Finite Group Databasebeta

← #29652428

Subgroups — #29652428

Order 15616.

1054 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1054 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×#16840388C211yesmaximal, characteristic
2#21455767C212yesmaximal, complemented
3#21444843C214yesmaximal, complemented
4#21444809C214yesmaximal, complemented
5#21444351C212yesmaximal, complemented
6#21444146C211yesmaximal, characteristic
7C61×#2482C211yesmaximal, characteristic
8#16840388—24no—
9#16840388—22no—
10#16840388C2214yes—
11C2×#14519756—21no—
12C2×#14519756C2212yes—
13C2×#14519754—22no—
14C2×#14519754—22no—
15C22×C22⋊Dic61—21no—
16#16840103—24no—
17#16840097C2214yescomplemented
18#16840092C2214yescomplemented
19#16839805—22no—
20#16839770—22no—
21#16839726C2212yes—
22#16839498C2218yescomplemented
23#16839486C2214yes—
24C2×#117×C61C2211yescharacteristic
25C22×C22⋊C4×C61C2211yescharacteristic
26C61×#399C2212yes—
27C61×#397C2212yes—
28C61×#388—22no—
29C61×#385—21no—
30C61×#381C2211yescharacteristic
31C61×#379—21no—
32C22×Dic122—21no—
33C22×Dic122D412yes—
34C23×Dic61—21no—
35C23×Dic61—21no—
36C2×#11758194—22no—
37#14519756—24no—
38#14519756—24no—
39#14519756—22no—
40#14519756—22no—
41#14519756D414yes—
42#14519756C2314yes—
43#14519754—28no—
44#14519754—28no—
45#14519754—28no—
46#14519754—28no—
47C2×C4⋊Dic61—21no—
48C2×C4⋊Dic61D412yes—
49C2×#11758196—22no—
50C2×#11758196—22no—
51C2×C22⋊Dic61—22no—
52C2×C22⋊Dic61—24no—
53C2×C22⋊Dic61—22no—
54C2×C22⋊Dic61—24no—
55#14519710—24no—
56#14519708—24no—
57#14519700—24no—
58#14519700—24no—
59C22⋊#92418—22no—
60C22⋊#92418C2314yes—
61C25×C61D411yescharacteristic
62C23×C4×C61D411yescharacteristic
63C23×C4×C61C2311yescharacteristic
64#117×C61—22no—
65#117×C61—22no—
66C2×C4⋊C4×C61—22no—
67C2×C4⋊C4×C61C2311yescharacteristic
68C2×C4⋊C4×C61C2311yescharacteristic
69C2×C22⋊C4×C61—22no—
70C2×C22⋊C4×C61—22no—
71C2×C22⋊C4×C61—21no—
72C2×C22⋊C4×C61—21no—
73C2×C22⋊C4×C61—22no—
74C2×C22⋊C4×C61—22no—
75C2×C22⋊C4×C61D412yes—
76C2×C22⋊C4×C61C2312yes—
77C2×C42×C61—21no—
78#95×C61—21no—
79#95×C61—22no—
80#95×C61C2312yes—
81C24×C61—21no—
82C24×C61—21no—
83C24×C61—24no—
84C24×C61—42no—
85C24×C61—81no—
86C24×C61C2×D411yescharacteristic
87C24×C61C2×D412yes—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—22no—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—42no—
94C22×C4×C61—81no—
95C22×C4×C61C2×D412yes—
96C22×C4×C61C2×D412yes—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D412yes—
99C22×C4×C61C2411yescharacteristic
100C22×Dic61—22no—
101C22×Dic61—22no—
102C22×Dic61—24no—
103C22×Dic61—24no—
104C22×Dic61—82no—
105C22×Dic61—82no—
106C2×Dic122—24no—
107C2×Dic122—22no—
108C2×Dic122—44no—
109C2×Dic122—82no—
110C2×Dic122C2×D414yes—
111C4⋊C4×C61—22no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—82no—
114C22⋊C4×C61—24no—
115C22⋊C4×C61—24no—
116C22⋊C4×C61—42no—
117C22⋊C4×C61—41no—
118C22⋊C4×C61—41no—
119C22⋊C4×C61—44no—
120C22⋊C4×C61—82no—
121C42×C61—42no—
122C22⋊Dic61—28no—
123C22⋊Dic61—24no—
124C22⋊Dic61—24no—
125C4⋊Dic61—24no—
126C4⋊Dic61—22no—
127C4⋊Dic61C2×D414yes—
128#11758196—28no—
129#11758196—24no—
130#11758196—24no—
131#11758194—44no—
132C2×#92418—24no—
133C23×C61—24no—
134C23×C61—24no—
135C23×C61—22no—
136C23×C61—22no—
137C23×C61—22no—
138C23×C61—22no—
139C23×C61—41no—
140C23×C61—42no—
141C23×C61—42no—
142C23×C61—42no—
143C23×C61—42no—
144C23×C61—81no—
145C23×C61—81no—
146C23×C61—88no—
147C23×C61#12011yescharacteristic
148C23×C61#12011yescharacteristic
149C23×C61#13714yes—
150C23×C61C22×D411yescharacteristic
151C2×C4×C61—24no—
152C2×C4×C61—22no—
153C2×C4×C61—42no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—42no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61—81no—
166C2×C4×C61#12012yes—
167C2×C4×C61C22×D412yes—
168C2×Dic61—24no—
169C2×Dic61—24no—
170C2×Dic61—44no—
171C2×Dic61—44no—
172C2×Dic61—84no—
173C2×Dic61—84no—
174Dic122—44no—
175Dic122—82no—
176Dic122—84no—
177#92418—44no—
178#18138—611nomaximal, Sylow
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—24no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—44no—
188C22×C61—44no—
189C22×C61—48no—
190C22×C61—42no—
191C22×C61—42no—
192C22×C61—82no—
193C22×C61—84no—
194C22×C61#45812yes—
195C22×C61C2×#12011yescharacteristic
196C22×C61C2×#13712yes—
197C4×C61—22no—
198C4×C61—22no—
199C4×C61—42no—
200C4×C61—82no—
201C4×C61—81no—
202C4×C61—81no—
203Dic61—44no—
204Dic61—44no—
205C2×#543—611no—
206#3095—612no—
207#2482D6111yescharacteristic
208#1614—614no—
209#1608—614no—
210#1511—612no—
211#1462—611no—
212C2×C61—22no—
213C2×C61—22no—
214C2×C61—41no—
215C2×C61—44no—
216C2×C61#300411yescharacteristic
217C2×C61#301911yescharacteristic
218C2×C61C2×#45811yescharacteristic
219#543—614no—
220#543—1224no—
221#543—1222no—
222C2×C22⋊Q8—612no—
223C2×C22⋊Q8—1222no—
224C2×C22⋊Q8—1222no—
225C2×C22⋊Q8—1221no—
226C2×#117C2×D6111yescharacteristic
227C22×C22⋊C4—1221no—
228C22×C22⋊C4C2×D6111yescharacteristic
229#485—1224no—
230#484—614no—
231#483—614no—
232#428—1222no—
233#419—1222no—
234#407—612no—
235#399C2×D6112yes—
236#397C2×D6112yes—
237#388—22no—
238#385—21no—
239#381C2×D6111yescharacteristic
240#379—21no—
241#328—618no—
242#323—614no—
243C61#1813811yescharacteristic, complemented, Sylow
244C25C22⋊D6111yescharacteristic
245C22×Q8—612no—
246C22×Q8—1221no—
247C23×C4—1221no—
248C23×C4—1221no—
249C23×C4C22⋊D6111yescharacteristic
250C23×C4C22×D6111yescharacteristic
251C2×M16—1222no—
252C22⋊Q8—614no—
253C22⋊Q8—614no—
254C22⋊Q8—1228no—
255C22⋊Q8—1228no—
256C22⋊Q8—1228no—
257C22⋊Q8—1228no—
258C22⋊Q8—1224no—
259C22⋊Q8—1222no—
260C22⋊Q8—1222no—
261C22⋊Q8—1224no—
262#117—22no—
263#117—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—612no—
266C2×C4⋊C4—1222no—
267C2×C4⋊C4—1221no—
268C2×C4⋊C4—1222no—
269C2×C4⋊C4C22×D6111yescharacteristic
270C2×C4⋊C4C22×D6111yescharacteristic
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—1222no—
278C2×C22⋊C4—1224no—
279C2×C22⋊C4—1222no—
280C2×C22⋊C4—1224no—
281C2×C22⋊C4C22⋊D6112yes—
282C2×C22⋊C4C22×D6112yes—
283C2×C42—21no—
284#107—1224no—
285#106—1224no—
286#103—1224no—
287#103—1224no—
288C22⋊C8—614no—
289C22⋊C8—1222no—
290#95—21no—
291#95—22no—
292#95C22×D6112yes—
293C24—21no—
294C24—21no—
295C24—24no—
296C24—42no—
297C24—81no—
298C24C2×C22⋊D6111yescharacteristic
299C24C2×C22⋊D6112yes—
300C2×Q8—614no—
301C2×Q8—1224no—
302C2×Q8—1222no—
303C2×Q8—4882no—
304C2×Q8—2444no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—42no—
311C22×C4—81no—
312C22×C4—1224no—
313C22×C4—1224no—
314C22×C4—1222no—
315C22×C4—1222no—
316C22×C4—4882no—
317C22×C4—4882no—
318C22×C4D4×D6112yes—
319C22×C4D4×D6112yes—
320C22×C4C2×C22⋊D6112yes—
321C22×C4C2×C22⋊D6111yescharacteristic
322C22×C4C23×D6111yescharacteristic
323M16—2444no—
324C2×C8—1224no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—82no—
328C4⋊C4—614no—
329C4⋊C4—1228no—
330C4⋊C4—1224no—
331C4⋊C4—1224no—
332C4⋊C4—1222no—
333C4⋊C4—1224no—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—44no—
340C22⋊C4—82no—
341C22⋊C4—1228no—
342C22⋊C4—1224no—
343C22⋊C4—1224no—
344C42—42no—
345C23—24no—
346C23—24no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—81no—
357C23—81no—
358C23—88no—
359C23#1451974511yescharacteristic
360C23#1451974511yescharacteristic
361C23#1451980714yes—
362C23C22×C22⋊D6111yescharacteristic
363Q8—4882no—
364Q8—4884no—
365Q8—2444no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—81no—
381C2×C4—1224no—
382C2×C4—1224no—
383C2×C4—4884no—
384C2×C4—4884no—
385C2×C4—2444no—
386C2×C4—2444no—
387C2×C4#1451974512yes—
388C2×C4C2×D4×D6112yes—
389C8—2444no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—24no—
398C22—44no—
399C22—44no—
400C22—48no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—84no—
405C22#1683995112yes—
406C22C2×#1451974511yescharacteristic
407C22C2×#1451980712yes—
408C4—22no—
409C4—22no—
410C4—42no—
411C4—82no—
412C4—81no—
413C4—81no—
414C4—2444no—
415C4—2444no—
416C2—22no—
417C2—22no—
418C2—41no—
419C2—44no—
420C2#2145505211yescharacteristic
421C2#2145517511yescharacteristic
422C2C2×#1683995111yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2482C2
socleC22×C61C2×#120
prime radicalC25×C61D4
hypercenter#2482D61
hypocenterC61#18138
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2482

Complements

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

NH ≅ G/Ntype
2. #21455767C2semidirect
3. #21444843C2semidirect
4. #21444809C2semidirect
5. #21444351C2semidirect
17. #16840097C22semidirect
18. #16840092C22semidirect
22. #16839498C22semidirect
243. C61#18138semidirect

Maximal subgroups

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

subgroupindex
C2×#168403882
#214557672
#214448432
#214448092
#214443512
#214441462
C61×#24822
#1813861