61—22no—
98C2×D4×C61—24no—
99C2×D4×C61—24no—
100C2×D4×C61—24no—
101C2×D4×C61—22no—
102C2×D4×C61—24no—
103C2×D4×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61—44no—
106C2×D4×C61—84no—
107C2×D4×C61—84no—
108C2×D4×C61—84no—
109C2×D4×C61C2×D412yes—
110C2×D4×C61C2×D412yes—
111C2×D4×C61C2×D412yes—
112C22×C4×C61—24no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D412yes—
117C22×Dic61—41no—
118C2×Dic122—22no—
119C2×Dic122—44no—
120C2×Dic122—41no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61C2×D412yes—
126C4⋊C4×C61C2×D412yes—
127C22⋊C4×C61—22no—
128C22⋊C4×C61—22no—
129C22⋊C4×C61—24no—
130C22⋊C4×C61—24no—
131C22⋊C4×C61—22no—
132C22⋊C4×C61—22no—
133C22⋊C4×C61—22no—
134C22⋊C4×C61—22no—
135C22⋊C4×C61—22no—
136C42×C61—22no—
137C42×C61C2×D412yes—
138C42×C61C2×D411yescharacteristic
139C42×C61C2411yescharacteristic
140#11758200—48no—
141#11758200—48no—
142C4⋊Dic61—24no—
143C4⋊Dic61—42no—
144C2×#92418—24no—
145C2×#92418—22no—
146C2×#92418—24no—
147C23×C61—22no—
148C23×C61—22no—
149C23×C61—22no—
150C23×C61—44no—
151C23×C61—84no—
152C23×C61—84no—
153C23×C61—82no—
154C23×C61#12011yescharacteristic
155Q8×C61—24no—
156D4×C61—24no—
157D4×C61—42no—
158D4×C61—42no—
159D4×C61—44no—
160D4×C61—44no—
161D4×C61—44no—
162D4×C61—82no—
163D4×C61—84no—
164D4×C61—84no—
165D4×C61—84no—
166D4×C61—84no—
167C2×C4×C61—22no—
168C2×C4×C61—24no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—44no—
173C2×C4×C61—82no—
174C2×C4×C61—84no—
175C2×C4×C61#12012yes—
176C2×C4×C61#12011yescharacteristic
177C2×C4×C61C22×D411yescharacteristic
178C2×C4×C61C22×D412yes—
179C2×Dic61—42no—
180C2×Dic61—82no—
181Dic122—44no—
182Dic122—82no—
183#92418—24no—
184#92418—44no—
185#18683—611nomaximal, Sylow
186C22×C61—22no—
187C22×C61—42no—
188C22×C61—44no—
189C22×C61—42no—
190C22×C61—44no—
191C22×C61—42no—
192C22×C61—42no—
193C22×C61—42no—
194C22×C61—164no—
195C22×C61—164no—
196C22×C61#45312yes—
197C22×C61C2×#12011yescharacteristic
198C4×C61—22no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—42no—
202C4×C61—42no—
203C4×C61—42no—
204C4×C61#57714yes—
205Dic61—82no—
206#3543D6111yescharacteristic
207#3238—612no—
208C2×#492—611no—
209#1631—614no—
210#1630—612no—
211#1629—612no—
212#1555—611no—
213#1494—612no—
214C2×C61—22no—
215C2×C61—82no—
216C2×C61—82no—
217C2×C61—82no—
218C2×C61#300912yes—
219C2×C61C2×#45311yescharacteristic
220C22×SD16—1222no—
221#560—22no—
222#559C2×D6111yescharacteristic
223#550C2×D6112yes—
224#546—22no—
225D42—22no—
226D42C2×D6111yescharacteristic
227#540—22no—
228#535—22no—
229#534—22no—
230C2×C4⋊Q8—611no—
231C2×#127C2×D6111yescharacteristic
232#529C2×D6112yes—
233#501—1222no—
234#492—612no—
235#492—612no—
236#492—1222no—
237#489—1222no—
238#483—1222no—
239#480—1222no—
240#477—614no—
241#476—614no—
242#432—612no—
243C2×C4×C8—611no—
244#331—612no—
245#331—612no—
246Q8⋊C8—614no—
247D4⋊C8—614no—
248C61#1868311yescharacteristic, complemented, Sylow
249C2×#41—22no—
250C22×Q8—1221no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4C22⋊D6112yes—
255C2×SD16—1224no—
256C2×SD16—1224no—
257C2×SD16—2448no—
258C22×C8—1221no—
259C4⋊Q8—612no—
260C4⋊Q8—1222no—
261C4⋊Q8—1221no—
262#127—22no—
263#127—22no—
264#127C22⋊D6112yes—
265#127C22×D6111yescharacteristic
266#127C22×D6111yescharacteristic
267#126—22no—
268#125C22⋊D6112yes—
269#124—22no—
270#124C22⋊D6112yes—
271#123—22no—
272#123—24no—
273#123—22no—
274#123—22no—
275C22⋊Q8C22⋊D6112yes—
276#121—24no—
277#121—24no—
278#121—24no—
279#121—24no—
280#121—22no—
281#121—22no—
282#121C22⋊D6112yes—
283#120—22no—
284#120—24no—
285#120—24no—
286#120—22no—
287C4×Q8C22×D6112yes—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4C22×D6112yes—
292#117—22no—
293C2×C4⋊C4—1222no—
294C2×C22⋊C4—22no—
295C2×C42C22×D6111yescharacteristic
296#107—1224no—
297#106—1224no—
298C4⋊C8—614no—
299#103—1224no—
300#102—1224no—
301C22⋊C8—1222no—
302#97—1222no—
303C4×C8—612no—
304C4×C8—1221no—
305C24—22no—
306C24—22no—
307#41—84no—
308C2×Q8—1222no—
309C2×Q8—2444no—
310C2×Q8—2441no—
311C2×Q8C2×C22⋊D6112yes—
312C2×D4—22no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—24no—
320C2×D4—44no—
321C2×D4—84no—
322C2×D4—84no—
323C2×D4—84no—
324C2×D4C2×C22⋊D6112yes—
325C2×D4C2×C22⋊D6112yes—
326C2×D4C2×C22⋊D6112yes—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—2441no—
331C22×C4C2×C22⋊D6112yes—
332C22×C4C2×C22⋊D6111yescharacteristic
333SD16—2448no—
334SD16—2448no—
335C2×C8—1224no—
336C2×C8—1222no—
337C2×C8—1224no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—1224no—
343C4⋊C4—2442no—
344C4⋊C4C2×C22⋊D6112yes—
345C4⋊C4C2×C22⋊D6112yes—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C42—22no—
356C42C2×C22⋊D6111yescharacteristic
357C42C2×C22⋊D6112yes—
358C42C23×D6111yescharacteristic
359C23—22no—
360C23—22no—
361C23—22no—
362C23—44no—
363C23—84no—
364C23—82no—
365C23—84no—
366C23#1451974611yescharacteristic
367Q8—24no—
368Q8—4882no—
369Q8—2444no—
370D4—24no—
371D4—42no—
372D4—42no—
373D4—44no—
374D4—44no—
375D4—44no—
376D4—82no—
377D4—84no—
378D4—84no—
379D4—84no—
380D4—84no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—44no—
387C2×C4—82no—
388C2×C4—84no—
389C2×C4—4882no—
390C2×C4—2442no—
391C2×C4#1451974612yes—
392C2×C4#1451974611yescharacteristic
393C2×C4C22×C22⋊D6111yescharacteristic
394C2×C4C22×C22⋊D6112yes—
395C8—1224no—
396C8—2444no—
397C22—22no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—44no—
403C22—42no—
404C22—44no—
405C22—164no—
406C22—164no—
407C22#1683993012yes—
408C22C2×#1451974611yescharacteristic
409C4—21no—
410C4—21no—
411C4—22no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—4882no—
416C4#1684057314yes—
417C2—22no—
418C2—82no—
419C2—82no—
420C2—82no—
421C2#2145509612yes—
422C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3543C2
socleC22×C61C2×#120
hypercenter#3543D61
hypocenterC61#18683
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3543

Complements

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

NH ≅ G/Ntype
1. #21457037C2semidirect
2. C2×#16840134C2semidirect
3. #21444946C2semidirect
4. #21444940C2semidirect
5. #21444935C2semidirect
6. #21444533C2semidirect
7. #21444279C2semidirect
10. C2×#14519777C22semidirect
14. #16840134C22semidirect
18. #16840057C22semidirect
19. #16840052C22semidirect
20. #16839819C22semidirect
21. C2×C4×#92418C22semidirect
23. #16839506C22semidirect
24. Q8⋊#92418C22semidirect
25. D4⋊#92418C22semidirect
248. C61#18683semidirect

Maximal subgroups

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

subgroupindex
#214570372
C2×#168401342
#214449462
#214449402
#214449352
#214445332
#214442792
C61×#35432
#1868361