Finite Group Databasebeta

← #29653939

Subgroups — #29653939

Order 15616.

940 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840393C211yesmaximal, characteristic
2#21456594C212yesmaximal
3#21451632C211yesmaximal, characteristic
4#21444863C214yesmaximal, complemented
5#21444795C214yesmaximal, complemented
6#21444392C212yesmaximal
7C2×C61×#324C211yesmaximal, characteristic
8#16840393—24no—
9#16840393—22no—
10#16840393C2214yes—
11C2×#14519776—22no—
12C2×#14519768—22no—
13C2×#14519756—21no—
14C2×#14519756C2212yes—
15C2×#14519756C2211yescharacteristic
16C2×#14519736—21no—
17#16840098—22no—
18#16840098C2214yes—
19#16840003—24no—
20#16839920—22no—
21#16839920C2214yes—
22#16839701C2212yes—
23#16839695—22no—
24#16839682C2212yes—
25#16839635—21no—
26#16839625—21no—
27#16839622C2211yescharacteristic
28#16839500C2218yes—
29C22×C4⋊C4×C61C2211yescharacteristic
30C2×C22⋊C8×C61C2212yes—
31C61×#324C2214yes—
32C22×Dic122—21no—
33C22×Dic122D412yes—
34C22×Dic122C2311yescharacteristic
35C2×Dic244—24no—
36#14519776—28no—
37#14519776—28no—
38#14519768—28no—
39#14519768—28no—
40#14519756—24no—
41#14519756—24no—
42#14519756—22no—
43#14519756—22no—
44#14519756—22no—
45#14519756—22no—
46#14519756D414yes—
47#14519756C2314yes—
48#14519736—24no—
49#14519736—24no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61D412yes—
55C2×C4⋊Dic61C2311yescharacteristic
56C2×#11758196—22no—
57C2×#11758196—22no—
58C2×#11758196—21no—
59C2×#11758196—22no—
60C2×C22⋊Dic61—22no—
61C2×C22⋊Dic61—21no—
62C2×C22⋊Dic61—21no—
63C2×C4×Dic61—22no—
64#14519711—24no—
65#14519702—24no—
66#14519702—24no—
67#14519684—22no—
68#14519684—22no—
69#14519684—22no—
70#14519684C2312yes—
71C23×C4×C61D411yescharacteristic
72C23×C4×C61D411yescharacteristic
73C23×C4×C61C2311yescharacteristic
74C22×C8×C61—22no—
75C2×C4⋊C4×C61—22no—
76C2×C4⋊C4×C61—22no—
77C2×C4⋊C4×C61D412yes—
78C2×C4⋊C4×C61C2312yes—
79C22⋊C8×C61—22no—
80C22⋊C8×C61C2314yes—
81C24×C61C2×D411yescharacteristic
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—22no—
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61—81no—
88C22×C4×C61—81no—
89C22×C4×C61Q1614yescomplemented
90C22×C4×C61C2×D412yes—
91C22×C4×C61C2×D412yes—
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D412yes—
94C22×C4×C61C2×D412yes—
95C22×C4×C61C2411yescharacteristic
96C22×Dic61—22no—
97C22×Dic61—22no—
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61—22no—
101C2×Dic122—22no—
102C2×Dic122—24no—
103C2×Dic122—22no—
104C2×Dic122—81no—
105C2×Dic122—82no—
106C2×Dic122—82no—
107C2×Dic122C2×D414yes—
108C2×C8×C61—24no—
109C2×C8×C61—82no—
110C4⋊C4×C61—24no—
111C4⋊C4×C61—24no—
112C22⋊Dic61—24no—
113C22⋊Dic61—24no—
114C22⋊Dic61—41no—
115C22⋊Dic61—41no—
116C22⋊Dic61—42no—
117C4⋊Dic61—24no—
118C4⋊Dic61—24no—
119C4⋊Dic61—24no—
120C4⋊Dic61—22no—
121C4⋊Dic61—22no—
122C4⋊Dic61—81no—
123C4⋊Dic61C2×D414yes—
124#11758196—28no—
125#11758196—24no—
126#11758196—24no—
127#11758196—42no—
128#11758196—82no—
129C4×Dic61—28no—
130Dic244—84no—
131C23×C61—41no—
132C23×C61—41no—
133C23×C61#12011yescharacteristic
134C23×C61#12011yescharacteristic
135C23×C61C2×Q1612yes—
136C23×C61#13712yes—
137C23×C61C22×D411yescharacteristic
138C2×C4×C61—22no—
139C2×C4×C61—22no—
140C2×C4×C61—24no—
141C2×C4×C61—24no—
142C2×C4×C61—22no—
143C2×C4×C61—41no—
144C2×C4×C61—42no—
145C2×C4×C61—42no—
146C2×C4×C61—41no—
147C2×C4×C61—42no—
148C2×C4×C61—42no—
149C2×C4×C61—81no—
150C2×C4×C61#12012yes—
151C2×C4×C61C2×Q1614yes—
152C2×C4×C61C22×D412yes—
153C2×Dic61—24no—
154C2×Dic61—24no—
155C2×Dic61—44no—
156C2×Dic61—44no—
157C2×Dic61—42no—
158C2×Dic61—42no—
159C2×Dic61—42no—
160C2×Dic61—42no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—42no—
164Dic122—44no—
165Dic122—44no—
166Dic122—82no—
167Dic122—84no—
168C8×C61—44no—
169#18295—611nomaximal, Sylow
170C22×C61—21no—
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—42no—
177C22×C61—42no—
178C22×C61#45112yes—
179C22×C61#45112yes—
180C22×C61C2×#12011yescharacteristic
181C22×C61C22×Q1611yescharacteristic
182C22×C61C2×#13711yescharacteristic
183C4×C61—24no—
184C4×C61—22no—
185C4×C61—22no—
186C4×C61—42no—
187Dic61—44no—
188Dic61—44no—
189Dic61—82no—
190Dic61—81no—
191Dic61—81no—
192C2×#544—611no—
193#3189—612no—
194#2627—611no—
195#1617—614no—
196#1605—614no—
197#1520—612no—
198C2×#324D6111yescharacteristic
199C2×C61—21no—
200C2×C61—21no—
201C2×C61—22no—
202C2×C61—22no—
203C2×C61C2×#45111yescharacteristic
204C2×C61#302011yescharacteristic
205C2×C61#302611yescharacteristic
206#544—614no—
207#544—1224no—
208#544—1222no—
209C2×C4⋊Q8—1222no—
210C2×#125—1222no—
211C2×C22⋊Q8—612no—
212C2×C22⋊Q8—611no—
213C2×C22⋊Q8—1221no—
214C2×#117—1221no—
215C22×C4⋊C4C2×D6111yescharacteristic
216#484—614no—
217#484—1222no—
218#467—1224no—
219#451—614no—
220#451—1222no—
221C2×C22⋊C8C2×D6112yes—
222#400—612no—
223#398—1222no—
224#395—612no—
225#385—1221no—
226#381—1221no—
227#380—611no—
228#328—618no—
229#324C2×D6114yes—
230C61#1829511yescharacteristic, complemented, Sylow
231C22×Q8—612no—
232C22×Q8—611no—
233C22×Q8—1221no—
234C23×C4D24411yescharacteristic
235C23×C4D24411yescharacteristic
236C23×C4C22×D6111yescharacteristic
237C2×Q16—1224no—
238C22×C8—22no—
239C4⋊Q8—1228no—
240C4⋊Q8—1228no—
241#125—1228no—
242#125—1228no—
243C22⋊Q8—614no—
244C22⋊Q8—614no—
245C22⋊Q8—1224no—
246C22⋊Q8—1222no—
247C22⋊Q8—1224no—
248C22⋊Q8—1222no—
249C22⋊Q8—1222no—
250C22⋊Q8—1222no—
251#117—1224no—
252#117—1224no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—612no—
256C2×C4⋊C4—611no—
257C2×C4⋊C4—1221no—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4—1221no—
260C2×C4⋊C4—1222no—
261C2×C4⋊C4—1222no—
262C2×C4⋊C4—1221no—
263C2×C4⋊C4—1222no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4D24412yes—
266C2×C4⋊C4C22×D6112yes—
267C2×C22⋊C4—1222no—
268C2×C22⋊C4—1221no—
269C2×C22⋊C4—1221no—
270C2×C42—1222no—
271#107—1224no—
272#103—1224no—
273#103—1224no—
274C22⋊C8—22no—
275C22⋊C8C22×D6114yes—
276#95—612no—
277#95—1222no—
278#95—1222no—
279#95—1222no—
280C24C2×D24411yescharacteristic
281C2×Q8—614no—
282C2×Q8—1224no—
283C2×Q8—1222no—
284C2×Q8—1222no—
285C2×Q8—4882no—
286C2×Q8—4882no—
287C2×Q8—4881no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—81no—
294C22×C4—81no—
295C22×C4—1222no—
296C22×C4—1222no—
297C22×C4—1222no—
298C22×C4—1221no—
299C22×C4—1221no—
300C22×C4Dic24414yescomplemented
301C22×C4C2×D24412yes—
302C22×C4C2×D24412yes—
303C22×C4C2×D24411yescharacteristic
304C22×C4D4×D6112yes—
305C22×C4D4×D6112yes—
306C22×C4C23×D6111yescharacteristic
307Q16—4884no—
308C2×C8—24no—
309C2×C8—82no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—614no—
313C4⋊C4—1228no—
314C4⋊C4—1224no—
315C4⋊C4—1224no—
316C4⋊C4—1224no—
317C4⋊C4—1224no—
318C4⋊C4—1222no—
319C4⋊C4—1224no—
320C4⋊C4—1222no—
321C4⋊C4—4882no—
322C4⋊C4—4881no—
323C4⋊C4—2442no—
324C22⋊C4—1224no—
325C22⋊C4—1224no—
326C22⋊C4—2442no—
327C22⋊C4—2441no—
328C22⋊C4—2441no—
329C42—1228no—
330C23—41no—
331C23—41no—
332C23#1451974711yescharacteristic
333C23#1451974711yescharacteristic
334C23C2×Dic24412yes—
335C23#1451981012yes—
336C23C22×D24411yescharacteristic
337Q8—4882no—
338Q8—4884no—
339Q8—2444no—
340Q8—2444no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—81no—
353C2×C4—1224no—
354C2×C4—1224no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4—2442no—
358C2×C4—2442no—
359C2×C4—2441no—
360C2×C4—2441no—
361C2×C4—2442no—
362C2×C4—2444no—
363C2×C4—2444no—
364C2×C4#1451974712yes—
365C2×C4C2×Dic24414yes—
366C2×C4C2×D4×D6112yes—
367C8—44no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#1683992012yes—
377C22#1683992012yes—
378C22C2×#1451974711yescharacteristic
379C22C22×Dic24411yescharacteristic
380C22C2×#1451981011yescharacteristic
381C4—22no—
382C4—22no—
383C4—24no—
384C4—42no—
385C4—4882no—
386C4—4881no—
387C4—4881no—
388C4—2444no—
389C4—2444no—
390C2—21no—
391C2—21no—
392C2—22no—
393C2—22no—
394C2C2×#1683992011yescharacteristic
395C2#2145518611yescharacteristic
396C2#2145521811yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#324C2
socleC22×C61C2×#120
prime radicalC24×C61C2×D4
hypercenterC2×#324D61
hypocenterC61#18295
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
4. #21444863C2semidirect
5. #21444795C2semidirect
89. C22×C4×C61Q16semidirect
230. C61#18295semidirect
300. C22×C4Dic244semidirect

Maximal subgroups

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

subgroupindex
C2×#168403932
#214565942
#214516322
#214448632
#214447952
#214443922
C2×C61×#3242
#1829561