Finite Group Databasebeta

← #29698511

Subgroups — #29698511

Order 15616.

862 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839878C211yesmaximal, characteristic
2#21454696C211yesmaximal, characteristic
3#21454693C211yesmaximal, characteristic
4#21454418C212yesmaximal, complemented
5#21446605C211yesmaximal, characteristic, complemented
6#21446048C212yesmaximal, complemented
7#21446042C212yesmaximal, complemented
8#21445977C211yesmaximal, characteristic, complemented
9#21445882C212yesmaximal, complemented
10#21445699C211yesmaximal, characteristic, complemented
11C61×#1952C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C2×#14519785C2212yes—
14#16840243C2212yes—
15#16840242C2212yes—
16#16839878C418yescomplemented
17#16839864—22no—
18#16839861—22no—
19C4×Dic244—21no—
20#16839856—21no—
21C4×D488—21no—
22#16839844—21no—
23C2×#14519712C2211yescharacteristic
24#16839804C2212yes—
25#16839803—22no—
26#16839801—22no—
27#16839798C2211yescharacteristic
28#16839797—22no—
29C2×#14519711C2211yescharacteristic
30C2×#14519710C2212yes—
31#107×D61—22no—
32C2×#14519703C2212yes—
33#16839764C2212yes—
34#16839761C2212yes—
35#16839717C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37#16839631C2212yes—
38#16839530C2212yes—
39#16839523C2211yescharacteristic
40#16839521C2211yescharacteristic
41C2×#108×C61C2211yescharacteristic
42C61×#428C2212yes—
43C2×#107×C61C2211yescharacteristic
44C61×#405C2211yescharacteristic
45C61×#343C2212yes—
46C2×#11758222C2312yes—
47#14519799C2×C414yes—
48#14519799Q814yes—
49C2×Dic244D411yescharacteristic
50C2×#11758190D412yes—
51C2×D488D411yescharacteristic
52#14519785—24no—
53#14519785C2×C418yescomplemented
54C2×#11758194C2311yescharacteristic
55C2×#11758189—22no—
56C22×#92418C2311yescharacteristic
57C2×C8×D61—22no—
58C4×Dic122—22no—
59#14519742—22no—
60C4×D244—22no—
61#14519739—22no—
62C4×C22⋊D61—42no—
63#14519737—42no—
64#14519736C2311yescharacteristic
65#14519735—22no—
66#14519735—22no—
67C2×C4⋊Dic61C2312yes—
68C2×#11758196—42no—
69C4⋊C4×D61—22no—
70C4⋊C4×D61—22no—
71C2×C4×Dic61C2311yescharacteristic
72#14519712—22no—
73#14519711—21no—
74#14519711—21no—
75#14519711—22no—
76#14519710—22no—
77#14519710—22no—
78#14519710D411yescharacteristic
79#14519710D411yescharacteristic
80#14519709—22no—
81#14519708D411yescharacteristic
82#14519708D411yescharacteristic
83#14519703C2×C418yescomplemented
84#14519702—22no—
85#14519700—22no—
86#14519699—22no—
87#14519697—22no—
88#14519687—21no—
89C8×Dic61—21no—
90C8×Dic61C2×C414yescomplemented
91C2×M16×C61C2311yescharacteristic
92C2×M16×C61C2312yes—
93C22×C8×C61C2311yescharacteristic
94#117×C61C2311yescharacteristic
95C2×C4⋊C4×C61C2312yes—
96#108×C61C2×C414yescomplemented
97#107×C61—21no—
98#107×C61—21no—
99#107×C61—22no—
100#106×C61—22no—
101#97×C61—21no—
102C4×C8×C61—21no—
103C22×C4×C61—22no—
104C22×C4×C61C2411yescharacteristic
105C2×C22⋊D61—22no—
106C22×Dic61—21no—
107C22×Dic61—22no—
108#11758222—22no—
109#11758222—22no—
110#11758222C22×C414yes—
111#11758222C2×Q814yes—
112C2×D244C2×D412yes—
113C2×C4×D61—22no—
114C2×C4×D61—42no—
115C2×C4×D61—42no—
116C2×Dic122C2×D412yes—
117M16×C61—22no—
118M16×C61—21no—
119M16×C61—21no—
120M16×C61C22×C414yes—
121C2×C8×C61—22no—
122C2×C8×C61—21no—
123C2×C8×C61C22×C412yes—
124C2×C8×C61C2×D411yescharacteristic
125C2×C8×C61C2×D411yescharacteristic
126C2×C8×C61C2×Q812yes—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61C2×D411yescharacteristic
130C4⋊C4×C61C2×D411yescharacteristic
131C22⋊C4×C61—21no—
132C42×C61—21no—
133C22⋊Dic61—21no—
134#11758198—42no—
135#11758198—42no—
136C4⋊Dic61—22no—
137C4⋊Dic61—22no—
138C4⋊Dic61C2×D411yescharacteristic
139C4⋊Dic61C2×D411yescharacteristic
140#11758196—42no—
141#11758196—42no—
142#11758196—84no—
143C4×Dic61—21no—
144C4×Dic61—21no—
145C4×Dic61—42no—
146C4×Dic61—42no—
147C4×Dic61C22×C412yes—
148#11758194—21no—
149#11758194—21no—
150C2×#92418—21no—
151C2×#92418C22×C412yes—
152C2×#92418C2×D411yescharacteristic
153C2×#92418C2×D411yescharacteristic
154C2×#92418#4112yes—
155Dic244C4⋊C414yescomplemented
156D488C4⋊C414yescomplemented
157#11758190C4⋊C418yescomplemented
158#11758189—28no—
159C8×D61—28no—
160C23×C61H(5,𝔽2)11yescharacteristic
161C22×D61—22no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—42no—
165C2×C4×C61—82no—
166C2×C4×C61C23×C411yescharacteristic
167C2×C4×C61C22×D411yescharacteristic
168C2×C4×C61C22×D411yescharacteristic
169C2×C4×C61C22×Q811yescharacteristic
170C2×C4×C61C2×#4111yescharacteristic
171C2×C4×C612−1+411yescharacteristic
172C22⋊D61—24no—
173C22⋊D61—24no—
174C2×Dic61—22no—
175C2×Dic61—22no—
176C2×Dic61—22no—
177C2×Dic61—22no—
178C2×Dic61—21no—
179C2×Dic61—21no—
180C2×Dic61—42no—
181C2×Dic61—82no—
182D244—22no—
183D244C2×C4⋊C414yes—
184C4×D61—24no—
185C4×D61—24no—
186C4×D61—162no—
187C4×D61—162no—
188Dic122—22no—
189Dic122C2×C4⋊C414yes—
190C8×C61—24no—
191C8×C61—41no—
192C8×C61C2×C4⋊C412yes—
193C8×C61C2×C4⋊C412yes—
194#92418—41no—
195#92418C4×D412yes—
196#92418C4×D412yes—
197#22692—611nomaximal, Sylow
198C22×C61—21no—
199C22×C61—21no—
200C22×C61#52011yescharacteristic
201C22×C61D4211yescharacteristic
202C22×C61#55411yescharacteristic
203C2×D61—24no—
204C2×D61—42no—
205C2×D61—42no—
206C4×C61—41no—
207C4×C61—41no—
208C4×C61—82no—
209C4×C61C22×C4⋊C411yescharacteristic
210C4×C61C2×C4×D411yescharacteristic
211C4×C61#54711yescharacteristic
212C4×C61D4×Q811yescharacteristic
213Dic61—24no—
214Dic61—24no—
215Dic61—41no—
216Dic61—41no—
217Dic61—82no—
218C2×#443—611no—
219#2951—611no—
220#2951—611no—
221#2913—612no—
222#1952—611no—
223#1952D6111yescharacteristic, complemented
224#1859—612no—
225#1859—612no—
226#1845—611no—
227#1816—612no—
228#1772—611no—
229C2×C61—21no—
230C2×C61—21no—
231C2×C61D4×C4⋊C411yescharacteristic
232C2×C61#329211yescharacteristic
233C2×C61#329311yescharacteristic
234D61—44no—
235C2×#135—611no—
236C2×#131—612no—
237#520—612no—
238#520—612no—
239#443—618no—
240#440—1222no—
241#440—1222no—
242C4×Q16—1221no—
243C4×Q16—1221no—
244C4×D8—1221no—
245C4×D8—1221no—
246C2×#108—611no—
247C2×#108C2×D6111yescharacteristic
248#428—612no—
249#428—1222no—
250#428—1222no—
251#428C2×D6112yes—
252#427—611no—
253#427—1222no—
254C2×#107—611no—
255C2×#107—612no—
256C2×#107—1222no—
257C2×#107C2×D6111yescharacteristic
258C2×#104—612no—
259#417—612no—
260#417—612no—
261#405—611no—
262#405C2×D6111yescharacteristic
263C2×C4×C8—611no—
264#384—612no—
265#343—612no—
266#343C2×D6112yes—
267#339—611no—
268#339—611no—
269C61#2269211yescharacteristic, complemented, Sylow
270C2×#41—612no—
271#135—614no—
272#135—614no—
273C2×Q16—611no—
274C2×SD16—612no—
275C2×D8—611no—
276#131—618no—
277#131—1224no—
278C2×M16—611no—
279C2×M16—1222no—
280C2×M16C22×D6111yescharacteristic
281C2×M16C22×D6112yes—
282C22×C8—611no—
283C22×C8—1222no—
284C22×C8C22×D6111yescharacteristic
285C4×Q8—1222no—
286C4×Q8—1222no—
287C4×D4—1222no—
288C4×D4—1222no—
289C4×D4—2442no—
290C4×D4—2442no—
291#117—611no—
292#117—1222no—
293#117—1222no—
294#117C22×D6111yescharacteristic
295C2×C4⋊C4—612no—
296C2×C4⋊C4—1222no—
297C2×C4⋊C4—1222no—
298C2×C4⋊C4—2442no—
299C2×C4⋊C4C22×D6112yes—
300C2×C42—611no—
301#108—1222no—
302#108C4×D6114yescomplemented
303#107—21no—
304#107—21no—
305#107—22no—
306#107—611no—
307#107—611no—
308#107—1222no—
309#107—1222no—
310#107—1222no—
311#107—1221no—
312#107—1221no—
313#106—22no—
314#106—611no—
315#106—611no—
316#106—1222no—
317#104—618no—
318#103—1222no—
319#103—1222no—
320#102—1222no—
321#102—1222no—
322#97—21no—
323#97—1221no—
324C4×C8—21no—
325C4×C8—614no—
326C4×C8—1221no—
327#41—614no—
328#41—614no—
329#41—1222no—
330#41—1222no—
331C2×Q8—612no—
332C2×D4—612no—
333C2×D4—1222no—
334C22×C4—22no—
335C22×C4—1221no—
336C22×C4—1222no—
337C22×C4—1222no—
338C22×C4—2442no—
339C22×C4—2442no—
340C22×C4C23×D6111yescharacteristic
341Q16—614no—
342SD16—618no—
343D8—614no—
344M16—22no—
345M16—21no—
346M16—21no—
347M16—1228no—
348M16—1221no—
349M16—1221no—
350M16C2×C4×D6114yes—
351C2×C8—22no—
352C2×C8—21no—
353C2×C8—612no—
354C2×C8—611no—
355C2×C8—611no—
356C2×C8—612no—
357C2×C8—1228no—
358C2×C8—1221no—
359C2×C8C2×C4×D6112yes—
360C2×C8D4×D6111yescharacteristic
361C2×C8D4×D6111yescharacteristic
362C2×C8Q8×D6112yes—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—611no—
366C4⋊C4—611no—
367C4⋊C4—1222no—
368C4⋊C4—1222no—
369C4⋊C4—4884no—
370C4⋊C4—2442no—
371C4⋊C4—2442no—
372C4⋊C4D4×D6111yescharacteristic
373C4⋊C4D4×D6111yescharacteristic
374C22⋊C4—21no—
375C22⋊C4—1221no—
376C22⋊C4—2442no—
377C22⋊C4—2442no—
378C42—21no—
379C42—612no—
380C42—1221no—
381C42—1221no—
382C42—2442no—
383C42—2442no—
384C23—1222no—
385C23#1451982311yescharacteristic
386Q8—614no—
387Q8—1222no—
388D4—614no—
389D4—1224no—
390D4—1224no—
391D4—1222no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—82no—
396C2×C4—1222no—
397C2×C4—1222no—
398C2×C4—1222no—
399C2×C4—1222no—
400C2×C4—1224no—
401C2×C4—1224no—
402C2×C4—1221no—
403C2×C4—1221no—
404C2×C4—9762no—
405C2×C4—9762no—
406C2×C4—4882no—
407C2×C4—2442no—
408C2×C4C22×C4×D6111yescharacteristic
409C2×C4C2×D4×D6111yescharacteristic
410C2×C4C2×D4×D6111yescharacteristic
411C2×C4C2×Q8×D6111yescharacteristic
412C2×C4#41×D6111yescharacteristic
413C2×C4#1451982511yescharacteristic
414C8—24no—
415C8—41no—
416C8—612no—
417C8—612no—
418C8—2441no—
419C8C4⋊C4×D6112yes—
420C8C4⋊C4×D6112yes—
421C22—21no—
422C22—21no—
423C22—1224no—
424C22—2442no—
425C22—2442no—
426C22#1684024411yescharacteristic
427C22#1684039611yescharacteristic
428C22#1684046811yescharacteristic
429C4—41no—
430C4—41no—
431C4—82no—
432C4—1224no—
433C4—1224no—
434C4—4882no—
435C4—2441no—
436C4—2441no—
437C4C2×C4⋊C4×D6111yescharacteristic
438C4C4×D4×D6111yescharacteristic
439C4#1684041411yescharacteristic
440C4#1684042711yescharacteristic
441C2—21no—
442C2—21no—
443C2—2444no—
444C2#2144872011yescharacteristic
445C2#2145761711yescharacteristic
446C2#2145762211yescharacteristic
subgroupG/H
centerC22#16840396
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1952C2
socleC22×C61D42
prime radicalC2×#14519799C22
cosocleC22×C4×C61C24
hypercenter#1952D61
hypocenterC61#22692
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1952

Complements

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

NH ≅ G/Ntype
4. #21454418C2semidirect
5. #21446605C2semidirect
6. #21446048C2semidirect
7. #21446042C2semidirect
8. #21445977C2semidirect
9. #21445882C2semidirect
10. #21445699C2semidirect
11. C61×#1952C2semidirect
16. #16839878C4semidirect
53. #14519785C2×C4semidirect
83. #14519703C2×C4semidirect
90. C8×Dic61C2×C4semidirect
96. #108×C61C2×C4semidirect
155. Dic244C4⋊C4semidirect
156. D488C4⋊C4semidirect
157. #11758190C4⋊C4semidirect
223. #1952D61semidirect
269. C61#22692semidirect
302. #108C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398782
#214546962
#214546932
#214544182
#214466052
#214460482
#214460422
#214459772
#214458822
#214456992
C61×#19522
#2269261