Finite Group Databasebeta

← #29689809

Subgroups — #29689809

Order 15616.

1022 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
1C2×#16840375C211yesmaximal, characteristic
2C2×#16839821C211yesmaximal, characteristic
3#21448444C211yesmaximal, characteristic
4#21446789C211yesmaximal, characteristic, complemented
5#21446638C211yesmaximal, characteristic, complemented
6#21446621C211yesmaximal, characteristic, complemented
7#21446130C212yesmaximal
8#21446125C212yesmaximal
9#21445833C212yesmaximal, complemented
10#21445823C212yesmaximal, complemented
11C61×#1991C211yesmaximal, characteristic, complemented
12#16840375—24no—
13#16840375C418yescomplemented
14C2×#14519777C2211yescharacteristic
15C2×#14519772—22no—
16C2×#14519768—22no—
17C2×#14519757C2212yes—
18C2×#14519742C2212yes—
19C2×#14519735C2211yescharacteristic
20C2×#14519734C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22#16839821—24no—
23C2×#14519706C2212yes—
24C2×#14519702C2212yes—
25C2×#14519702C2212yes—
26C2×#14519700C2212yes—
27C2×#14519700C2212yes—
28C2×#14519689C2212yes—
29C2×#14519688C2211yescharacteristic
30#16839659—22no—
31#16839636—22no—
32#16839632C2211yescharacteristic
33#16839631C2211yescharacteristic
34#16839627C2211yescharacteristic
35#16839524C2212yes—
36#16839524C2212yes—
37C2×#125×C61C2211yescharacteristic
38C2×C4×C8×C61C2211yescharacteristic
39C61×#382C2211yescharacteristic
40C61×#340C2212yes—
41C61×#340C2212yes—
42C22×Dic122C2312yes—
43C22×C4×D61—21no—
44C22×C4×D61C2311yescharacteristic
45C22×#92418C2312yes—
46#14519777—22no—
47#14519777C2×C414yescomplemented
48#14519772—216no—
49#14519768—216no—
50#14519757—24no—
51#14519757C2×C418yes—
52#14519742—24no—
53#14519742C2×C418yescomplemented
54#14519735—42no—
55#14519734C2×C414yes—
56#14519734D414yes—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61—22no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2311yescharacteristic
62C2×#11758196—22no—
63C2×#11758196—21no—
64C2×#11758196—22no—
65C2×#11758196D412yes—
66C4⋊C4×D61—42no—
67C2×#11758198—22no—
68C2×#11758198—21no—
69C2×#11758198D412yes—
70C2×C4×Dic61—22no—
71C2×C4×Dic61—21no—
72C2×C4×Dic61C2311yescharacteristic
73#14519706—24no—
74#14519702—42no—
75#14519702—42no—
76#14519700—42no—
77#14519700—42no—
78#14519689—24no—
79#14519688—22no—
80#14519684—22no—
81C22×C8×C61C2312yes—
82#125×C61—22no—
83#125×C61C2×C414yescomplemented
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C4⋊C4×C61C2312yes—
88C2×C42×C61C2311yescharacteristic
89C4×C8×C61—22no—
90#95×C61—22no—
91C23×D61C2×D411yescharacteristic
92C22×C4×C61—22no—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61C2×D412yes—
96C22×C4×C61C2411yescharacteristic
97C22×Dic61—22no—
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61C2×D411yescharacteristic
101C22×Dic61C2×D412yes—
102C2×C4×D61—22no—
103C2×C4×D61—22no—
104C2×C4×D61—42no—
105C2×C4×D61—82no—
106C2×C4×D61C22×C412yes—
107C2×C4×D61C2×D412yes—
108C2×Dic122—22no—
109C2×Dic122—22no—
110C2×Dic122—22no—
111C2×Dic122—22no—
112C2×Dic122—22no—
113C2×Dic122C22×C414yes—
114C2×C8×C61—22no—
115C2×C8×C61—22no—
116C2×C8×C61—22no—
117C4⋊C4×C61—28no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61C22×C414yes—
122C42×C61C22×C412yes—
123C42×C61C2×D412yes—
124#11758198—28no—
125#11758198—42no—
126#11758198C22⋊C418yes—
127C4⋊Dic61—28no—
128C4⋊Dic61—28no—
129C4⋊Dic61—22no—
130C4⋊Dic61—22no—
131C4⋊Dic61—82no—
132#11758196—28no—
133#11758196—42no—
134#11758196—82no—
135#11758196C22⋊C418yes—
136C4×Dic61—28no—
137C4×Dic61—42no—
138C4×Dic61C22×C412yes—
139C4×Dic61C2×D412yes—
140C2×#92418—22no—
141C2×#92418—22no—
142C2×#92418—22no—
143C23×C61C22×D411yescharacteristic
144C22×D61—21no—
145C22×D61—21no—
146C22×D61—22no—
147C22×D61—22no—
148C22×D61C2×C22⋊C412yes—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—24no—
152C2×C4×C61—42no—
153C2×C4×C61—42no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—42no—
157C2×C4×C61—42no—
158C2×C4×C61C2×C22⋊C414yes—
159C2×C4×C61C23×C411yescharacteristic
160C2×C4×C61C22×D411yescharacteristic
161C2×C4×C61H(5,𝔽2)12yes—
162C2×C4×C612−1+412yes—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—22no—
166C2×Dic61—22no—
167C2×Dic61—24no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—42no—
173C2×Dic61—42no—
174C2×Dic61C2×C22⋊C414yes—
175C2×Dic61C2×C22⋊C412yes—
176C4×D61—24no—
177C4×D61—24no—
178C4×D61—84no—
179Dic122—24no—
180Dic122—24no—
181Dic122—42no—
182Dic122—42no—
183C8×C61—42no—
184#92418—42no—
185#21873—611nomaximal, Sylow
186C22×C61C22×C22⋊C411yescharacteristic
187C22×C61#53511yescharacteristic
188C22×C61#53611yescharacteristic
189C22×C61#57712yes—
190C22×C61#57812yes—
191C2×D61—22no—
192C2×D61—24no—
193C2×D61—41no—
194C2×D61—41no—
195C2×D61—42no—
196C4×C61—24no—
197C4×C61—44no—
198C4×C61—81no—
199C4×C61—81no—
200C4×C61#52012yes—
201C4×C61#53712yes—
202Dic61—22no—
203Dic61—24no—
204Dic61—44no—
205Dic61—81no—
206Dic61—81no—
207C2×#541—611no—
208C2×#433—611no—
209#2317—611no—
210#1992—611no—
211#1991D6111yescharacteristic, complemented
212#1960—611no—
213#1956—611no—
214#1873—612no—
215#1872—612no—
216#1805—612no—
217#1803—612no—
218C2×C61#231111yescharacteristic
219C2×C61#289812yes—
220C2×C61#322211yescharacteristic
221C2×C61#322311yescharacteristic
222C2×C61#324212yes—
223D61—42no—
224#541—618no—
225#541—1224no—
226C2×C4⋊Q8—611no—
227C2×#126—1222no—
228C2×#125—1222no—
229C2×#125C2×D6111yescharacteristic
230C2×C22⋊Q8—612no—
231C2×C4×Q8—612no—
232C2×#117—611no—
233C2×#117—611no—
234C22×C4⋊C4—611no—
235#433—1224no—
236C2×C4⋊C8—612no—
237C2×#103—612no—
238C2×#103—612no—
239C2×#103—612no—
240C2×#103—612no—
241C2×C22⋊C8—612no—
242C2×#97—611no—
243C2×C4×C8C2×D6111yescharacteristic
244#389—1222no—
245#385—1222no—
246#384—611no—
247#384—611no—
248#382—611no—
249#382C2×D6111yescharacteristic
250#340—612no—
251#340—612no—
252#340C2×D6112yes—
253#340C2×D6112yes—
254C61#2187311yescharacteristic, complemented, Sylow
255C22×Q8—612no—
256C23×C4—611no—
257C23×C4—1221no—
258C22×C8—612no—
259C22×C8C22×D6112yes—
260C4⋊Q8—614no—
261C4⋊Q8—1222no—
262#126—12216no—
263#125—22no—
264#125—12216no—
265#125C4×D6114yescomplemented
266C22⋊Q8—618no—
267C22⋊Q8—1224no—
268C4×Q8—618no—
269C4×Q8—1224no—
270#117—614no—
271#117—614no—
272#117—2442no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—612no—
275C2×C4⋊C4—611no—
276C2×C4⋊C4—611no—
277C2×C4⋊C4—1222no—
278C2×C4⋊C4—1222no—
279C2×C4⋊C4—1222no—
280C2×C4⋊C4—1222no—
281C2×C4⋊C4—1222no—
282C2×C4⋊C4—1221no—
283C2×C4⋊C4—2442no—
284C2×C4⋊C4C22×D6111yescharacteristic
285C2×C4⋊C4C22×D6111yescharacteristic
286C2×C4⋊C4C22×D6112yes—
287C2×C22⋊C4—612no—
288C2×C22⋊C4—1222no—
289C2×C22⋊C4—1221no—
290C2×C42—611no—
291C2×C42—1222no—
292C2×C42—1221no—
293C2×C42C22×D6111yescharacteristic
294C4⋊C8—1224no—
295#103—2442no—
296#103—2442no—
297#103—2442no—
298#103—2442no—
299C22⋊C8—1224no—
300#97—1222no—
301C4×C8—22no—
302#95—22no—
303#95—1222no—
304C24—611no—
305C2×Q8—614no—
306C2×Q8—1222no—
307C2×Q8—1222no—
308C2×Q8—1222no—
309C2×Q8—1222no—
310C2×Q8—1222no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—612no—
315C22×C4—611no—
316C22×C4—612no—
317C22×C4—612no—
318C22×C4—1222no—
319C22×C4—1222no—
320C22×C4—1222no—
321C22×C4—1221no—
322C22×C4—1221no—
323C22×C4—4882no—
324C22×C4—2442no—
325C22×C4D4×D6112yes—
326C22×C4C23×D6111yescharacteristic
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—1222no—
331C2×C8—1222no—
332C2×C8—1222no—
333C4⋊C4—22no—
334C4⋊C4—28no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—618no—
338C4⋊C4—1228no—
339C4⋊C4—1228no—
340C4⋊C4—1228no—
341C4⋊C4—1222no—
342C4⋊C4—1222no—
343C4⋊C4—4882no—
344C4⋊C4—4882no—
345C4⋊C4—2442no—
346C4⋊C4C2×C4×D6114yes—
347C22⋊C4—618no—
348C22⋊C4—1228no—
349C22⋊C4—2442no—
350C42—612no—
351C42—612no—
352C42—1228no—
353C42—2442no—
354C42C2×C4×D6112yes—
355C42D4×D6112yes—
356C23—612no—
357C23—1222no—
358C23—1222no—
359C23—1221no—
360C23—1221no—
361C23C2×D4×D6111yescharacteristic
362Q8—1224no—
363Q8—1224no—
364Q8—2442no—
365Q8—2442no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—614no—
376C2×C4—612no—
377C2×C4—1222no—
378C2×C4—1222no—
379C2×C4—1221no—
380C2×C4—1221no—
381C2×C4—1224no—
382C2×C4—1224no—
383C2×C4—1224no—
384C2×C4—4884no—
385C2×C4—2442no—
386C2×C4—2442no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4—2442no—
390C2×C4—2442no—
391C2×C4C22⋊C4×D6114yes—
392C2×C4C22×C4×D6111yescharacteristic
393C2×C4C2×D4×D6111yescharacteristic
394C2×C4#1451982312yes—
395C2×C4#1451982512yes—
396C8—42no—
397C8—2442no—
398C22—1224no—
399C22—1222no—
400C22—2441no—
401C22—2442no—
402C22—2441no—
403C22C2×C22⋊C4×D6111yescharacteristic
404C22#1684033011yescharacteristic
405C22#1684033711yescharacteristic
406C22#1684056912yes—
407C22#1684057612yes—
408C4—24no—
409C4—44no—
410C4—81no—
411C4—81no—
412C4—1224no—
413C4—1222no—
414C4—4881no—
415C4—4881no—
416C4—2444no—
417C4#1684024412yes—
418C4#1684034212yes—
419C2—2442no—
420C2#2144841011yescharacteristic
421C2#2145434812yes—
422C2#2145689411yescharacteristic
423C2#2145690511yescharacteristic
424C2#2145709012yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1991C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1991D61
hypocenterC61#21873
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1991

Complements

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

NH ≅ G/Ntype
4. #21446789C2semidirect
5. #21446638C2semidirect
6. #21446621C2semidirect
9. #21445833C2semidirect
10. #21445823C2semidirect
11. C61×#1991C2semidirect
13. #16840375C4semidirect
47. #14519777C2×C4semidirect
53. #14519742C2×C4semidirect
83. #125×C61C2×C4semidirect
211. #1991D61semidirect
254. C61#21873semidirect
265. #125C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403752
C2×#168398212
#214484442
#214467892
#214466382
#214466212
#214461302
#214461252
#214458332
#214458232
C61×#19912
#2187361