Finite Group Databasebeta

← #29577819

Subgroups — #29577819

Order 15616.

734 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 97 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#21450089C211yesmaximal, characteristic
2#21448171C211yesmaximal, characteristic
3#21445918C211yesmaximal, characteristic, complemented
4#21444145C211yesmaximal, characteristic
5#21443673C211yesmaximal, characteristic, complemented
6#21443672C211yesmaximal, characteristic, complemented
7C61×#1826C211yesmaximal, characteristic, complemented
8C22×#11758198C2211yescharacteristic
9C2×C22⋊C4×D61—21no—
10#16839726C2211yescharacteristic
11#16839697C412yescomplemented
12#16839688C412yescomplemented
13#16839653—22no—
14#16839634—21no—
15#16839625C2211yescharacteristic
16#16839617—21no—
17C22⋊C4×Dic61—21no—
18C4×#11758198—22no—
19C4×C22⋊Dic61C2211yescharacteristic
20#16839609—22no—
21#16839521—22no—
22#16839485—22no—
23C2×#117×C61C2211yescharacteristic
24C61×#407C2211yescharacteristic
25C4×C22⋊C4×C61C2211yescharacteristic
26C61×#339—22no—
27C24×D61D411yescharacteristic
28C23×Dic61D411yescharacteristic
29C22×C4×D61—22no—
30C2×#11758194—21no—
31C2×#11758196—22no—
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61—21no—
34C2×#11758198—22no—
35C2×#11758198—21no—
36C2×#11758198—21no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×#11758198—22no—
40C2×#11758198—22no—
41C2×#11758198C2×C412yes—
42C2×#11758198D412yes—
43C22⋊C4×D61—28no—
44C2×C4×Dic61—22no—
45C2×C4×Dic61D412yes—
46C22⋊#92418—21no—
47C22⋊#92418—21no—
48#14519684—22no—
49#14519684—21no—
50#14519684—21no—
51#14519684—22no—
52#14519684C2×C412yescomplemented
53C23×C4×C61C2311yescharacteristic
54C2×M16×C61—21no—
55#117×C61—21no—
56#117×C61—21no—
57#117×C61—21no—
58#117×C61—21no—
59C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
60C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
61C2×C22⋊C4×C61—21no—
62C2×C22⋊C4×C61—21no—
63C2×C42×C61—21no—
64C2×C42×C61D412yes—
65C22⋊C8×C61—21no—
66C22⋊C8×C61—21no—
67#95×C61—21no—
68C24×C61C2×D411yescharacteristic
69C23×D61—21no—
70C23×D61—21no—
71C23×D61—24no—
72C23×D61—42no—
73C23×D61—81no—
74C23×D61C22⋊C412yes—
75C22×C4×C61—22no—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—42no—
79C22×C4×C61C22×C411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61#4111yescharacteristic
84C22×C4×C61#4111yescharacteristic
85C22×Dic61—22no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61—22no—
89C22×Dic61—81no—
90C22×Dic61C22⋊C412yes—
91C2×C4×D61—28no—
92C2×C4×D61—82no—
93C2×C4×D61—82no—
94M16×C61—42no—
95C2×C8×C61—41no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—21no—
100C22⋊C4×C61—24no—
101C22⋊C4×C61—41no—
102C22⋊C4×C61—41no—
103C42×C61—42no—
104C42×C61—41no—
105C42×C61—41no—
106C22⋊Dic61—24no—
107C22⋊Dic61—42no—
108#11758198—28no—
109#11758198—22no—
110#11758198—22no—
111#11758198—22no—
112#11758198—22no—
113#11758198—44no—
114#11758198—82no—
115#11758196—82no—
116C4×Dic61—42no—
117C4×Dic61—82no—
118#11758194—42no—
119C2×#92418—41no—
120C23×C61—22no—
121C23×C61—22no—
122C23×C61—41no—
123C23×C61C2×C22⋊C411yescharacteristic
124C23×C61#12011yescharacteristic
125C23×C61#12311yescharacteristic
126C22×D61—22no—
127C22×D61—22no—
128C22×D61—22no—
129C22×D61—24no—
130C22×D61—22no—
131C22×D61—42no—
132C22×D61—42no—
133C22×D61—42no—
134C22×D61—42no—
135C22×D61—88no—
136C22×D61—81no—
137C22×D61—81no—
138C22×D61#10414yes—
139C2×C4×C61—22no—
140C2×C4×C61—22no—
141C2×C4×C61—24no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—42no—
147C2×C4×C61—42no—
148C2×C4×C61—42no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—42no—
152C2×C4×C61C4×D411yescharacteristic
153C2×C4×C61C4×D411yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C2×Dic61—24no—
157C2×Dic61—22no—
158C2×Dic61—42no—
159C2×Dic61—42no—
160C2×Dic61—42no—
161C2×Dic61—42no—
162C2×Dic61—42no—
163C2×Dic61—42no—
164C2×Dic61—42no—
165C2×Dic61—81no—
166C4×D61—88no—
167C8×C61—81no—
168#92418—81no—
169#9477—611nomaximal, Sylow
170C22×C61—21no—
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—44no—
178C22×C61#38611yescharacteristic
179C22×C61C2×#10412yes—
180C22×C61#45812yes—
181C2×D61—24no—
182C2×D61—44no—
183C2×D61—44no—
184C2×D61—42no—
185C2×D61—42no—
186C2×D61—44no—
187C2×D61—82no—
188C2×D61—84no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—44no—
194C4×C61—81no—
195C4×C61—81no—
196Dic61—44no—
197Dic61—42no—
198Dic61—82no—
199#2482—611no—
200#2273—611no—
201#1826—611no—
202#1826D6111yescharacteristic, complemented
203#1462—611no—
204#1284—611no—
205#1284—611no—
206C2×C61—22no—
207C2×C61—22no—
208C2×C61—41no—
209C2×C61#189011yescharacteristic
210C2×C61#189611yescharacteristic
211C2×C61#191111yescharacteristic
212D61—44no—
213C2×#117C2×D6111yescharacteristic
214C22×C22⋊C4—611no—
215C22×C22⋊C4—1221no—
216#407—611no—
217#407C2×D6111yescharacteristic
218#399—612no—
219#397—612no—
220#388—1222no—
221#385—1221no—
222#381—611no—
223#379—1221no—
224C4×C22⋊C4—611no—
225C4×C22⋊C4—1222no—
226C4×C22⋊C4—1221no—
227C4×C22⋊C4—1222no—
228C4×C22⋊C4C2×D6111yescharacteristic
229#339—22no—
230#339—1222no—
231#323—1222no—
232C61#947711yescharacteristic, complemented, Sylow
233C25—611no—
234C23×C4—611no—
235C23×C4—1222no—
236C23×C4C22×D6111yescharacteristic
237C2×M16—21no—
238C2×M16—1221no—
239#117—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243C2×C4⋊C4—1222no—
244C2×C4⋊C4C4×D6111yescharacteristic, complemented
245C2×C4⋊C4C4×D6111yescharacteristic, complemented
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—612no—
249C2×C22⋊C4—612no—
250C2×C22⋊C4—1228no—
251C2×C22⋊C4—1222no—
252C2×C22⋊C4—1221no—
253C2×C22⋊C4—1221no—
254C2×C22⋊C4—1221no—
255C2×C22⋊C4—1221no—
256C2×C22⋊C4—1221no—
257C2×C22⋊C4—1222no—
258C2×C22⋊C4—1222no—
259C2×C22⋊C4—1221no—
260C2×C42—21no—
261C2×C42—612no—
262C2×C42—1222no—
263C2×C42C22⋊D6112yes—
264C22⋊C8—21no—
265C22⋊C8—21no—
266C22⋊C8—1221no—
267C22⋊C8—1221no—
268#95—21no—
269#95—612no—
270#95—1222no—
271#95—1222no—
272#95—1221no—
273#95—1221no—
274C24—612no—
275C24—1224no—
276C24—1221no—
277C24—1221no—
278C24—4881no—
279C24—2442no—
280C24D4×D6111yescharacteristic
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—42no—
285C22×C4—612no—
286C22×C4—1228no—
287C22×C4—1221no—
288C22×C4—1222no—
289C22×C4—1221no—
290C22×C4—1222no—
291C22×C4—4882no—
292C22×C4—4882no—
293C22×C4—4881no—
294C22×C4C2×C4×D6111yescharacteristic
295C22×C4#1175822211yescharacteristic
296C22×C4D4×D6111yescharacteristic
297C22×C4D4×D6111yescharacteristic
298C22×C4#1175822411yescharacteristic
299C22×C4C2×C22⋊D6111yescharacteristic
300M16—42no—
301M16—2442no—
302C2×C8—41no—
303C2×C8—2441no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—4882no—
309C22⋊C4—24no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—1228no—
313C22⋊C4—1224no—
314C22⋊C4—1222no—
315C22⋊C4—1222no—
316C22⋊C4—1222no—
317C22⋊C4—1222no—
318C22⋊C4—4882no—
319C22⋊C4—2444no—
320C22⋊C4—2442no—
321C42—42no—
322C42—41no—
323C42—41no—
324C42—4882no—
325C42—2442no—
326C23—22no—
327C23—22no—
328C23—41no—
329C23—614no—
330C23—1224no—
331C23—1222no—
332C23—1222no—
333C23—1222no—
334C23—1222no—
335C23—4881no—
336C23—4881no—
337C23—4888no—
338C23—2442no—
339C23—2442no—
340C23—2442no—
341C23—2442no—
342C23C22⋊C4×D6111yescharacteristic
343C23#1451974511yescharacteristic
344C23#1451975911yescharacteristic
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—1224no—
359C2×C4—1222no—
360C2×C4—4888no—
361C2×C4—4881no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4—2442no—
365C2×C4—2442no—
366C2×C4—2442no—
367C2×C4—2442no—
368C2×C4—2442no—
369C2×C4#1451973711yescharacteristic
370C2×C4C4×C22⋊D6111yescharacteristic
371C2×C4#1451974811yescharacteristic
372C2×C4#1451975011yescharacteristic
373C8—81no—
374C8—4881no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—44no—
383C22—1224no—
384C22—4882no—
385C22—4884no—
386C22—2444no—
387C22—2444no—
388C22—2444no—
389C22—2442no—
390C22—2442no—
391C22#1683963811yescharacteristic
392C22#104×D6112yes—
393C22#1683995112yes—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—44no—
399C4—81no—
400C4—81no—
401C4—4882no—
402C4—2444no—
403C4—2442no—
404C2—22no—
405C2—22no—
406C2—41no—
407C2—2444no—
408C2#2144622911yescharacteristic
409C2#2144625911yescharacteristic
410C2#2144634811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1826C2
socleC22×C61#386
prime radicalC24×D61D4
cosocleC23×C4×C61C23
hypercenter#1826D61
hypocenterC61#9477
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
3. #21445918C2semidirect
5. #21443673C2semidirect
6. #21443672C2semidirect
7. C61×#1826C2semidirect
11. #16839697C4semidirect
12. #16839688C4semidirect
52. #14519684C2×C4semidirect
59. C2×C4⋊C4×C61C2×C4semidirect
60. C2×C4⋊C4×C61C2×C4semidirect
202. #1826D61semidirect
232. C61#9477semidirect
244. C2×C4⋊C4C4×D61semidirect
245. C2×C4⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214500892
#214481712
#214459182
#214441452
#214436732
#214436722
C61×#18262
#947761