Finite Group Databasebeta

← #29663623

Subgroups — #29663623

Order 15616.

636 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457161C211yesmaximal, characteristic, complemented
2#21456760C211yesmaximal, characteristic
3#21454766C211yesmaximal, characteristic, complemented
4#21445384C212yesmaximal, complemented
5#21445356C212yesmaximal, complemented
6#21445260C212yesmaximal, complemented
7#21445232C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444476C211yesmaximal, characteristic, complemented
10C2×#16839506C211yesmaximal, characteristic
11C61×#3138C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840134—21no—
14#16840108—21no—
15#16840057—21no—
16#16840054C2212yescomplemented
17#16840001—22no—
18#16839972—21no—
19#16839969C2212yescomplemented
20#16839959C2212yes—
21#16839959C2212yes—
22#16839913—22no—
23#16839892C2212yescomplemented
24#16839890—22no—
25C8×Dic122C2212yescomplemented
26#16839815C2211yescharacteristic
27#16839790—21no—
28C2×C4⋊#92418C2211yescharacteristic
29C2×C4⋊#92418C2211yescharacteristic
30#16839720—21no—
31#16839515C2212yescomplemented
32#16839513C2212yescomplemented
33#16839511C2211yescharacteristic, complemented
34#16839511C2211yescharacteristic, complemented
35#16839508C2211yescharacteristic, complemented
36#16839508C2211yescharacteristic, complemented
37#16839506C2212yes—
38#16839506C2212yes—
39C61×#533C2211yescharacteristic
40C2×#127×C61C2211yescharacteristic
41C61×#490—22no—
42C61×#486C2212yes—
43C61×#486C2212yes—
44C61×#418—21no—
45C61×#418—21no—
46C4×M16×C61C2211yescharacteristic
47C2×#11758202—22no—
48C2×#11758200—22no—
49C2×#11758200—22no—
50C22×#92418—21no—
51C22×#92418—21no—
52#14519777D411yescharacteristic
53#14519777D411yescharacteristic
54#14519769—21no—
55#14519756—21no—
56#14519756D412yes—
57C4×Dic122—21no—
58C4×Dic122C2312yes—
59#14519736—21no—
60#14519710—22no—
61C4⋊#92418—21no—
62C4⋊#92418—21no—
63C4⋊#92418C2311yescharacteristic
64C4⋊#92418C2311yescharacteristic
65C4⋊#92418C2312yes—
66C4⋊#92418C2311yescharacteristic
67C4⋊#92418C2311yescharacteristic
68#14519706—22no—
69#14519706—22no—
70#14519701—22no—
71#14519700—22no—
72#14519698—22no—
73#14519698—22no—
74C22⋊#92418—21no—
75#14519687—22no—
76C4×#92418—21no—
77C8×Dic61—22no—
78C22×D4×C61—21no—
79C22×D4×C61—21no—
80C22×D4×C61—21no—
81C2×M16×C61D411yescharacteristic
82C2×M16×C61D411yescharacteristic
83C4⋊Q8×C61C2311yescharacteristic
84C4⋊Q8×C61C2311yescharacteristic
85#127×C61—21no—
86#127×C61—21no—
87#127×C61—21no—
88#127×C61C2312yes—
89#125×C61—21no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94#117×C61—21no—
95C2×C42×C61C2311yescharacteristic
96#102×C61—22no—
97#102×C61—22no—
98#102×C61—22no—
99#102×C61—22no—
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—41no—
109C2×D4×C61—41no—
110C2×D4×C61—41no—
111C2×D4×C61—42no—
112C2×D4×C61—42no—
113C2×D4×C61—81no—
114C2×D4×C61—81no—
115C2×D4×C61—82no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61#4111yescharacteristic
119C2×Dic122—21no—
120C2×Dic122C2×D412yes—
121M16×C61—22no—
122M16×C61—22no—
123C2×C8×C61C2×D412yes—
124C2×C8×C61C2×D412yes—
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×C61#4111yescharacteristic
136C42×C61C2411yescharacteristic
137C22⋊Dic61—22no—
138#11758202—82no—
139#11758200—82no—
140#11758200—82no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61C2×D412yes—
145#11758196—22no—
146#11758196—22no—
147C4×Dic61—22no—
148C2×#92418—21no—
149C2×#92418—21no—
150C2×#92418—21no—
151C2×#92418—21no—
152C2×#92418—22no—
153C2×#92418—42no—
154C2×#92418—81no—
155C23×C61—42no—
156C23×C61—82no—
157C23×C61—82no—
158C23×C61—82no—
159C23×C61#12111yescharacteristic
160Q8×C61—41no—
161Q8×C61—41no—
162Q8×C61—81no—
163D4×C61—42no—
164D4×C61—42no—
165D4×C61—42no—
166D4×C61—42no—
167D4×C61—82no—
168D4×C61—82no—
169D4×C61—82no—
170D4×C61—82no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61#13512yes—
180C2×C4×C61#13712yes—
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C2×#4111yescharacteristic
184C2×Dic61—22no—
185C2×Dic61—22no—
186Dic122—22no—
187Dic122—22no—
188Dic122—81no—
189C8×C61—22no—
190C8×C61—22no—
191#92418—41no—
192#92418—41no—
193#92418—42no—
194#92418—41no—
195#92418—41no—
196#19242—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—82no—
200C22×C61—82no—
201C22×C61—82no—
202C22×C61—164no—
203C22×C61#46412yes—
204C22×C61C2×#12111yescharacteristic
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—21no—
208C4×C61—81no—
209C4×C61—81no—
210C4×C61#46412yes—
211C4×C61C2×#13511yescharacteristic
212C4×C61C2×#13711yescharacteristic
213C4×C61#57611yescharacteristic
214C4×C61#57711yescharacteristic
215Dic61—42no—
216Dic61—42no—
217#3249—611no—
218#3208—611no—
219#3138D6111yescharacteristic
220#2962—611no—
221#1702—612no—
222#1698—612no—
223#1682—612no—
224#1678—612no—
225#1565—611no—
226#1542—611no—
227C2×#331—611no—
228C2×C61—22no—
229C2×C61—162no—
230C2×C61C2×#46411yescharacteristic
231C2×C61#306211yescharacteristic
232C2×C61#306811yescharacteristic
233#533—611no—
234#533C2×D6111yescharacteristic
235C2×#127C2×D6111yescharacteristic
236#492—1221no—
237#490—22no—
238#486—1221no—
239#486C2×D6112yes—
240#486C2×D6112yes—
241#477—612no—
242#477—1221no—
243#467—1222no—
244#462—612no—
245#462—1221no—
246#460—612no—
247#460—612no—
248#450—1222no—
249#446—612no—
250#446—1222no—
251C8×Q8—612no—
252#431—611no—
253#424—1221no—
254C2×C4⋊C8—611no—
255C2×C4⋊C8—611no—
256#418—21no—
257#418—21no—
258#405—1221no—
259C4×M16C2×D6111yescharacteristic
260#335—612no—
261#334—612no—
262#333—611no—
263#333—611no—
264#332—611no—
265#332—611no—
266#331—612no—
267#331—612no—
268C61#1924211yescharacteristic, complemented, Sylow
269C22×D4—21no—
270C22×D4—21no—
271C22×D4—21no—
272C2×Q16—1222no—
273C2×SD16—1222no—
274C2×SD16—1222no—
275C2×M16C22⋊D6111yescharacteristic
276C2×M16C22⋊D6111yescharacteristic
277C22×C8—1221no—
278C22×C8—1221no—
279C4⋊Q8—611no—
280C4⋊Q8—611no—
281C4⋊Q8C22×D6111yescharacteristic
282C4⋊Q8C22×D6111yescharacteristic
283#127—21no—
284#127—21no—
285#127—21no—
286#127C22×D6112yes—
287#125—21no—
288#125—1221no—
289C22⋊Q8—21no—
290C22⋊Q8—21no—
291C22⋊Q8—612no—
292C22⋊Q8—1221no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8—612no—
296C4×Q8—1221no—
297#117—21no—
298#117—1221no—
299C2×C42C22×D6111yescharacteristic
300#107—1222no—
301C4⋊C8—611no—
302C4⋊C8—611no—
303C4⋊C8—612no—
304C4⋊C8—611no—
305C4⋊C8—611no—
306C4⋊C8—1222no—
307C4⋊C8—1222no—
308C4⋊C8—1221no—
309C4⋊C8—1221no—
310#103—1222no—
311#103—1222no—
312#102—22no—
313#102—22no—
314#102—22no—
315#102—22no—
316#102—1222no—
317#102—1222no—
318C22⋊C8—1221no—
319#97—1222no—
320#97C22⋊D6112yes—
321C4×C8—1222no—
322C4×C8—1221no—
323C4×C8C22×D6112yes—
324C24—41no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—612no—
328C2×Q8—1221no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—81no—
338C2×D4—81no—
339C2×D4—82no—
340C22×C4D4×D6111yescharacteristic
341C22×C4#1175822411yescharacteristic
342C22×C4C2×C22⋊D6111yescharacteristic
343Q16—4882no—
344SD16—4882no—
345SD16—4882no—
346M16—22no—
347M16—22no—
348C2×C8—1221no—
349C2×C8—1221no—
350C2×C8—1222no—
351C2×C8—1221no—
352C2×C8—1221no—
353C2×C8—4881no—
354C2×C8—2442no—
355C2×C8C2×C22⋊D6112yes—
356C2×C8C2×C22⋊D6112yes—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—612no—
364C4⋊C4—1222no—
365C4⋊C4—1222no—
366C4⋊C4—1221no—
367C4⋊C4—1221no—
368C4⋊C4—1221no—
369C22⋊C4—41no—
370C22⋊C4—1222no—
371C42—41no—
372C42—1222no—
373C42D4×D6111yescharacteristic
374C42#1175822411yescharacteristic
375C42C2×C22⋊D6111yescharacteristic
376C42C23×D6111yescharacteristic
377C23—42no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23#1451974811yescharacteristic
382Q8—41no—
383Q8—41no—
384Q8—81no—
385Q8—1222no—
386Q8—1222no—
387Q8—4881no—
388D4—42no—
389D4—42no—
390D4—42no—
391D4—42no—
392D4—82no—
393D4—82no—
394D4—82no—
395D4—82no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—1222no—
402C2×C4—1222no—
403C2×C4#1451974811yescharacteristic
404C2×C4#1451974811yescharacteristic
405C2×C4#1451974811yescharacteristic
406C2×C4#1451979712yes—
407C2×C4#1451980712yes—
408C2×C4C2×D4×D6111yescharacteristic
409C2×C4C22×C22⋊D6111yescharacteristic
410C2×C4C2×#1175822411yescharacteristic
411C8—22no—
412C8—22no—
413C8—2441no—
414C8—2441no—
415C8—2442no—
416C8—2441no—
417C8—2441no—
418C22—21no—
419C22—21no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—164no—
424C22#1683998112yes—
425C22C2×#1451974811yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—81no—
430C4—81no—
431C4—2442no—
432C4—2442no—
433C4#1683998112yes—
434C4C2×#1451979711yescharacteristic
435C4C2×#1451980711yescharacteristic
436C4#1684056611yescharacteristic
437C4#1684057211yescharacteristic
438C2—22no—
439C2—162no—
440C2C2×#1683998111yescharacteristic
441C2#2145549311yescharacteristic
442C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3138C2
socleC22×C61C2×#121
prime radicalC2×#127×C61C22
hypercenter#3138D61
hypocenterC61#19242
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. #21457161C2semidirect
3. #21454766C2semidirect
4. #21445384C2semidirect
5. #21445356C2semidirect
6. #21445260C2semidirect
7. #21445232C2semidirect
8. #21444571C2semidirect
9. #21444476C2semidirect
16. #16840054C22semidirect
19. #16839969C22semidirect
23. #16839892C22semidirect
25. C8×Dic122C22semidirect
31. #16839515C22semidirect
32. #16839513C22semidirect
33. #16839511C22semidirect
34. #16839511C22semidirect
35. #16839508C22semidirect
36. #16839508C22semidirect
268. C61#19242semidirect

Maximal subgroups

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

subgroupindex
#214571612
#214567602
#214547662
#214453842
#214453562
#214452602
#214452322
#214445712
#214444762
C2×#168395062
C61×#31382
#1924261