Finite Group Databasebeta

← #29568258

Subgroups — #29568258

Order 15616.

590 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 105 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#21448300C211yesmaximal, characteristic
2#21445511C211yesmaximal, characteristic
3#21445511C211yesmaximal, characteristic
4#21443643C211yesmaximal, characteristic, complemented
5#21443642C211yesmaximal, characteristic, complemented
6#21443641C211yesmaximal, characteristic, complemented
7C61×#1271C211yesmaximal, characteristic, complemented
8C22×#11758189—21no—
9C22×#11758189—21no—
10C2×#14519735C411yescharacteristic
11C2×#14519735C411yescharacteristic
12C2×C42×D61C2211yescharacteristic
13#16839661C411yescharacteristic
14#16839660C2211yescharacteristic
15#16839658C411yescharacteristic
16#16839636—21no—
17#16839633—21no—
18C4⋊C4×Dic61—21no—
19#16839613—21no—
20#16839517C2211yescharacteristic
21#16839517C2211yescharacteristic
22#16839516—21no—
23#16839516—21no—
24C61×#389C2211yescharacteristic
25C61×#336C2211yescharacteristic
26C61×#336C2211yescharacteristic
27C22×C4×D61C2×C411yescharacteristic
28C22×C4×D61D411yescharacteristic
29C22×C4×D61D411yescharacteristic
30C2×#11758189—24no—
31C2×#11758189—24no—
32C2×#11758189—42no—
33C2×#11758189—42no—
34C2×#11758189—42no—
35C2×#11758189—42no—
36C22×#92418—21no—
37C22×#92418—21no—
38#14519735—22no—
39#14519735—22no—
40#14519735—42no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×C4⋊Dic61C2×C411yescharacteristic
44C2×#11758196—21no—
45C2×#11758196—21no—
46C2×#11758198—21no—
47C2×#11758198—21no—
48C2×C4×Dic61—21no—
49C2×C4×Dic61—21no—
50C2×C4×Dic61C2×C411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53C42×D61—42no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57C22×C8×C61—21no—
58C22×C8×C61—21no—
59C2×C4⋊C4×C61—21no—
60C2×C4⋊C4×C61C2×C411yescharacteristic
61C2×C4⋊C4×C61C2×C411yescharacteristic
62C2×C42×C61C2311yescharacteristic
63#95×C61—21no—
64C23×D61C22⋊C411yescharacteristic
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61C22×C411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×Dic61—21no—
71C22×Dic61—21no—
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C22⋊C411yescharacteristic
75C2×C4×D61—22no—
76C2×C4×D61—22no—
77C2×C4×D61—22no—
78C2×C4×D61—21no—
79C2×C4×D61—21no—
80C2×C4×D61—21no—
81C2×C4×D61—21no—
82C2×C4×D61—21no—
83C2×C4×D61—21no—
84C2×C4×D61—22no—
85C2×C4×D61—22no—
86C2×C4×D61—22no—
87C2×C8×C61—22no—
88C2×C8×C61—22no—
89C2×C8×C61—41no—
90C2×C8×C61—41no—
91C2×C8×C61—41no—
92C2×C8×C61—41no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—41no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97C4⋊C4×C61M1612yescomplemented
98C42×C61—41no—
99#11758198—24no—
100#11758198—42no—
101C4⋊Dic61—21no—
102C4⋊Dic61—42no—
103C4⋊Dic61—41no—
104C4⋊Dic61M1612yescomplemented
105#11758196—41no—
106#11758196—41no—
107#11758196—42no—
108C4×Dic61—24no—
109C4×Dic61—41no—
110C4×Dic61—41no—
111C4×Dic61—41no—
112C4×Dic61—42no—
113C2×#92418—22no—
114C2×#92418—22no—
115C2×#92418—41no—
116C2×#92418—41no—
117C2×#92418—41no—
118C2×#92418—41no—
119#11758189—48no—
120#11758189—48no—
121C23×C61C2×C22⋊C411yescharacteristic
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—21no—
125C22×D61—21no—
126C22×D61—21no—
127C22×D61—21no—
128C22×D61#9912yes—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—22no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61C2×M1611yescharacteristic
142C2×C4×C61C2×M1611yescharacteristic
143C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C61#13711yescharacteristic
146C2×C4×C61#13711yescharacteristic
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—21no—
152C2×Dic61—21no—
153C2×Dic61—22no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61#10112yes—
160C2×Dic61#10412yes—
161C2×Dic61#10412yes—
162C4×D61—24no—
163C4×D61—24no—
164C4×D61—24no—
165C4×D61—24no—
166C4×D61—24no—
167C4×D61—24no—
168C8×C61—42no—
169C8×C61—42no—
170#92418—42no—
171#92418—42no—
172#7632—611nomaximal, Sylow
173C22×C61#40711yescharacteristic
174C22×C61C2×#9911yescharacteristic
175C22×C61C2×#10111yescharacteristic
176C22×C61#41711yescharacteristic
177C22×C61#41711yescharacteristic
178C22×C61C2×#10411yescharacteristic
179C22×C61C2×#10411yescharacteristic
180C2×D61—22no—
181C2×D61—22no—
182C2×D61—22no—
183C2×D61—41no—
184C2×D61—41no—
185C2×D61—41no—
186C2×D61—41no—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—42no—
194C4×C61—81no—
195Dic61—22no—
196Dic61—22no—
197Dic61—22no—
198Dic61—42no—
199Dic61—81no—
200#2296—611no—
201#1728—611no—
202#1728—611no—
203#1271—611no—
204#1271—611no—
205#1271—611no—
206#1271D6111yescharacteristic, complemented
207C2×C61#146311yescharacteristic
208C2×C61#148911yescharacteristic
209C2×C61#148911yescharacteristic
210C2×C61#150411yescharacteristic
211C2×C61#150411yescharacteristic
212C2×C61#152611yescharacteristic
213C2×C61#156111yescharacteristic
214D61—42no—
215C22×M16—1221no—
216C22×M16—1221no—
217C2×#117—611no—
218C2×#117—611no—
219C22×C42—611no—
220#389—611no—
221#389—611no—
222#389—611no—
223#389C2×D6111yescharacteristic
224#385—1221no—
225#385—1221no—
226C4×C4⋊C4—1221no—
227C4×C4⋊C4—1221no—
228#336—611no—
229#336—611no—
230#336—1221no—
231#336—1221no—
232#336C2×D6111yescharacteristic
233#336C2×D6111yescharacteristic
234C61#763211yescharacteristic, complemented, Sylow
235C23×C4—611no—
236C23×C4—611no—
237C23×C4—611no—
238C2×M16—1224no—
239C2×M16—1224no—
240C2×M16—2442no—
241C2×M16—2442no—
242C2×M16—2442no—
243C2×M16—2442no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—1221no—
247C22×C8—1221no—
248#117—1222no—
249#117—1222no—
250#117—2442no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—611no—
253C2×C4⋊C4—611no—
254C2×C4⋊C4—1221no—
255C2×C4⋊C4—1221no—
256C2×C4⋊C4—1221no—
257C2×C4⋊C4C4×D6111yescharacteristic
258C2×C4⋊C4C4×D6111yescharacteristic
259C2×C22⋊C4—1221no—
260C2×C22⋊C4—1221no—
261C2×C42—611no—
262C2×C42—611no—
263C2×C42—611no—
264C2×C42—1221no—
265C2×C42—1221no—
266C2×C42—2442no—
267C2×C42C22×D6111yescharacteristic
268#95—21no—
269#95—1221no—
270#95—1221no—
271#95—1221no—
272C24—611no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—611no—
276C22×C4—611no—
277C22×C4—611no—
278C22×C4—1222no—
279C22×C4—1222no—
280C22×C4—1222no—
281C22×C4—1222no—
282C22×C4—1222no—
283C22×C4—1222no—
284C22×C4—1221no—
285C22×C4—1221no—
286C22×C4—1221no—
287C22×C4—1221no—
288C22×C4—1221no—
289C22×C4—1221no—
290C22×C4—1221no—
291C22×C4—1221no—
292C22×C4C2×C4×D6111yescharacteristic
293C22×C4D4×D6111yescharacteristic
294C22×C4D4×D6111yescharacteristic
295M16—2448no—
296M16—2448no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—1222no—
304C2×C8—1222no—
305C2×C8—2441no—
306C2×C8—2441no—
307C2×C8—2441no—
308C2×C8—2441no—
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—612no—
314C4⋊C4—1221no—
315C4⋊C4—2441no—
316C4⋊C4—2442no—
317C4⋊C4—2442no—
318C4⋊C4—2441no—
319C4⋊C4—2441no—
320C4⋊C4#1175818912yescomplemented
321C22⋊C4—1224no—
322C22⋊C4—2442no—
323C42—41no—
324C42—1224no—
325C42—2441no—
326C42—2441no—
327C42—2441no—
328C42—2442no—
329C23—612no—
330C23—1221no—
331C23—1221no—
332C23—1221no—
333C23—1221no—
334C23—1221no—
335C23—1221no—
336C23C22⋊C4×D6111yescharacteristic
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—612no—
350C2×C4—612no—
351C2×C4—612no—
352C2×C4—1224no—
353C2×C4—1224no—
354C2×C4—1224no—
355C2×C4—1224no—
356C2×C4—1224no—
357C2×C4—1224no—
358C2×C4—1221no—
359C2×C4—1221no—
360C2×C4—1221no—
361C2×C4—1221no—
362C2×C4—1221no—
363C2×C4—1221no—
364C2×C4—1222no—
365C2×C4—2441no—
366C2×C4—2441no—
367C2×C4—2441no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4M16×D6111yescharacteristic
371C2×C4C2×#1175818911yescharacteristic
372C2×C4#1451980111yescharacteristic
373C2×C4#1451980411yescharacteristic
374C2×C4#1451980611yescharacteristic
375C2×C4#1451980911yescharacteristic
376C8—42no—
377C8—42no—
378C8—2442no—
379C8—2442no—
380C22—1222no—
381C22—1222no—
382C22—1222no—
383C22—2441no—
384C22—2441no—
385C22—2441no—
386C22—2441no—
387C22#1683972411yescharacteristic
388C22#99×D6111yescharacteristic
389C22#101×D6111yescharacteristic
390C22#1683976011yescharacteristic
391C22#1683976211yescharacteristic
392C22#104×D6111yescharacteristic
393C22#104×D6111yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—42no—
401C4—81no—
402C4—1222no—
403C4—1222no—
404C4—1222no—
405C4—4881no—
406C4—2442no—
407C2—2442no—
408C2#2144414711yescharacteristic
409C2#2144425511yescharacteristic
410C2#2144425711yescharacteristic
411C2#2144432011yescharacteristic
412C2#2144432211yescharacteristic
413C2#2144441211yescharacteristic
414C2#2144455511yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1271C2
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1271D61
hypocenterC61#7632
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #21443643C2semidirect
5. #21443642C2semidirect
6. #21443641C2semidirect
7. C61×#1271C2semidirect
97. C4⋊C4×C61M16semidirect
104. C4⋊Dic61M16semidirect
206. #1271D61semidirect
234. C61#7632semidirect
320. C4⋊C4#11758189semidirect

Maximal subgroups

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

subgroupindex
#214483002
#214455112
#214436432
#214436422
#214436412
C61×#12712
#763261