Finite Group Databasebeta

← #29652429

Subgroups — #29652429

Order 15616.

1054 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1054 conjugacy classes of subgroups — too many to draw. Only its 183 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
1#21455763C212yesmaximal, complemented
2#21450093C211yesmaximal, characteristic
3#21444844C214yesmaximal, complemented
4#21444810C214yesmaximal, complemented
5#21444352C212yesmaximal, complemented
6#21444146C211yesmaximal, characteristic
7C2×C61×#543C211yesmaximal, characteristic
8C2×#14519736C2211yescharacteristic
9#16840100—24no—
10#16840095C2214yescomplemented
11#16840090C2214yescomplemented
12#16839802—22no—
13#16839771—22no—
14#16839726C2212yes—
15#16839698C2212yes—
16#16839687C2212yes—
17#16839650—22no—
18#16839635—21no—
19#16839625C2211yescharacteristic
20#16839618—21no—
21#16839499C2218yescomplemented
22#16839486C2214yes—
23C61×#543—24no—
24C61×#543—22no—
25C61×#543C2214yes—
26C2×C22⋊Q8×C61—22no—
27C2×C22⋊Q8×C61—21no—
28C2×C22⋊Q8×C61—22no—
29C2×C22⋊Q8×C61C2212yes—
30C22×C22⋊C4×C61—21no—
31C22×C22⋊C4×C61C2211yescharacteristic
32C2×#11758194—22no—
33#14519736—22no—
34#14519736—22no—
35C2×C4⋊Dic61C2311yescharacteristic
36C2×C4⋊Dic61C2311yescharacteristic
37C2×#11758196—22no—
38C2×C22⋊Dic61—22no—
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—22no—
42C2×C4×Dic61—21no—
43#14519710—24no—
44#14519708—24no—
45#14519701—24no—
46#14519701—24no—
47C22⋊#92418—22no—
48C22⋊#92418C2314yes—
49#14519684—22no—
50#14519684—21no—
51#14519684C2312yes—
52C25×C61D411yescharacteristic
53C22×Q8×C61—21no—
54C22×Q8×C61D412yes—
55C23×C4×C61—21no—
56C23×C4×C61—21no—
57C23×C4×C61D411yescharacteristic
58C23×C4×C61C2311yescharacteristic
59C22⋊Q8×C61—28no—
60C22⋊Q8×C61—28no—
61C22⋊Q8×C61—24no—
62C22⋊Q8×C61—28no—
63C22⋊Q8×C61—22no—
64C22⋊Q8×C61—22no—
65C22⋊Q8×C61—24no—
66C22⋊Q8×C61—28no—
67C22⋊Q8×C61D414yes—
68C22⋊Q8×C61C2314yes—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—22no—
71C2×C4⋊C4×C61—22no—
72C2×C4⋊C4×C61D412yes—
73C2×C22⋊C4×C61—24no—
74C2×C22⋊C4×C61—22no—
75C2×C22⋊C4×C61—24no—
76C2×C22⋊C4×C61—22no—
77C2×C22⋊C4×C61—22no—
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61D412yes—
80C2×C22⋊C4×C61C2312yes—
81C24×C61—21no—
82C24×C61—21no—
83C24×C61—24no—
84C24×C61—42no—
85C24×C61—81no—
86C24×C61C2×D411yescharacteristic
87C24×C61C2×D412yes—
88C2×Q8×C61—24no—
89C2×Q8×C61—22no—
90C2×Q8×C61—44no—
91C2×Q8×C61—82no—
92C2×Q8×C61C2×D414yes—
93C22×C4×C61—22no—
94C22×C4×C61—21no—
95C22×C4×C61—22no—
96C22×C4×C61—24no—
97C22×C4×C61—24no—
98C22×C4×C61—21no—
99C22×C4×C61—42no—
100C22×C4×C61—81no—
101C22×C4×C61—82no—
102C22×C4×C61—82no—
103C22×C4×C61C2×D412yes—
104C22×C4×C61C2×D412yes—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2411yescharacteristic
108C22×Dic61—22no—
109C22×Dic61—21no—
110C22×Dic61—21no—
111C4⋊C4×C61—24no—
112C4⋊C4×C61—28no—
113C4⋊C4×C61—24no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61—24no—
116C4⋊C4×C61C2×D414yes—
117C22⋊C4×C61—28no—
118C22⋊C4×C61—24no—
119C22⋊C4×C61—24no—
120C22⋊C4×C61—24no—
121C22⋊C4×C61—24no—
122C22⋊Dic61—42no—
123C22⋊Dic61—41no—
124C22⋊Dic61—41no—
125C22⋊Dic61—44no—
126C22⋊Dic61—82no—
127C4⋊Dic61—22no—
128C4⋊Dic61—22no—
129#11758196—82no—
130C4×Dic61—42no—
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
151Q8×C61—44no—
152Q8×C61—82no—
153Q8×C61—84no—
154C2×C4×C61—24no—
155C2×C4×C61—24no—
156C2×C4×C61—22no—
157C2×C4×C61—24no—
158C2×C4×C61—42no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—42no—
162C2×C4×C61—44no—
163C2×C4×C61—44no—
164C2×C4×C61—42no—
165C2×C4×C61—81no—
166C2×C4×C61—84no—
167C2×C4×C61—84no—
168C2×C4×C61#12012yes—
169C2×C4×C61C22×D412yes—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61—42no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—42no—
176C2×Dic61—42no—
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—44no—
201C4×C61—44no—
202Dic61—82no—
203Dic61—81no—
204Dic61—81no—
205C2×#543D6111yescharacteristic
206#3095—612no—
207#2482—611no—
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—24no—
220#543—22no—
221#543C2×D6114yes—
222C2×C22⋊Q8—22no—
223C2×C22⋊Q8—21no—
224C2×C22⋊Q8—22no—
225C2×C22⋊Q8C2×D6112yes—
226C2×#117—611no—
227C22×C22⋊C4—21no—
228C22×C22⋊C4C2×D6111yescharacteristic
229#485—1224no—
230#484—614no—
231#483—614no—
232#428—1222no—
233#419—1222no—
234#407—612no—
235#399—612no—
236#397—612no—
237#388—1222no—
238#385—1221no—
239#381—611no—
240#379—1221no—
241#328—618no—
242#323—614no—
243C61#1813811yescharacteristic, complemented, Sylow
244C25C22⋊D6111yescharacteristic
245C22×Q8—21no—
246C22×Q8C22⋊D6112yes—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4C22⋊D6111yescharacteristic
250C23×C4C22×D6111yescharacteristic
251C2×M16—1222no—
252C22⋊Q8—28no—
253C22⋊Q8—28no—
254C22⋊Q8—24no—
255C22⋊Q8—28no—
256C22⋊Q8—24no—
257C22⋊Q8—22no—
258C22⋊Q8—22no—
259C22⋊Q8—28no—
260C22⋊Q8C22⋊D6114yes—
261C22⋊Q8C22×D6114yes—
262#117—1222no—
263#117—1222no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—611no—
268C2×C4⋊C4—611no—
269C2×C4⋊C4—1222no—
270C2×C4⋊C4C22⋊D6112yes—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—1222no—
278C2×C22⋊C4—1221no—
279C2×C22⋊C4—1221no—
280C2×C22⋊C4—1222no—
281C2×C22⋊C4C22⋊D6112yes—
282C2×C22⋊C4C22×D6112yes—
283C2×C42—1221no—
284#107—1224no—
285#106—1224no—
286#103—1224no—
287#103—1224no—
288C22⋊C8—614no—
289C22⋊C8—1222no—
290#95—612no—
291#95—1221no—
292#95—1222no—
293C24—21no—
294C24—24no—
295C24—21no—
296C24—42no—
297C24—81no—
298C24C2×C22⋊D6111yescharacteristic
299C24C2×C22⋊D6112yes—
300C2×Q8—24no—
301C2×Q8—22no—
302C2×Q8—44no—
303C2×Q8—82no—
304C2×Q8C2×C22⋊D6114yes—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—21no—
311C22×C4—42no—
312C22×C4—81no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—1222no—
316C22×C4—1221no—
317C22×C4—1221no—
318C22×C4C2×C22⋊D6112yes—
319C22×C4C2×C22⋊D6112yes—
320C22×C4C2×C22⋊D6112yes—
321C22×C4C2×C22⋊D6111yescharacteristic
322C22×C4C23×D6111yescharacteristic
323M16—2444no—
324C2×C8—1224no—
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—1222no—
331C4⋊C4—1222no—
332C4⋊C4—4882no—
333C4⋊C4C2×C22⋊D6114yes—
334C22⋊C4—28no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—4882no—
340C22⋊C4—2442no—
341C22⋊C4—2444no—
342C22⋊C4—2441no—
343C22⋊C4—2441no—
344C42—2442no—
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#1451974611yescharacteristic
360C23#1451974611yescharacteristic
361C23#1451980814yes—
362C23C22×C22⋊D6111yescharacteristic
363Q8—44no—
364Q8—82no—
365Q8—84no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4—2442no—
386C2×C4—2442no—
387C2×C4#1451974612yes—
388C2×C4C22×C22⋊D6112yes—
389C8—2444no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—24no—
397C22—21no—
398C22—44no—
399C22—44no—
400C22—48no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—84no—
405C22#1683995212yes—
406C22C2×#1451974611yescharacteristic
407C22C2×#1451980812yes—
408C4—22no—
409C4—22no—
410C4—42no—
411C4—44no—
412C4—44no—
413C4—4882no—
414C4—4881no—
415C4—4881no—
416C2—22no—
417C2—22no—
418C2—41no—
419C2—44no—
420C2#2145505311yescharacteristic
421C2#2145517611yescharacteristic
422C2C2×#1683995211yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#543C2
socleC22×C61C2×#120
prime radicalC25×C61D4
hypercenterC2×#543D61
hypocenterC61#18138
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
1. #21455763C2semidirect
3. #21444844C2semidirect
4. #21444810C2semidirect
5. #21444352C2semidirect
10. #16840095C22semidirect
11. #16840090C22semidirect
21. #16839499C22semidirect
243. C61#18138semidirect

Maximal subgroups

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

subgroupindex
#214557632
#214500932
#214448442
#214448102
#214443522
#214441462
C2×C61×#5432
#1813861