Finite Group Databasebeta

← #29734403

Subgroups — #29734403

Order 15616.

960 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840375C211yesmaximal, characteristic
2C2×#16840132C211yesmaximal, characteristic, complemented
3#21451650C211yesmaximal, characteristic
4#21447024C212yesmaximal
5#21447017C212yesmaximal
6#21446789C211yesmaximal, characteristic, complemented
7#21446621C211yesmaximal, characteristic, complemented
8#21446084C212yesmaximal, complemented
9#21446070C212yesmaximal, complemented
10#21445586C211yesmaximal, characteristic
11C2×C61×#499C211yesmaximal, characteristic, complemented
12#16840375—24no—
13#16840375—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519772—22no—
16C2×#14519768—22no—
17C2×#14519757C2212yes—
18C2×#14519742C2212yes—
19C2×#14519734C2211yescharacteristic
20#16840132—24no—
21C2×#14519702C2212yes—
22C2×#14519702C2212yes—
23C2×#14519700C2212yes—
24C2×#14519700C2212yes—
25C2×#14519689C2211yescharacteristic
26C2×#14519689C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28#16839695—22no—
29#16839684—22no—
30#16839658C2212yes—
31#16839632C2211yescharacteristic
32#16839619C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35#16839516C2211yescharacteristic
36#16839516C2211yescharacteristic
37C2×#125×C61C2211yescharacteristic
38C2×#125×C61C2211yescharacteristic
39C61×#499—24no—
40C2×#107×C61C2212yes—
41C2×#106×C61C2212yes—
42C2×C4×C8×C61C2211yescharacteristic
43C22×Dic122C2312yes—
44C22×C4×D61C2311yescharacteristic
45C22×#92418C2311yescharacteristic
46C22×#92418C2311yescharacteristic
47#14519777—22no—
48#14519777D414yes—
49#14519772—216no—
50#14519768—216no—
51#14519757—24no—
52#14519757—24no—
53#14519742—24no—
54#14519742—24no—
55#14519734—22no—
56#14519734—22no—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61—22no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61C2312yes—
61C2×#11758196—22no—
62C2×#11758196—22no—
63C2×#11758196D412yes—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C2×#11758198D412yes—
67C2×C4×Dic61—22no—
68C2×C4×Dic61C2311yescharacteristic
69#14519702—24no—
70#14519702—24no—
71#14519700—24no—
72#14519700—24no—
73#14519689—22no—
74#14519689—22no—
75#14519688—22no—
76#14519684—22no—
77#14519684—22no—
78C22×C8×C61C2311yescharacteristic
79C22×C8×C61C2311yescharacteristic
80#125×C61—22no—
81#125×C61—22no—
82#125×C61—22no—
83#125×C61D414yes—
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61—22no—
86C2×C4⋊C4×C61C2312yes—
87C2×C4⋊C4×C61C2312yes—
88C2×C42×C61C2311yescharacteristic
89#107×C61—24no—
90#106×C61—24no—
91C4×C8×C61—22no—
92C23×D61C2×D411yescharacteristic
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61C2×D412yes—
96C22×C4×C61C2411yescharacteristic
97C22×Dic61—22no—
98C22×Dic61—22no—
99C22×Dic61C2×D411yescharacteristic
100C22×Dic61C2×D412yes—
101C2×C4×D61—21no—
102C2×C4×D61—21no—
103C2×C4×D61—21no—
104C2×C4×D61—21no—
105C2×C4×D61—21no—
106C2×C4×D61—21no—
107C2×Dic122—22no—
108C2×Dic122—22no—
109C2×Dic122—22no—
110C2×Dic122—22no—
111C2×Dic122—22no—
112C2×Dic122C2×D414yes—
113C2×C8×C61—21no—
114C2×C8×C61—21no—
115C2×C8×C61—21no—
116C2×C8×C61—21no—
117C2×C8×C61#4112yes—
118C2×C8×C61#4112yes—
119C4⋊C4×C61—28no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—44no—
124C4⋊C4×C61C2×D414yes—
125C42×C61C2×D412yes—
126C42×C61C2×D412yes—
127#11758198—28no—
128#11758198—24no—
129#11758198—82no—
130C4⋊Dic61—28no—
131C4⋊Dic61—28no—
132C4⋊Dic61—22no—
133C4⋊Dic61—22no—
134C4⋊Dic61—82no—
135#11758196—28no—
136#11758196—24no—
137#11758196—82no—
138C4×Dic61—28no—
139C4×Dic61—21no—
140C4×Dic61—21no—
141C2×#92418—21no—
142C2×#92418—21no—
143C2×#92418—21no—
144C2×#92418—21no—
145C2×#92418—21no—
146C2×#92418—21no—
147C23×C61C22×D411yescharacteristic
148C22×D61—21no—
149C22×D61—21no—
150C22×D61—21no—
151C22×D61—22no—
152C22×D61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—22no—
155C2×C4×C61—24no—
156C2×C4×C61—42no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61#12014yes—
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C2×#4111yescharacteristic
165C2×C4×C61C2×#4111yescharacteristic
166C2×C4×C612−1+411yescharacteristic
167C2×C4×C612−1+411yescharacteristic
168C2×Dic61—22no—
169C2×Dic61—24no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—21no—
173C2×Dic61—22no—
174C2×Dic61—22no—
175C2×Dic61—42no—
176C2×Dic61—42no—
177C2×Dic61—42no—
178C2×Dic61—42no—
179C2×Dic61—42no—
180C4×D61—41no—
181C4×D61—41no—
182C4×D61—41no—
183C4×D61—41no—
184Dic122—24no—
185Dic122—24no—
186Dic122—42no—
187Dic122—42no—
188C8×C61—22no—
189C8×C61—22no—
190#92418—41no—
191#92418—41no—
192#25683—611nomaximal, Sylow
193C22×C61C2×#12011yescharacteristic
194C22×C61#53211yescharacteristic
195C22×C61#53611yescharacteristic
196C22×C61#57712yes—
197C22×C61#57812yes—
198C2×D61—41no—
199C2×D61—41no—
200C2×D61—41no—
201C2×D61—42no—
202C2×D61—42no—
203C4×C61—24no—
204C4×C61—44no—
205C4×C61—82no—
206C4×C61#54711yescharacteristic
207C4×C61#54711yescharacteristic
208C4×C61#54811yescharacteristic
209C4×C61#54811yescharacteristic
210Dic61—44no—
211Dic61—41no—
212Dic61—42no—
213Dic61—82no—
214C2×#541—611no—
215C2×#499D6111yescharacteristic, complemented
216C2×#491—611no—
217#2629—611no—
218#2036—612no—
219#2035—612no—
220#1992—611no—
221#1956—611no—
222#1865—612no—
223#1863—612no—
224#1750—611no—
225C2×C61#243211yescharacteristic
226C2×C61#301312yes—
227C2×C61#317411yescharacteristic
228C2×C61#317711yescharacteristic
229C2×C61#324212yes—
230D61—81no—
231#541—1224no—
232#541—1224no—
233C2×C4⋊Q8—611no—
234C2×#126—1222no—
235C2×#125—1222no—
236C2×#125C2×D6111yescharacteristic
237C2×#125C2×D6111yescharacteristic
238C2×C22⋊Q8—612no—
239C2×C4×Q8—612no—
240C2×#117—611no—
241#499—24no—
242#491—1224no—
243C2×#107C2×D6112yes—
244C2×#106C2×D6112yes—
245C2×#103—612no—
246C2×#103—612no—
247C2×#103—612no—
248C2×#103—612no—
249C2×C22⋊C8—611no—
250C2×C22⋊C8—611no—
251C2×#97—611no—
252C2×C4×C8C2×D6111yescharacteristic
253#398—1222no—
254#396—1222no—
255#389—612no—
256#384—611no—
257#380—612no—
258#340—612no—
259#340—612no—
260#336—611no—
261#336—611no—
262C61#2568311yescharacteristic, complemented, Sylow
263C22×Q8—612no—
264C23×C4—611no—
265C22×C8—611no—
266C22×C8—611no—
267C22×C8C22×D6111yescharacteristic
268C22×C8C22×D6111yescharacteristic
269C4⋊Q8—614no—
270C4⋊Q8—1222no—
271#126—12216no—
272#125—22no—
273#125—22no—
274#125—22no—
275#125—12216no—
276#125C22⋊D6114yes—
277C22⋊Q8—1224no—
278C22⋊Q8—1224no—
279C4×Q8—1224no—
280C4×Q8—1224no—
281#117—1222no—
282#117—1222no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—612no—
286C2×C4⋊C4—612no—
287C2×C4⋊C4—1222no—
288C2×C4⋊C4—1222no—
289C2×C4⋊C4—1222no—
290C2×C4⋊C4—1222no—
291C2×C4⋊C4—1222no—
292C2×C4⋊C4C22×D6112yes—
293C2×C4⋊C4C22×D6112yes—
294C2×C22⋊C4—612no—
295C2×C22⋊C4—1222no—
296C2×C22⋊C4—1222no—
297C2×C42—611no—
298C2×C42—1222no—
299C2×C42C22×D6111yescharacteristic
300#107—24no—
301#106—24no—
302#103—1224no—
303#103—1224no—
304#103—1224no—
305#103—1224no—
306C22⋊C8—1222no—
307C22⋊C8—1222no—
308#97—1222no—
309C4×C8—22no—
310#95—1222no—
311#95—1222no—
312C24—611no—
313C2×Q8—614no—
314C2×Q8—1222no—
315C2×Q8—1222no—
316C2×Q8—1222no—
317C2×Q8—1222no—
318C2×Q8—1222no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—611no—
322C22×C4—612no—
323C22×C4—1222no—
324C22×C4—1221no—
325C22×C4—1221no—
326C22×C4—1221no—
327C22×C4—1221no—
328C22×C4—1221no—
329C22×C4—1221no—
330C22×C4—1222no—
331C22×C4D4×D6112yes—
332C22×C4C23×D6111yescharacteristic
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—1221no—
338C2×C8—1221no—
339C2×C8—1221no—
340C2×C8—1221no—
341C2×C8—1221no—
342C2×C8—1221no—
343C2×C8#1175822612yes—
344C2×C8#1175822612yes—
345C4⋊C4—22no—
346C4⋊C4—28no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—44no—
350C4⋊C4—1228no—
351C4⋊C4—1228no—
352C4⋊C4—1224no—
353C4⋊C4—1228no—
354C4⋊C4—1222no—
355C4⋊C4—1222no—
356C4⋊C4—4882no—
357C4⋊C4—4882no—
358C4⋊C4C2×C22⋊D6114yes—
359C22⋊C4—1228no—
360C22⋊C4—1224no—
361C22⋊C4—4882no—
362C42—1228no—
363C42—1221no—
364C42—1221no—
365C42D4×D6112yes—
366C42C2×C22⋊D6112yes—
367C23—1221no—
368C23—1221no—
369C23—1222no—
370C23—1222no—
371C23—1221no—
372C23C2×D4×D6111yescharacteristic
373Q8—1224no—
374Q8—1224no—
375Q8—2442no—
376Q8—2442no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—1222no—
386C2×C4—1224no—
387C2×C4—1221no—
388C2×C4—1221no—
389C2×C4—1222no—
390C2×C4—1222no—
391C2×C4—1221no—
392C2×C4—2442no—
393C2×C4—2442no—
394C2×C4—2442no—
395C2×C4—2442no—
396C2×C4—2442no—
397C2×C4—2441no—
398C2×C4—2441no—
399C2×C4—2441no—
400C2×C4—2441no—
401C2×C4#1451974514yes—
402C2×C4C2×D4×D6111yescharacteristic
403C2×C4C22×C22⋊D6111yescharacteristic
404C2×C4C2×#1175822611yescharacteristic
405C2×C4C2×#1175822611yescharacteristic
406C2×C4#1451982511yescharacteristic
407C2×C4#1451982511yescharacteristic
408C8—22no—
409C8—22no—
410C8—2441no—
411C8—2441no—
412C22—2442no—
413C22—2441no—
414C22—2441no—
415C22—2442no—
416C22—2441no—
417C22C2×#1451974511yescharacteristic
418C22#1684031311yescharacteristic
419C22#1684033711yescharacteristic
420C22#1684056912yes—
421C22#1684057612yes—
422C4—24no—
423C4—44no—
424C4—82no—
425C4—4882no—
426C4—2442no—
427C4—2441no—
428C4—2444no—
429C4#1684041511yescharacteristic
430C4#1684041511yescharacteristic
431C4#1684042311yescharacteristic
432C4#1684042311yescharacteristic
433C2—4881no—
434C2#2144944411yescharacteristic
435C2#2145513012yes—
436C2#2145644911yescharacteristic
437C2#2145647911yescharacteristic
438C2#2145709012yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#499C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#499D61
hypocenterC61#25683
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#499

Complements

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

NH ≅ G/Ntype
2. C2×#16840132C2semidirect
6. #21446789C2semidirect
7. #21446621C2semidirect
8. #21446084C2semidirect
9. #21446070C2semidirect
11. C2×C61×#499C2semidirect
215. C2×#499D61semidirect
262. C61#25683semidirect

Maximal subgroups

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

subgroupindex
C2×#168403752
C2×#168401322
#214516502
#214470242
#214470172
#214467892
#214466212
#214460842
#214460702
#214455862
C2×C61×#4992
#2568361