Finite Group Databasebeta

← #29834934

Subgroups — #29834934

Order 15616.

654 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456393C211yesmaximal, characteristic, complemented
2C2×#16839844C211yesmaximal, characteristic, complemented
3#21447999C211yesmaximal, characteristic, complemented
4#21447955C212yesmaximal, complemented
5#21447853C212yesmaximal, complemented
6#21447733C211yesmaximal, characteristic, complemented
7#21447730C211yesmaximal, characteristic, complemented
8#21447665C212yesmaximal, complemented
9#21447663C212yesmaximal, complemented
10#21447615C211yesmaximal, characteristic, complemented
11C61×#2232C211yesmaximal, characteristic, complemented
12C22×D488C2211yescharacteristic
13#16840307—21no—
14C2×#14519739—21no—
15C2×D976C2212yes—
16C2×#14519718C2212yes—
17#16840166C2211yescharacteristic, complemented
18#16840137C2211yescharacteristic, complemented
19#16840107—21no—
20#16839998C2211yescharacteristic
21#16839996—21no—
22D8×Dic61C2211yescharacteristic, complemented
23#16839844—21no—
24#16839844—21no—
25#16839844C2212yes—
26#16839844C2212yes—
27#16839844C2211yescharacteristic, complemented
28#16839842—21no—
29C2×#14519697—21no—
30C2×C8×Dic61C2211yescharacteristic
31#16839587C2212yescomplemented
32#16839580C2212yescomplemented
33#16839564C2212yes—
34#16839562C2212yes—
35#16839562C2212yescomplemented
36C22⋊#11758185C2211yescharacteristic
37C16×Dic61C2212yescomplemented
38C61×#466C2211yescharacteristic
39C2×#107×C61C2211yescharacteristic
40C61×#366C2212yescomplemented
41C61×#357C2212yescomplemented
42C61×C22⋊C16C2211yescharacteristic
43C2×#11758224—41no—
44C22×D244—21no—
45C22×C4×D61—41no—
46C2×D488—21no—
47C2×D488—24no—
48C2×D488C2311yescharacteristic
49C2×D488C2312yes—
50C2×#11758199—21no—
51C22×#92418D411yescharacteristic
52#14519776—21no—
53#14519773—21no—
54#14519763—41no—
55#14519751—21no—
56#14519749—41no—
57D4×Dic61—21no—
58#14519739—22no—
59#14519739—22no—
60#14519739—21no—
61#14519739—41no—
62#14519739—41no—
63#14519737—41no—
64C2×#11758198—41no—
65C2×C4×Dic61D411yescharacteristic
66D976—24no—
67#14519718—24no—
68C2×#11758185C2312yes—
69#14519711C2311yescharacteristic
70#14519710—21no—
71#14519699—21no—
72#14519698—21no—
73#14519697—21no—
74#14519697—22no—
75#14519697—22no—
76C8×Dic61D411yescharacteristic
77C8×Dic61D411yescharacteristic
78C8×Dic61C2311yescharacteristic
79C8×Dic61C2311yescharacteristic
80C2×D8×C61C2311yescharacteristic
81C22×C8×C61C2311yescharacteristic
82#121×C61—21no—
83C2×C4⋊C4×C61—21no—
84C2×C16×C61C2312yes—
85#107×C61—21no—
86#107×C61C2312yes—
87#107×C61C2311yescharacteristic
88#102×C61—21no—
89C23×D61—41no—
90C2×D4×C61—21no—
91C2×D4×C61—41no—
92C22×C4×C61—41no—
93C22×C4×C61C2×D411yescharacteristic
94C2×C22⋊D61—41no—
95C2×C22⋊D61—41no—
96C2×C22⋊D61—41no—
97C22×Dic61—41no—
98C22×Dic61D811yescharacteristic
99C22×Dic61D811yescharacteristic
100#11758224—84no—
101C2×D244—24no—
102C2×D244—22no—
103C2×D244—21no—
104C2×D244—42no—
105C2×D244—41no—
106C2×C4×D61—42no—
107C2×C4×D61—42no—
108C2×C4×D61—41no—
109C2×C4×D61—84no—
110C2×Dic122—41no—
111D8×C61—22no—
112C2×C8×C61—22no—
113C2×C8×C61C2×D411yescharacteristic
114C2×C8×C61C2411yescharacteristic
115C4⋊C4×C61—21no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—41no—
118C22⋊C4×C61—41no—
119C22⋊Dic61—41no—
120#11758199—42no—
121#11758198—41no—
122#11758198—42no—
123#11758198—42no—
124#11758198—41no—
125C4⋊Dic61—21no—
126#11758196—41no—
127C4×Dic61D811yescharacteristic
128C4×Dic61D811yescharacteristic
129C4×Dic61C2×D411yescharacteristic
130C4×Dic61C2×D411yescharacteristic
131C2×#92418—41no—
132C2×#92418C2×D411yescharacteristic
133C2×#92418C2×D411yescharacteristic
134D488—24no—
135D488—22no—
136D488—22no—
137D488#4114yescomplemented
138C16×C61—22no—
139#11758185—22no—
140C23×C61—41no—
141C23×C61C2×D811yescharacteristic
142C22×D61—44no—
143C22×D61—42no—
144C22×D61—41no—
145C22×D61—82no—
146C22×D61—82no—
147D4×C61—41no—
148D4×C61—41no—
149D4×C61—82no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—41no—
153C2×C4×C61—42no—
154C2×C4×C61—82no—
155C2×C4×C61C2×D811yescharacteristic
156C2×C4×C61C22×D411yescharacteristic
157C22⋊D61—82no—
158C22⋊D61—82no—
159C22⋊D61—82no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—82no—
163C2×Dic61D1614yescomplemented
164C2×Dic61C2×D811yescharacteristic
165C2×Dic61C2×D811yescharacteristic
166C2×Dic61C2×D811yescharacteristic
167C2×Dic61C2×D811yescharacteristic
168D244—24no—
169D244—44no—
170D244—41no—
171D244—42no—
172D244—41no—
173C4×D61—84no—
174C4×D61—84no—
175C4×D61—82no—
176Dic122—82no—
177C8×C61—21no—
178C8×C61C2×#4112yes—
179#92418—21no—
180#92418—21no—
181#33660—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—81no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61C2×D1612yes—
188C22×C61C22×D811yescharacteristic
189C2×D61—44no—
190C2×D61—88no—
191C2×D61—82no—
192C2×D61—82no—
193C2×D61—81no—
194C2×D61—81no—
195C2×D61—81no—
196C4×C61—21no—
197C4×C61—81no—
198C4×C61—82no—
199C4×C61#52511yescharacteristic
200C4×C61#57511yescharacteristic
201Dic61—21no—
202Dic61—21no—
203Dic61—81no—
204Dic61C2×D1612yes—
205Dic61C2×D1612yes—
206#3169—611no—
207C2×C4×D8—611no—
208#2232—611no—
209#2232D6111yescharacteristic, complemented
210#2224—612no—
211#2206—612no—
212#2184—611no—
213#2184—611no—
214C4×D16—612no—
215C4×D16—612no—
216#2162—611no—
217C2×C61—22no—
218C2×C61—161no—
219C2×C61#310011yescharacteristic
220C2×C61C22×D1611yescharacteristic
221C2×C61#341411yescharacteristic
222D61—84no—
223D61—161no—
224C22×D8—611no—
225#532—1221no—
226C2×C4×D4—1221no—
227C2×D16—612no—
228C2×D16—612no—
229#500—611no—
230#493—611no—
231#486—1221no—
232#466—611no—
233#466—1221no—
234#466C2×D6111yescharacteristic
235C4×D8—611no—
236C4×D8—611no—
237C4×D8—612no—
238C4×D8—612no—
239C4×D8—1221no—
240C4×D8—1221no—
241C4×D8—1221no—
242C2×#107C2×D6111yescharacteristic
243C2×#102—1221no—
244C2×C4×C8—611no—
245#366—612no—
246#366C2×D6112yescomplemented
247C4⋊C16—612no—
248#357—612no—
249#357—612no—
250#357—612no—
251#357C2×D6112yescomplemented
252C22⋊C16—611no—
253C22⋊C16C2×D6111yescharacteristic
254C4×C16—612no—
255C61#3366011yescharacteristic, complemented, Sylow
256C2×#41—2441no—
257C22×D4—1221no—
258C23×C4—2441no—
259C2×D8—612no—
260C2×D8—611no—
261C2×D8—1224no—
262C2×D8—1221no—
263C2×D8—1221no—
264C2×D8C22×D6111yescharacteristic
265C22×C8—611no—
266C22×C8C22×D6111yescharacteristic
267C4⋊Q8—1221no—
268#127—1221no—
269#124—2441no—
270#121—21no—
271#121—1221no—
272#121—2441no—
273C4×D4—1222no—
274C4×D4—1222no—
275C4×D4—1221no—
276C4×D4—1221no—
277C4×D4—2441no—
278C4×D4—2441no—
279C4×D4—2441no—
280C2×C4⋊C4—21no—
281C2×C22⋊C4—2441no—
282C2×C42—611no—
283D16—1224no—
284D16—1224no—
285C2×C16—612no—
286C2×C16C22×D6112yes—
287#107—21no—
288#107—611no—
289#107—1221no—
290#107C22×D6111yescharacteristic
291#107C22×D6112yes—
292#102—21no—
293#102—1222no—
294#102—1222no—
295#102—1221no—
296#102—1221no—
297#102—1221no—
298C4×C8—611no—
299C4×C8—611no—
300C4×C8—611no—
301C4×C8—611no—
302C24—2441no—
303#41—4884no—
304C2×Q8—2441no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—1224no—
308C2×D4—1222no—
309C2×D4—1221no—
310C2×D4—2442no—
311C2×D4—2441no—
312C2×D4—2441no—
313C2×D4—2441no—
314C2×D4—2441no—
315C22×C4—41no—
316C22×C4—611no—
317C22×C4—611no—
318C22×C4—4884no—
319C22×C4—2442no—
320C22×C4—2442no—
321C22×C4—2441no—
322C22×C4—2441no—
323C22×C4D4×D6111yescharacteristic
324D8—22no—
325D8—614no—
326D8—1224no—
327D8—1222no—
328D8—1222no—
329D8—2442no—
330C2×C8—22no—
331C2×C8—611no—
332C2×C8—611no—
333C2×C8—2441no—
334C2×C8D4×D6111yescharacteristic
335C2×C8C23×D6111yescharacteristic
336C4⋊C4—21no—
337C4⋊C4—22no—
338C4⋊C4—41no—
339C4⋊C4—1221no—
340C4⋊C4—2441no—
341C22⋊C4—41no—
342C22⋊C4—2442no—
343C22⋊C4—2442no—
344C22⋊C4—2441no—
345C22⋊C4—2441no—
346C22⋊C4—2441no—
347C42—611no—
348C42—611no—
349C42—611no—
350C42—611no—
351C16—22no—
352C16—1222no—
353C23—41no—
354C23—4882no—
355C23—4882no—
356C23—2444no—
357C23—2441no—
358C23—2442no—
359C23D8×D6111yescharacteristic
360Q8—4882no—
361D4—41no—
362D4—41no—
363D4—82no—
364D4—1224no—
365D4—4882no—
366D4—4882no—
367D4—4882no—
368D4—2444no—
369D4—2441no—
370D4—2441no—
371D4—2442no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—82no—
377C2×C4—611no—
378C2×C4—611no—
379C2×C4—614no—
380C2×C4—611no—
381C2×C4—611no—
382C2×C4—4884no—
383C2×C4—4884no—
384C2×C4—4882no—
385C2×C4—4882no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4D8×D6111yescharacteristic
389C2×C4C2×D4×D6111yescharacteristic
390C8—21no—
391C8—1221no—
392C8—1221no—
393C8#41×D6112yes—
394C22—21no—
395C22—21no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—4888no—
400C22—4881no—
401C22—4881no—
402C22—4881no—
403C22—4882no—
404C22—4882no—
405C22—2444no—
406C22D16×D6112yes—
407C22C2×D8×D6111yescharacteristic
408C4—21no—
409C4—82no—
410C4—81no—
411C4—612no—
412C4—612no—
413C4—1221no—
414C4—1221no—
415C4—4881no—
416C4#1684027011yescharacteristic
417C4#1684056111yescharacteristic
418C2—22no—
419C2—161no—
420C2—9761no—
421C2—4884no—
422C2#2145581611yescharacteristic
423C2C2×D16×D6111yescharacteristic
424C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2232C2
socleC22×C61C22×D8
hypercenter#2232D61
hypocenterC61#33660
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2232

Complements

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

NH ≅ G/Ntype
1. #21456393C2semidirect
2. C2×#16839844C2semidirect
3. #21447999C2semidirect
4. #21447955C2semidirect
5. #21447853C2semidirect
6. #21447733C2semidirect
7. #21447730C2semidirect
8. #21447665C2semidirect
9. #21447663C2semidirect
10. #21447615C2semidirect
11. C61×#2232C2semidirect
17. #16840166C22semidirect
18. #16840137C22semidirect
22. D8×Dic61C22semidirect
27. #16839844C22semidirect
31. #16839587C22semidirect
32. #16839580C22semidirect
35. #16839562C22semidirect
37. C16×Dic61C22semidirect
40. C61×#366C22semidirect
41. C61×#357C22semidirect
137. D488#41semidirect
163. C2×Dic61D16semidirect
209. #2232D61semidirect
246. #366C2×D61semidirect
251. #357C2×D61semidirect
255. C61#33660semidirect

Maximal subgroups

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

subgroupindex
#214563932
C2×#168398442
#214479992
#214479552
#214478532
#214477332
#214477302
#214476652
#214476632
#214476152
C61×#22322
#3366061