Finite Group Databasebeta

← #29568198

Subgroups — #29568198

Order 15616.

494 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 101 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#21448366C211yesmaximal, characteristic
2#21445934C211yesmaximal, characteristic, complemented
3#21445566C211yesmaximal, characteristic, complemented
4#21443644C211yesmaximal, characteristic, complemented
5#21443644C211yesmaximal, characteristic, complemented
6#21443640C211yesmaximal, characteristic
7C61×#1270C211yesmaximal, characteristic
8C2×#14519764C411yescharacteristic
9C2×#14519742C411yescharacteristic
10C2×D4×Dic61C411yescharacteristic
11C2×#14519706—21no—
12C2×#14519687—21no—
13#16839668C2211yescharacteristic
14#16839650—21no—
15#16839650—21no—
16#16839627C411yescharacteristic
17C22⋊C4×Dic61—21no—
18C22⋊C4×Dic61—21no—
19C42×Dic61C2211yescharacteristic
20#16839517C2211yescharacteristic
21#16839517C2211yescharacteristic
22#16839516—21no—
23#16839516—21no—
24C61×#390C2211yescharacteristic
25C61×#336C2211yescharacteristic
26C61×#336C2211yescharacteristic
27C22×Dic122C2×C411yescharacteristic
28C23×Dic61—21no—
29C22×#92418—21no—
30C22×#92418—21no—
31#14519764—22no—
32#14519764—22no—
33#14519742—42no—
34D4×Dic61—42no—
35C2×C4⋊Dic61C2×C411yescharacteristic
36C2×#11758196—21no—
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—21no—
39C2×C22⋊Dic61—21no—
40C2×C4×Dic61—21no—
41C2×C4×Dic61C2×C411yescharacteristic
42C2×C4×Dic61C2×C411yescharacteristic
43C2×C4×Dic61D411yescharacteristic
44C2×C4×Dic61D411yescharacteristic
45C2×C4×Dic61D411yescharacteristic
46C2×C4×Dic61D411yescharacteristic
47#14519706—42no—
48#14519687—24no—
49#14519684—21no—
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53C22×D4×C61C2×C411yescharacteristic
54C22×C8×C61—21no—
55C22×C8×C61—21no—
56C2×C4⋊C4×C61C2×C411yescharacteristic
57C2×C22⋊C4×C61—21no—
58C2×C22⋊C4×C61—21no—
59C2×C42×C61C2311yescharacteristic
60C24×C61—21no—
61C2×D4×C61—21no—
62C2×D4×C61—41no—
63C2×D4×C61—41no—
64C2×D4×C61M1612yes—
65C22×C4×C61—21no—
66C22×C4×C61C22×C411yescharacteristic
67C22×C4×C61C2×D411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×Dic61—21no—
70C22×Dic61—21no—
71C22×Dic61—42no—
72C22×Dic61—42no—
73C22×Dic61—42no—
74C22×Dic61C22⋊C411yescharacteristic
75C22×Dic61C22⋊C411yescharacteristic
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C22⋊C411yescharacteristic
78C2×Dic122—21no—
79C2×Dic122—41no—
80C2×Dic122—41no—
81C2×Dic122M1612yes—
82C2×C8×C61—22no—
83C2×C8×C61—41no—
84C2×C8×C61—41no—
85C2×C8×C61—41no—
86C2×C8×C61—41no—
87C2×C8×C61—41no—
88C4⋊C4×C61—41no—
89C22⋊C4×C61—41no—
90C22⋊C4×C61—41no—
91C22⋊C4×C61—41no—
92C22⋊C4×C61—41no—
93C42×C61—41no—
94C22⋊Dic61—24no—
95C22⋊Dic61—24no—
96C22⋊Dic61—42no—
97C4⋊Dic61—41no—
98#11758196—42no—
99C4×Dic61—21no—
100C4×Dic61—21no—
101C4×Dic61—21no—
102C4×Dic61—21no—
103C4×Dic61—21no—
104C4×Dic61—21no—
105C4×Dic61—22no—
106C4×Dic61—22no—
107C4×Dic61—41no—
108C4×Dic61—42no—
109C2×#92418—22no—
110C2×#92418—41no—
111C2×#92418—41no—
112C2×#92418—41no—
113C2×#92418—41no—
114C2×#92418—41no—
115C23×C61—22no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61C2×C22⋊C411yescharacteristic
123D4×C61—42no—
124D4×C61—42no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61C2×M1611yescharacteristic
135C2×C4×C61C2×M1611yescharacteristic
136C2×C4×C61#13511yescharacteristic
137C2×C4×C61#13511yescharacteristic
138C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C61#13711yescharacteristic
140C2×Dic61—21no—
141C2×Dic61—21no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—22no—
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—84no—
157C2×Dic61#10012yes—
158C2×Dic61#10412yes—
159Dic122—42no—
160Dic122—42no—
161C8×C61—42no—
162C8×C61—81no—
163#92418—42no—
164#92418—81no—
165#7618—611nomaximal, Sylow
166C22×C61—42no—
167C22×C61—42no—
168C22×C61—42no—
169C22×C61—81no—
170C22×C61—81no—
171C22×C61—81no—
172C22×C61—81no—
173C22×C61#40711yescharacteristic
174C22×C61#41011yescharacteristic
175C22×C61C2×#10011yescharacteristic
176C22×C61#41611yescharacteristic
177C22×C61#41711yescharacteristic
178C22×C61C2×#10411yescharacteristic
179C22×C61#42111yescharacteristic
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—21no—
185C4×C61—21no—
186C4×C61—81no—
187Dic61—22no—
188Dic61—22no—
189Dic61—42no—
190Dic61—41no—
191Dic61—41no—
192Dic61—81no—
193C4×#124—611no—
194#1831—611no—
195#1745—611no—
196#1272—611no—
197#1272—611no—
198#1270—611no—
199#1270D6111yescharacteristic
200C2×C61—82no—
201C2×C61#146611yescharacteristic
202C2×C61#148811yescharacteristic
203C2×C61#148911yescharacteristic
204C2×C61#150011yescharacteristic
205C2×C61#151211yescharacteristic
206C2×C61#152711yescharacteristic
207C2×C61#155111yescharacteristic
208C2×#124—611no—
209C2×C4×Q8—611no—
210C2×C4×D4—611no—
211C2×C4⋊C8—1221no—
212C2×#97—1221no—
213#391—611no—
214#390C2×D6111yescharacteristic
215#388—1221no—
216#388—1221no—
217#382—611no—
218C4×C22⋊C4—1221no—
219C4×C22⋊C4—1221no—
220C43—611no—
221#336—611no—
222#336—611no—
223#336—1221no—
224#336—1221no—
225#336C2×D6111yescharacteristic
226#336C2×D6111yescharacteristic
227C61#761811yescharacteristic, complemented, Sylow
228C22×Q8—611no—
229C22×D4C4×D6111yescharacteristic
230C23×C4—1221no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—1221no—
234C22×C8—1221no—
235#124—1222no—
236#124—1222no—
237C4×Q8—2442no—
238C4×D4—2442no—
239C2×C4⋊C4—611no—
240C2×C4⋊C4—1221no—
241C2×C4⋊C4C4×D6111yescharacteristic
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—1221no—
245C2×C22⋊C4—1221no—
246C2×C22⋊C4—1221no—
247C2×C42—611no—
248C2×C42—611no—
249C2×C42—611no—
250C2×C42—611no—
251C2×C42—611no—
252C2×C42—611no—
253C2×C42—1221no—
254C2×C42C22×D6111yescharacteristic
255C4⋊C8—2442no—
256#97—1224no—
257#95—1221no—
258#95—1221no—
259#95—1221no—
260#95—1221no—
261C24—21no—
262C2×Q8—612no—
263C2×Q8—1221no—
264C2×Q8—2441no—
265C2×Q8—2441no—
266C2×D4—21no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4#1175818912yes—
270C22×C4—21no—
271C22×C4—611no—
272C22×C4—611no—
273C22×C4—611no—
274C22×C4—611no—
275C22×C4—1221no—
276C22×C4—1221no—
277C22×C4—2442no—
278C22×C4—2442no—
279C22×C4—2442no—
280C22×C4C2×C4×D6111yescharacteristic
281C22×C4D4×D6111yescharacteristic
282C22×C4D4×D6111yescharacteristic
283C2×C8—22no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—1222no—
290C2×C8—2441no—
291C2×C8—2441no—
292C2×C8—2441no—
293C2×C8—2441no—
294C2×C8—2441no—
295C4⋊C4—41no—
296C4⋊C4—2441no—
297C4⋊C4—2442no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—1224no—
303C22⋊C4—1224no—
304C22⋊C4—2442no—
305C42—41no—
306C42—1222no—
307C42—1222no—
308C42—1221no—
309C42—1221no—
310C42—1221no—
311C42—1221no—
312C42—1221no—
313C42—1221no—
314C42—2441no—
315C42—2442no—
316C23—22no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23C22⋊C4×D6111yescharacteristic
324Q8—2442no—
325Q8—2442no—
326D4—42no—
327D4—42no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—612no—
338C2×C4—612no—
339C2×C4—1221no—
340C2×C4—1221no—
341C2×C4—1221no—
342C2×C4—1221no—
343C2×C4—1221no—
344C2×C4—1221no—
345C2×C4—1222no—
346C2×C4—1221no—
347C2×C4—1221no—
348C2×C4—1221no—
349C2×C4—1221no—
350C2×C4—4884no—
351C2×C4—2441no—
352C2×C4—2441no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2441no—
356C2×C4M16×D6111yescharacteristic
357C2×C4C2×#1175818911yescharacteristic
358C2×C4#1451979611yescharacteristic
359C2×C4#1451979811yescharacteristic
360C2×C4#1451980111yescharacteristic
361C2×C4#1451980611yescharacteristic
362C8—42no—
363C8—81no—
364C8—4881no—
365C8—2442no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22#1683972411yescharacteristic
374C22#1683973411yescharacteristic
375C22#100×D6111yescharacteristic
376C22#1683975511yescharacteristic
377C22#1683976011yescharacteristic
378C22#104×D6111yescharacteristic
379C22#1683977711yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—81no—
387C4—1222no—
388C4—1222no—
389C4—4881no—
390C4—2442no—
391C4—2441no—
392C4—2441no—
393C2—82no—
394C2#2144415811yescharacteristic
395C2#2144425011yescharacteristic
396C2#2144425511yescharacteristic
397C2#2144430211yescharacteristic
398C2#2144435411yescharacteristic
399C2#2144441511yescharacteristic
400C2#2144451111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1270C2
socleC23×C61C2×C22⋊C4
prime radicalC22×D4×C61C2×C4
cosocleC2×C42×C61C23
hypercenter#1270D61
hypocenterC61#7618
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #21445934C2semidirect
3. #21445566C2semidirect
4. #21443644C2semidirect
5. #21443644C2semidirect
227. C61#7618semidirect

Maximal subgroups

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

subgroupindex
#214483662
#214459342
#214455662
#214436442
#214436402
C61×#12702
#761861