Finite Group Databasebeta

← #29652506

Subgroups — #29652506

Order 15616.

930 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 167 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#21455767C212yesmaximal
3#21444860C214yesmaximal, complemented
4#21444826C214yesmaximal, complemented
5#21444351C212yesmaximal
6#21444149C211yesmaximal, characteristic
7C61×#2484C211yesmaximal, characteristic
8#16840393—24no—
9#16840393—22no—
10#16840393C2214yes—
11C2×#14519776—22no—
12C2×#14519768—22no—
13C2×#14519756—21no—
14C2×#14519756C2212yes—
15C2×#14519736—21no—
16#16840103—24no—
17#16840097C2214yes—
18#16840092C2214yes—
19#16839805—22no—
20#16839770—22no—
21#16839726C2212yes—
22#16839498C2218yes—
23#16839488C2214yes—
24C2×#117×C61C2211yescharacteristic
25C22×C4⋊C4×C61C2211yescharacteristic
26C61×#400C2212yes—
27C61×#398C2212yes—
28C61×#387—22no—
29C61×#385—21no—
30C61×#381—21no—
31C61×#381C2211yescharacteristic
32C22×Dic122—21no—
33C22×Dic122D412yes—
34C2×#11758194—22no—
35#14519776—28no—
36#14519776—28no—
37#14519768—28no—
38#14519768—28no—
39#14519756—24no—
40#14519756—24no—
41#14519756—22no—
42#14519756—22no—
43#14519756D414yes—
44#14519756C2314yes—
45#14519736—24no—
46#14519736—24no—
47C2×C4⋊Dic61—21no—
48C2×C4⋊Dic61—21no—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61D412yes—
51C2×#11758196—22no—
52C2×#11758196—22no—
53C2×C22⋊Dic61—22no—
54C2×C4×Dic61—22no—
55#14519710—24no—
56#14519708—24no—
57#14519700—24no—
58#14519700—24no—
59C22⋊#92418—22no—
60C22⋊#92418C2314yes—
61C23×C4×C61D411yescharacteristic
62C23×C4×C61D411yescharacteristic
63C23×C4×C61C2311yescharacteristic
64#117×C61—22no—
65#117×C61—22no—
66C2×C4⋊C4×C61—22no—
67C2×C4⋊C4×C61—22no—
68C2×C4⋊C4×C61—22no—
69C2×C4⋊C4×C61D412yes—
70C2×C4⋊C4×C61C2311yescharacteristic
71C2×C4⋊C4×C61C2311yescharacteristic
72C2×C4⋊C4×C61C2312yes—
73C2×C22⋊C4×C61—21no—
74C2×C22⋊C4×C61—21no—
75C2×C42×C61—21no—
76#95×C61—22no—
77#95×C61—22no—
78#95×C61—21no—
79#95×C61—22no—
80#95×C61C2312yes—
81C24×C61C2×D411yescharacteristic
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—24no—
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61—22no—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—42no—
91C22×C4×C61—42no—
92C22×C4×C61C2×D412yes—
93C22×C4×C61C2×D412yes—
94C22×C4×C61C2×D412yes—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D412yes—
97C22×C4×C61C2411yescharacteristic
98C22×Dic61—22no—
99C22×Dic61—22no—
100C2×Dic122—24no—
101C2×Dic122—22no—
102C2×Dic122—44no—
103C2×Dic122—82no—
104C2×Dic122C2×D414yes—
105C4⋊C4×C61—24no—
106C4⋊C4×C61—24no—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—22no—
109C4⋊C4×C61—44no—
110C22⋊C4×C61—42no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113C42×C61—42no—
114C22⋊Dic61—24no—
115C22⋊Dic61—24no—
116C4⋊Dic61—24no—
117C4⋊Dic61—24no—
118C4⋊Dic61—24no—
119C4⋊Dic61—22no—
120C4⋊Dic61C2×D414yes—
121#11758196—28no—
122#11758196—24no—
123#11758196—24no—
124C4×Dic61—28no—
125#11758194—44no—
126C2×#92418—24no—
127C23×C61—22no—
128C23×C61—22no—
129C23×C61—41no—
130C23×C61#12011yescharacteristic
131C23×C61#12011yescharacteristic
132C23×C61C22×D411yescharacteristic
133C2×C4×C61—24no—
134C2×C4×C61—24no—
135C2×C4×C61—22no—
136C2×C4×C61—22no—
137C2×C4×C61—24no—
138C2×C4×C61—22no—
139C2×C4×C61—42no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—42no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61—42no—
146C2×C4×C61—42no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61—81no—
152C2×C4×C61#12012yes—
153C2×C4×C61#13714yes—
154C2×C4×C61C22×D412yes—
155C2×Dic61—24no—
156C2×Dic61—24no—
157C2×Dic61—44no—
158C2×Dic61—44no—
159Dic122—44no—
160Dic122—82no—
161Dic122—84no—
162#92418—44no—
163#18148—611nomaximal, Sylow
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—44no—
172C22×C61#45812yes—
173C22×C61C2×#12011yescharacteristic
174C22×C61C2×#13712yes—
175C4×C61—24no—
176C4×C61—22no—
177C4×C61—22no—
178C4×C61—42no—
179C4×C61—82no—
180C4×C61—81no—
181C4×C61—81no—
182Dic61—44no—
183Dic61—44no—
184C2×#544—611no—
185#3095—612no—
186#2484D6111yescharacteristic
187#1617—614no—
188#1611—614no—
189#1511—612no—
190#1463—611no—
191C2×C61—22no—
192C2×C61—22no—
193C2×C61—41no—
194C2×C61#300411yescharacteristic
195C2×C61#302011yescharacteristic
196C2×C61C2×#45811yescharacteristic
197#544—614no—
198#544—1224no—
199#544—1222no—
200C2×C4⋊Q8—1222no—
201C2×#125—1222no—
202C2×C22⋊Q8—612no—
203C2×C22⋊Q8—1221no—
204C2×#117—1221no—
205C2×#117C2×D6111yescharacteristic
206C22×C4⋊C4C2×D6111yescharacteristic
207#485—1224no—
208#484—614no—
209#483—614no—
210#428—1222no—
211#419—1222no—
212#407—612no—
213#400C2×D6112yes—
214#398C2×D6112yes—
215#387—22no—
216#385—21no—
217#381—21no—
218#381C2×D6111yescharacteristic
219#328—618no—
220#324—614no—
221C61#1814811yescharacteristic, complemented, Sylow
222C22×Q8—612no—
223C22×Q8—1221no—
224C23×C4C22⋊D6111yescharacteristic
225C23×C4C22⋊D6111yescharacteristic
226C23×C4C22×D6111yescharacteristic
227C2×M16—1222no—
228C4⋊Q8—1228no—
229C4⋊Q8—1228no—
230#125—1228no—
231#125—1228no—
232C22⋊Q8—614no—
233C22⋊Q8—614no—
234C22⋊Q8—1224no—
235C22⋊Q8—1222no—
236C22⋊Q8—1222no—
237C22⋊Q8—1224no—
238#117—22no—
239#117—22no—
240#117—1224no—
241#117—1224no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—612no—
246C2×C4⋊C4—1221no—
247C2×C4⋊C4—1221no—
248C2×C4⋊C4—1221no—
249C2×C4⋊C4—1222no—
250C2×C4⋊C4—1222no—
251C2×C4⋊C4C22⋊D6112yes—
252C2×C4⋊C4C22×D6111yescharacteristic
253C2×C4⋊C4C22×D6111yescharacteristic
254C2×C4⋊C4C22×D6112yes—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—1222no—
258C2×C42—21no—
259C2×C42—1222no—
260#107—1224no—
261#106—1224no—
262#103—1224no—
263#103—1224no—
264C22⋊C8—614no—
265C22⋊C8—1222no—
266#95—22no—
267#95—22no—
268#95—21no—
269#95—22no—
270#95C22×D6112yes—
271C24C2×C22⋊D6111yescharacteristic
272C2×Q8—614no—
273C2×Q8—1224no—
274C2×Q8—1222no—
275C2×Q8—4882no—
276C2×Q8—2444no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—1222no—
288C22×C4—1222no—
289C22×C4D4×D6112yes—
290C22×C4D4×D6112yes—
291C22×C4C2×C22⋊D6112yes—
292C22×C4C2×C22⋊D6112yes—
293C22×C4C2×C22⋊D6111yescharacteristic
294C22×C4C23×D6111yescharacteristic
295M16—2444no—
296C2×C8—1224no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—44no—
302C4⋊C4—614no—
303C4⋊C4—1228no—
304C4⋊C4—1224no—
305C4⋊C4—1224no—
306C4⋊C4—1224no—
307C4⋊C4—1224no—
308C4⋊C4—1222no—
309C4⋊C4—1224no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—1224no—
314C22⋊C4—1224no—
315C42—42no—
316C42—1228no—
317C23—22no—
318C23—22no—
319C23—41no—
320C23#1451974511yescharacteristic
321C23#1451974511yescharacteristic
322C23C22×C22⋊D6111yescharacteristic
323Q8—4882no—
324Q8—4884no—
325Q8—2444no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—81no—
345C2×C4—1224no—
346C2×C4—1224no—
347C2×C4—2444no—
348C2×C4—2444no—
349C2×C4#1451974512yes—
350C2×C4#1451980714yes—
351C2×C4C2×D4×D6112yes—
352C8—2444no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—44no—
361C22#1683995112yes—
362C22C2×#1451974511yescharacteristic
363C22C2×#1451980712yes—
364C4—22no—
365C4—22no—
366C4—24no—
367C4—42no—
368C4—82no—
369C4—81no—
370C4—81no—
371C4—2444no—
372C4—2444no—
373C2—22no—
374C2—22no—
375C2—41no—
376C2#2145505211yescharacteristic
377C2#2145518211yescharacteristic
378C2C2×#1683995111yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2484C2
socleC22×C61C2×#120
prime radicalC24×C61C2×D4
hypercenter#2484D61
hypocenterC61#18148
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2484

Complements

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

NH ≅ G/Ntype
3. #21444860C2semidirect
4. #21444826C2semidirect
221. C61#18148semidirect

Maximal subgroups

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

subgroupindex
C2×#168403932
#214557672
#214448602
#214448262
#214443512
#214441492
C61×#24842
#1814861