Finite Group Databasebeta

← #29662454

Subgroups — #29662454

Order 15616.

618 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#21456403C211yesmaximal, characteristic, complemented
2#21455600C212yesmaximal
3C8×#11758224C211yesmaximal, characteristic
4#21445276C212yesmaximal, complemented
5#21445268C212yesmaximal, complemented
6#21445224C211yesmaximal, characteristic, complemented
7#21445218C211yesmaximal, characteristic, complemented
8#21445186C211yesmaximal, characteristic, complemented
9#335×D61C211yesmaximal, characteristic, complemented
10#21444475C212yesmaximal, complemented
11C61×#1669C211yesmaximal, characteristic, complemented
12#16840317C2212yes—
13C4×#11758224C2211yescharacteristic
14#16840167C2211yescharacteristic
15#16840112C2212yes—
16#16840068—22no—
17#16840060—22no—
18#16840058C2212yes—
19#16840052C2212yes—
20#16840044C2212yes—
21#16840000—21no—
22#16840000—21no—
23C8×Dic122C2211yescharacteristic
24#16839886C2211yescharacteristic
25Q16×Dic61—21no—
26Q16×Dic61—21no—
27D4×#92418C2211yescharacteristic
28C8×C22⋊D61—21no—
29#16839825—21no—
30#16839814C2211yescharacteristic
31#16839812—21no—
32#16839799—22no—
33C4⋊C8×D61C2212yes—
34#16839755—22no—
35C2×C8×Dic61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839514C2212yes—
39#16839510C2212yes—
40#16839510C2212yes—
41#16839508C2212yes—
42C61×#494C2211yescharacteristic
43C61×#474C2212yes—
44C8×D4×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#332C2212yes—
47C2×#11758224D411yescharacteristic
48C22×#92418—21no—
49C2×C8×D61—22no—
50C2×C8×D61D411yescharacteristic
51C2×C8×D61D411yescharacteristic
52#14519777C2312yes—
53#14519768—22no—
54#14519757—22no—
55#14519753—22no—
56Q8×Dic61—22no—
57C4×Dic122C2311yescharacteristic
58#14519742—22no—
59#14519742D411yescharacteristic
60D4×Dic61D411yescharacteristic
61C4×C22⋊D61—21no—
62#14519737—21no—
63#14519735—22no—
64#14519735D411yescharacteristic
65#14519733—21no—
66C2×C4×Dic61—21no—
67C42×D61C2311yescharacteristic
68#14519711—21no—
69#14519711—21no—
70#14519711—22no—
71#14519708—22no—
72#14519708—22no—
73C4⋊#92418C2311yescharacteristic
74C4⋊#92418C2312yes—
75#14519706—21no—
76#14519706—22no—
77#14519706—21no—
78#14519702—22no—
79#14519701—22no—
80#14519701—22no—
81#14519700—22no—
82#14519698—22no—
83C22⋊#92418—21no—
84#14519689—21no—
85C4×#92418C2311yescharacteristic
86C8×Dic61—21no—
87C8×Dic61—21no—
88C8×Dic61—21no—
89C8×Dic61D411yescharacteristic
90C8×Dic61D411yescharacteristic
91C2×Q16×C61—21no—
92C2×Q16×C61—21no—
93C22×C8×C61—21no—
94C4⋊Q8×C61C2312yes—
95C4×D4×C61C2311yescharacteristic
96#107×C61—22no—
97C4⋊C8×C61C2311yescharacteristic
98C4⋊C8×C61C2312yes—
99#102×C61—22no—
100C22⋊C8×C61—21no—
101C4×C8×C61C2311yescharacteristic
102C2×Q8×C61—22no—
103C2×D4×C61C2×D411yescharacteristic
104C22×C4×C61—21no—
105C2×C22⋊D61—21no—
106C22×Dic61—21no—
107C22×Dic61—21no—
108#11758224—22no—
109#11758224Q1614yescomplemented
110C2×C4×D61C2×D411yescharacteristic
111C2×C4×D61C2×D411yescharacteristic
112C2×C4×D61#4111yescharacteristic
113C2×Dic122—22no—
114C2×Dic122C2×D411yescharacteristic
115Q16×C61—42no—
116Q16×C61—42no—
117C2×C8×C61—21no—
118C2×C8×C61—22no—
119C2×C8×C61—42no—
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61C2×D411yescharacteristic
122C4⋊C4×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61C2×D411yescharacteristic
125C22⋊C4×C61—21no—
126C42×C61C2411yescharacteristic
127C22⋊Dic61—21no—
128#11758198—21no—
129#11758198—42no—
130C4⋊Dic61—22no—
131C4⋊Dic61—22no—
132C4⋊Dic61—42no—
133C4⋊Dic61C2×D411yescharacteristic
134#11758196—21no—
135#11758196—21no—
136#11758196—42no—
137C4×Dic61—21no—
138C4×Dic61—21no—
139C4×Dic61—21no—
140C4×Dic61—42no—
141C4×Dic61C2×D411yescharacteristic
142C4×Dic61C2×D411yescharacteristic
143C4×Dic61#4111yescharacteristic
144C2×#92418—21no—
145C2×#92418—22no—
146C2×#92418—42no—
147C2×#92418C2×D411yescharacteristic
148C2×#92418C2×D411yescharacteristic
149C8×D61—22no—
150C8×D61—41no—
151C8×D61—44no—
152C23×C61—21no—
153C22×D61#12111yescharacteristic
154Q8×C61—42no—
155Q8×C61—42no—
156D4×C61—21no—
157D4×C61C2×Q1612yes—
158C2×C4×C61—21no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—42no—
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C2×#4111yescharacteristic
165C22⋊D61—24no—
166C2×Dic61—21no—
167C2×Dic61—24no—
168C2×Dic61—21no—
169C2×Dic61—42no—
170C2×Dic61—42no—
171C2×Dic61#12111yescharacteristic
172C2×Dic61#12111yescharacteristic
173C2×Dic61#12111yescharacteristic
174C4×D61—21no—
175C4×D61—21no—
176C4×D61—41no—
177C4×D61C2×Q1612yes—
178C4×D61#13512yes—
179Dic122—21no—
180Dic122—82no—
181Dic122C2×Q1612yes—
182C8×C61—21no—
183C8×C61—42no—
184C8×C61—41no—
185C8×C61#13512yes—
186#92418—21no—
187#92418—21no—
188#92418—42no—
189#92418—41no—
190#19126—611nomaximal, Sylow
191C22×C61—22no—
192C22×C61—41no—
193C22×C61—41no—
194C22×C61C2×#12111yescharacteristic
195C2×D61—21no—
196C2×D61—21no—
197C2×D61#46712yes—
198C4×C61—21no—
199C4×C61—41no—
200C4×C61—82no—
201C4×C61C22×Q1611yescharacteristic
202C4×C61C2×#13511yescharacteristic
203C4×C61C2×#13511yescharacteristic
204C4×C61#57511yescharacteristic
205Dic61—21no—
206Dic61—22no—
207Dic61—21no—
208Dic61—41no—
209Dic61—82no—
210Dic61#46712yes—
211#3170—611no—
212#3076—612no—
213C8×#41—611no—
214#1684—612no—
215#1683—612no—
216#1676—611no—
217#1675—611no—
218#1670—611no—
219#1669D6111yescharacteristic, complemented
220C2×#335—611no—
221#1542—612no—
222C2×C61—42no—
223C2×C61#303511yescharacteristic
224C2×C61C2×#46711yescharacteristic
225C2×C61#306711yescharacteristic
226D61—22no—
227#533—612no—
228C4×#41—611no—
229#500—611no—
230#494C2×D6111yescharacteristic
231#487—612no—
232#479—1222no—
233#478—1222no—
234#477—612no—
235#476—612no—
236#475—612no—
237#474C2×D6112yes—
238#467—1221no—
239#467—1221no—
240C8×Q8—611no—
241C8×Q8—611no—
242C4×Q16—1221no—
243C4×Q16—1221no—
244C8×D4—611no—
245C8×D4—1221no—
246C8×D4—1221no—
247C8×D4C2×D6111yescharacteristic
248#431—611no—
249#431—1221no—
250#427—1222no—
251C2×C4⋊C8—612no—
252#416—1222no—
253C2×C4×C8—611no—
254C2×C4×C8—1221no—
255#335—612no—
256#335—611no—
257#335C2×D6111yescharacteristic
258#333—612no—
259#333—612no—
260#332—612no—
261#332C2×D6112yes—
262C61#1912611yescharacteristic, complemented, Sylow
263C2×#41—611no—
264C2×Q16—21no—
265C2×Q16—21no—
266C22×C8—21no—
267C22×C8—611no—
268C22×C8—611no—
269C22×C8—1222no—
270C22×C8—1221no—
271C4⋊Q8—612no—
272C4⋊Q8C22×D6112yes—
273#125—1222no—
274C22⋊Q8—1222no—
275C22⋊Q8—1222no—
276C4×Q8—611no—
277C4×Q8—611no—
278C4×Q8—1222no—
279C4×Q8—1222no—
280C4×D4—611no—
281C4×D4—1221no—
282C4×D4—1221no—
283C4×D4C22×D6111yescharacteristic
284#117—611no—
285#117—1221no—
286#117—1222no—
287C2×C42—611no—
288C2×C42—1221no—
289#107—22no—
290#107—1222no—
291#107—1221no—
292#107—1221no—
293#106—1222no—
294#106—1222no—
295C4⋊C8—611no—
296C4⋊C8—612no—
297C4⋊C8—1222no—
298C4⋊C8—1221no—
299C4⋊C8—1221no—
300C4⋊C8C22×D6111yescharacteristic
301C4⋊C8C22×D6112yes—
302#103—1222no—
303#103—1222no—
304#103—1222no—
305#103—1222no—
306#102—22no—
307#102—1222no—
308C22⋊C8—21no—
309C22⋊C8—1221no—
310C22⋊C8—1221no—
311C4×C8—611no—
312C4×C8—611no—
313C4×C8—611no—
314C4×C8—1221no—
315C4×C8—1221no—
316C4×C8—1221no—
317C4×C8C22×D6111yescharacteristic
318#41—614no—
319#41—1222no—
320C2×Q8—22no—
321C2×Q8—611no—
322C2×Q8—1222no—
323C2×D4—1221no—
324C2×D4D4×D6111yescharacteristic
325C22×C4—21no—
326C22×C4—611no—
327C22×C4—611no—
328C22×C4—611no—
329C22×C4—1221no—
330C22×C4—1221no—
331Q16—42no—
332Q16—42no—
333C2×C8—21no—
334C2×C8—22no—
335C2×C8—42no—
336C2×C8—611no—
337C2×C8—611no—
338C2×C8—1222no—
339C2×C8—1221no—
340C2×C8—1222no—
341C2×C8—2441no—
342C2×C8—2444no—
343C2×C8—2442no—
344C2×C8D4×D6111yescharacteristic
345C2×C8D4×D6111yescharacteristic
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—611no—
349C4⋊C4—1221no—
350C4⋊C4—1221no—
351C4⋊C4—1222no—
352C4⋊C4—1222no—
353C4⋊C4—2442no—
354C4⋊C4—2442no—
355C4⋊C4D4×D6111yescharacteristic
356C22⋊C4—21no—
357C22⋊C4—1221no—
358C22⋊C4—1221no—
359C22⋊C4—2442no—
360C42—611no—
361C42—611no—
362C42—611no—
363C42—1221no—
364C42—1221no—
365C42—1221no—
366C42—2442no—
367C42C23×D6111yescharacteristic
368C23—21no—
369C23—611no—
370Q8—42no—
371Q8—42no—
372Q8—612no—
373Q8—1221no—
374Q8—4882no—
375D4—21no—
376D4—1224no—
377D4Q16×D6112yes—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—611no—
383C2×C4—612no—
384C2×C4—612no—
385C2×C4—611no—
386C2×C4—611no—
387C2×C4—1224no—
388C2×C4—1221no—
389C2×C4—1221no—
390C2×C4—1221no—
391C2×C4—1221no—
392C2×C4—2441no—
393C2×C4—2442no—
394C2×C4—2442no—
395C2×C4C2×D4×D6111yescharacteristic
396C2×C4C2×D4×D6111yescharacteristic
397C2×C4#41×D6111yescharacteristic
398C8—21no—
399C8—42no—
400C8—41no—
401C8—1221no—
402C8—1221no—
403C8—2442no—
404C8—2441no—
405C8#1451979612yes—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—612no—
410C22—1221no—
411C22—1221no—
412C22#121×D6111yescharacteristic
413C4—21no—
414C4—41no—
415C4—82no—
416C4—612no—
417C4—1222no—
418C4—1221no—
419C4—1221no—
420C4—4882no—
421C4—2441no—
422C4C2×Q16×D6111yescharacteristic
423C4#135×D6111yescharacteristic
424C4C2×#1451979611yescharacteristic
425C4#1684055811yescharacteristic
426C2—42no—
427C2—1222no—
428C2#2145526711yescharacteristic
429C2#467×D6111yescharacteristic
430C2#2145554111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1669C2
socleC22×C61C2×#121
prime radicalC2×#11758224D4
hypercenter#1669D61
hypocenterC61#19126
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1669

Complements

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

NH ≅ G/Ntype
1. #21456403C2semidirect
4. #21445276C2semidirect
5. #21445268C2semidirect
6. #21445224C2semidirect
7. #21445218C2semidirect
8. #21445186C2semidirect
9. #335×D61C2semidirect
10. #21444475C2semidirect
11. C61×#1669C2semidirect
109. #11758224Q16semidirect
219. #1669D61semidirect
262. C61#19126semidirect

Maximal subgroups

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

subgroupindex
#214564032
#214556002
C8×#117582242
#214452762
#214452682
#214452242
#214452182
#214451862
#335×D612
#214444752
C61×#16692
#1912661