Finite Group Databasebeta

← #29589706

Subgroups — #29589706

Order 15616.

552 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 73 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#21449183C211yesmaximal, characteristic
2#21444154C211yesmaximal, characteristic
3#21444138C211yesmaximal, characteristic
4#21443768C211yesmaximal, characteristic, complemented
5#21443767C211yesmaximal, characteristic, complemented
6#21443766C211yesmaximal, characteristic, complemented
7C61×#1326C211yesmaximal, characteristic, complemented
8C23×Dic122C411yescharacteristic
9C2×#14519755—21no—
10C2×#14519737C2211yescharacteristic
11#16839812—21no—
12#16839724—21no—
13#16839682—21no—
14#16839675—21no—
15#16839666—21no—
16#16839641C411yescharacteristic
17#16839639C411yescharacteristic
18#16839634C2211yescharacteristic
19#16839620C411yescharacteristic
20#16839487—21no—
21#16839486C2211yescharacteristic
22#16839486C2211yescharacteristic
23#16839485—21no—
24C61×#385C2211yescharacteristic
25C61×#323C2211yescharacteristic
26C61×#323C2211yescharacteristic
27C22×Dic122—21no—
28C22×Dic122—21no—
29C22×Dic122—44no—
30C22×Dic122—41no—
31C22×Dic122—41no—
32C22×C22⋊D61C2×C411yescharacteristic
33C23×Dic61C2×C411yescharacteristic
34C23×Dic61C2×C411yescharacteristic
35C22×C4×D61D411yescharacteristic
36C2×#11758189—41no—
37#14519755—44no—
38#14519737—41no—
39#14519737—41no—
40C2×#11758196—21no—
41C2×#11758196—21no—
42C2×#11758196D411yescharacteristic
43C2×C22⋊Dic61C2×C411yescharacteristic
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×#11758198D411yescharacteristic
47C2×C4×Dic61D411yescharacteristic
48#14519706—41no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51#14519689—41no—
52C8×Dic61—41no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57C23×C4×C61C2×C411yescharacteristic
58C2×C4⋊C4×C61—21no—
59C2×C22⋊C4×C61C2×C411yescharacteristic
60C2×C22⋊C4×C61C2311yescharacteristic
61C22⋊C8×C61—21no—
62C22⋊C8×C61—21no—
63#95×C61—21no—
64C24×C61C22×C411yescharacteristic
65C23×D61C22⋊C411yescharacteristic
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—42no—
69C22×C4×C61—41no—
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C2×C22⋊D61—21no—
73C2×C22⋊D61—21no—
74C2×C22⋊D61—22no—
75C2×C22⋊D61—41no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—42no—
81C22×Dic61—42no—
82C22×Dic61—41no—
83C22×Dic61C22⋊C411yescharacteristic
84C22×Dic61C22⋊C411yescharacteristic
85C22×Dic61C22⋊C411yescharacteristic
86C2×C4×D61—21no—
87C2×C4×D61—21no—
88C2×C4×D61—42no—
89C2×Dic122—24no—
90C2×Dic122—42no—
91C2×Dic122—42no—
92C2×Dic122—42no—
93C2×Dic122—42no—
94C2×Dic122—44no—
95C2×Dic122—44no—
96C2×Dic122—44no—
97C2×Dic122—41no—
98C2×Dic122—41no—
99C2×Dic122—44no—
100C2×C8×C61—41no—
101C2×C8×C61—41no—
102C4⋊C4×C61—42no—
103C22⋊C4×C61—21no—
104C22⋊C4×C61—21no—
105C22⋊C4×C61—41no—
106C22⋊Dic61—21no—
107C22⋊Dic61—21no—
108#11758198—41no—
109#11758198—42no—
110#11758198—42no—
111#11758196—41no—
112#11758196—42no—
113#11758196—42no—
114C4×Dic61—21no—
115C4×Dic61—21no—
116C2×#92418—41no—
117C2×#92418—41no—
118#11758189—82no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61C2×C22⋊C411yescharacteristic
124C23×C61#13111yescharacteristic
125C23×C61#13111yescharacteristic
126C22×D61—21no—
127C22×D61—21no—
128C22×D61—21no—
129C22×D61—41no—
130C22×D61—41no—
131C2×C4×C61—22no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—41no—
135C2×C4×C61—42no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—42no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C22⋊D61—42no—
144C22⋊D61—42no—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—22no—
148C2×Dic61—22no—
149C2×Dic61—21no—
150C2×Dic61—42no—
151C2×Dic61—41no—
152C2×Dic61—42no—
153C2×Dic61—41no—
154C2×Dic61—42no—
155C2×Dic61—41no—
156C2×Dic61—42no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61#9911yescharacteristic
161C2×Dic61#9911yescharacteristic
162C2×Dic61#10112yes—
163C4×D61—42no—
164C4×D61—81no—
165Dic122—44no—
166Dic122—48no—
167Dic122—44no—
168C8×C61—81no—
169C8×C61—81no—
170#92418—81no—
171#92418—81no—
172#11580—611nomaximal, Sylow
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—41no—
178C22×C61—41no—
179C22×C61—42no—
180C22×C61—41no—
181C22×C61—41no—
182C22×C61#40811yescharacteristic
183C22×C61C2×#9911yescharacteristic
184C22×C61C2×#10111yescharacteristic
185C2×D61—42no—
186C2×D61—41no—
187C2×D61—41no—
188C2×D61—41no—
189C2×D61—81no—
190C4×C61—41no—
191C4×C61—42no—
192C4×C61—41no—
193C4×C61—81no—
194Dic61—21no—
195Dic61—22no—
196Dic61—42no—
197Dic61—41no—
198Dic61—81no—
199Dic61AΣL(1,𝔽16)12yes—
200#2409—611no—
201#1465—611no—
202#1460—611no—
203#1326—611no—
204#1326—611no—
205#1326—611no—
206#1326D6111yescharacteristic, complemented
207C2×C61—21no—
208C2×C61—21no—
209C2×C61—41no—
210C2×C61—41no—
211C2×C61#147211yescharacteristic
212C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
213C2×C61#212311yescharacteristic
214D61—81no—
215C23×Q8—611no—
216C2×C22⋊Q8—1221no—
217C2×C4×D4—611no—
218#431—1221no—
219#407—1221no—
220#395—1221no—
221#393—1221no—
222#391—1221no—
223#386—611no—
224#386—611no—
225#385—611no—
226#385C2×D6111yescharacteristic
227#380—611no—
228#324—1221no—
229#323—611no—
230#323—611no—
231#323—1221no—
232#323C2×D6111yescharacteristic
233#323C2×D6111yescharacteristic
234C61#1158011yescharacteristic, complemented, Sylow
235C22×Q8—1221no—
236C22×Q8—1221no—
237C22×Q8—2441no—
238C22×Q8—2441no—
239C22×Q8—2444no—
240C22×D4—611no—
241C23×C4—611no—
242C23×C4—611no—
243C23×C4—611no—
244C23×C4C4×D6111yescharacteristic
245C2×M16—2441no—
246C22⋊Q8—2444no—
247C4×D4—2441no—
248C4×D4—2441no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—611no—
251C2×C4⋊C4—1221no—
252C2×C4⋊C4—1221no—
253C2×C22⋊C4—611no—
254C2×C22⋊C4—611no—
255C2×C22⋊C4—1221no—
256C2×C22⋊C4—1221no—
257C2×C22⋊C4C4×D6111yescharacteristic
258C2×C22⋊C4C22×D6111yescharacteristic
259C2×C42—611no—
260C4⋊C8—2441no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263C22⋊C8—1221no—
264C22⋊C8—1221no—
265C22⋊C8—2441no—
266C4×C8—2441no—
267#95—21no—
268#95—1221no—
269#95—1221no—
270#95—1221no—
271#95—1221no—
272C24—611no—
273C24C2×C4×D6111yescharacteristic
274C2×Q8—1224no—
275C2×Q8—2442no—
276C2×Q8—2441no—
277C2×Q8—2441no—
278C2×Q8—2442no—
279C2×Q8—2442no—
280C2×Q8—2442no—
281C2×Q8—2444no—
282C2×Q8—2444no—
283C2×Q8—2444no—
284C2×Q8—2444no—
285C2×D4—1222no—
286C2×D4—1221no—
287C2×D4—1221no—
288C2×D4—2441no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—611no—
294C22×C4—611no—
295C22×C4—611no—
296C22×C4—1221no—
297C22×C4—1221no—
298C22×C4—1221no—
299C22×C4—1221no—
300C22×C4—1221no—
301C22×C4—1221no—
302C22×C4—2442no—
303C22×C4—2442no—
304C22×C4—2442no—
305C22×C4—2441no—
306C22×C4D4×D6111yescharacteristic
307C22×C4D4×D6111yescharacteristic
308M16—4882no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—2441no—
312C2×C8—2441no—
313C4⋊C4—42no—
314C4⋊C4—2441no—
315C4⋊C4—2442no—
316C4⋊C4—2442no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—41no—
320C22⋊C4—1221no—
321C22⋊C4—1221no—
322C22⋊C4—2441no—
323C22⋊C4—2442no—
324C22⋊C4—2442no—
325C42—1221no—
326C42—1221no—
327C23—21no—
328C23—21no—
329C23—41no—
330C23—41no—
331C23—1221no—
332C23—1221no—
333C23—1221no—
334C23—2441no—
335C23—2441no—
336C23C22⋊C4×D6111yescharacteristic
337C23#1451978311yescharacteristic
338C23#1451978511yescharacteristic
339Q8—2444no—
340Q8—2444no—
341Q8—2448no—
342D4—2442no—
343D4—2442no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—611no—
357C2×C4—611no—
358C2×C4—612no—
359C2×C4—1222no—
360C2×C4—1221no—
361C2×C4—1221no—
362C2×C4—1222no—
363C2×C4—1221no—
364C2×C4—4881no—
365C2×C4—2442no—
366C2×C4—2442no—
367C2×C4—2442no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4—2442no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2442no—
376C8—81no—
377C8—81no—
378C8—4881no—
379C8—4881no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—4881no—
390C22—2442no—
391C22—2441no—
392C22—2441no—
393C22—2441no—
394C22#1683972711yescharacteristic
395C22#99×D6111yescharacteristic
396C22#101×D6111yescharacteristic
397C4—41no—
398C4—42no—
399C4—41no—
400C4—81no—
401C4—612no—
402C4—1221no—
403C4—1222no—
404C4—4881no—
405C4—2442no—
406C4—2441no—
407C2—21no—
408C2—21no—
409C2—41no—
410C2—41no—
411C2—4881no—
412C2#2144418111yescharacteristic
413C2AΣL(1,𝔽16)×D6111yescharacteristic
414C2#2144747111yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC24×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1326C2
socleC22×C61C2×#99
prime radicalC22×C22⋊D61C2×C4
cosocleC2×C22⋊C4×C61C23
hypercenter#1326D61
hypocenterC61#11580
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1326

Complements

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

NH ≅ G/Ntype
4. #21443768C2semidirect
5. #21443767C2semidirect
6. #21443766C2semidirect
7. C61×#1326C2semidirect
206. #1326D61semidirect
234. C61#11580semidirect

Maximal subgroups

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

subgroupindex
#214491832
#214441542
#214441382
#214437682
#214437672
#214437662
C61×#13262
#1158061