Finite Group Databasebeta

← #29577182

Subgroups — #29577182

Order 15616.

500 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 93 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#21455768C211yesmaximal, characteristic
2#21448710C211yesmaximal, characteristic
3#21446804C211yesmaximal, characteristic
4#21445919C211yesmaximal, characteristic
5#21444338C211yesmaximal, characteristic
6#21443724C211yesmaximal, characteristic
7C61×#1297C211yesmaximal, characteristic
8C2×#14519756C2211yescharacteristic
9C2×#14519736C2211yescharacteristic
10#16840104—22no—
11#16840098—21no—
12#16840093—21no—
13#16839804C412yescomplemented
14#16839772C412yescomplemented
15#16839726C2211yescharacteristic
16#16839660—21no—
17#16839656—22no—
18#16839629—21no—
19#16839622C2211yescharacteristic
20C4×C4⋊Dic61—21no—
21#16839613—21no—
22#16839528—22no—
23#16839522—22no—
24#16839498—21no—
25#16839498—21no—
26C61×#407C2211yescharacteristic
27C4×C22⋊C4×C61C2211yescharacteristic
28C4×C22⋊C4×C61C2211yescharacteristic
29C22×Dic122C2×C411yescharacteristic
30C2×#11758194—21no—
31#14519756—21no—
32#14519756—21no—
33#14519756—21no—
34#14519756—21no—
35#14519736—21no—
36#14519736—21no—
37#14519736C2×C412yes—
38#14519736D412yes—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×C4⋊Dic61D411yescharacteristic
42C2×C4⋊Dic61D411yescharacteristic
43C2×#11758196—22no—
44C2×#11758196—21no—
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61—21no—
47C2×C22⋊Dic61D412yes—
48C2×C4×Dic61—21no—
49C2×C4×Dic61—22no—
50#14519711—22no—
51#14519709—22no—
52#14519702—22no—
53#14519702—22no—
54C22⋊#92418—21no—
55C22⋊#92418—21no—
56#14519684—22no—
57#14519684—22no—
58#14519684—21no—
59C23×C4×C61C2311yescharacteristic
60C2×M16×C61C2×C412yes—
61C2×C22⋊C4×C61—21no—
62C2×C22⋊C4×C61—21no—
63C2×C42×C61—21no—
64C2×C42×C61—21no—
65C22⋊C8×C61—21no—
66C22⋊C8×C61—21no—
67#95×C61—21no—
68#95×C61D411yescharacteristic
69#95×C61D411yescharacteristic
70C24×C61C2×D411yescharacteristic
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—42no—
74C22×C4×C61—41no—
75C22×C4×C61C22×C411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61#4111yescharacteristic
80C22×C4×C61#4111yescharacteristic
81C22×Dic61—21no—
82C22×Dic61—21no—
83C22×Dic61—22no—
84C22×Dic61—21no—
85C22×Dic61—21no—
86C2×Dic122—21no—
87C2×Dic122—21no—
88C2×Dic122—42no—
89M16×C61—42no—
90C2×C8×C61—22no—
91C22⋊C4×C61—22no—
92C22⋊C4×C61—42no—
93C22⋊C4×C61—41no—
94C42×C61—42no—
95C42×C61—42no—
96C22⋊Dic61—22no—
97C22⋊Dic61—22no—
98C22⋊Dic61—41no—
99C22⋊Dic61—41no—
100C22⋊Dic61—42no—
101C4⋊Dic61—21no—
102C4⋊Dic61—21no—
103C4⋊Dic61—21no—
104C4⋊Dic61—21no—
105C4⋊Dic61—42no—
106C4⋊Dic61C22⋊C412yes—
107C4⋊Dic61C22⋊C412yes—
108#11758196—44no—
109#11758196—41no—
110#11758196—41no—
111C4×Dic61—22no—
112C4×Dic61—22no—
113C4×Dic61—82no—
114#11758194—42no—
115C2×#92418—41no—
116C23×C61—21no—
117C23×C61—21no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61C4×D411yescharacteristic
121C23×C61#12011yescharacteristic
122C23×C61#12111yescharacteristic
123C2×C4×C61—22no—
124C2×C4×C61—22no—
125C2×C4×C61—22no—
126C2×C4×C61—22no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—42no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—81no—
137C2×C4×C61—81no—
138C2×C4×C61C2×C22⋊C411yescharacteristic
139C2×C4×C61C4×D411yescharacteristic
140C2×C4×C61#12111yescharacteristic
141C2×C4×C61#12311yescharacteristic
142C2×Dic61—22no—
143C2×Dic61—22no—
144C2×Dic61—42no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—42no—
148C2×Dic61—42no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—42no—
152C2×Dic61—81no—
153C2×Dic61—81no—
154Dic122—42no—
155Dic122—81no—
156C8×C61—42no—
157#92418—81no—
158#9362—611nomaximal, Sylow
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—81no—
168C22×C61—81no—
169C22×C61#38611yescharacteristic
170C22×C61#45511yescharacteristic
171C22×C61#45611yescharacteristic
172C4×C61—21no—
173C4×C61—21no—
174C4×C61—21no—
175C4×C61—42no—
176C4×C61—81no—
177C4×C61—82no—
178C4×C61#41012yes—
179Dic61—42no—
180Dic61—42no—
181Dic61—82no—
182Dic61—81no—
183Dic61—81no—
184#3095—611no—
185#2347—611no—
186#1996—611no—
187#1826—611no—
188#1508—611no—
189#1305—611no—
190#1297D6111yescharacteristic
191C2×C61—21no—
192C2×C61—21no—
193C2×C61—41no—
194C2×C61—41no—
195C2×C61#188211yescharacteristic
196C2×C61#189911yescharacteristic
197C2×C61#190611yescharacteristic
198C2×C22⋊Q8—611no—
199C2×#117—611no—
200#485—1222no—
201#484—1221no—
202#483—1221no—
203#428—612no—
204#419—612no—
205#407—611no—
206#407C2×D6111yescharacteristic
207#389—1221no—
208#388—1222no—
209#383—1221no—
210#380—611no—
211C4×C4⋊C4—1221no—
212C4×C4⋊C4—1221no—
213C4×C22⋊C4C2×D6111yescharacteristic
214C4×C22⋊C4C2×D6111yescharacteristic
215#342—1222no—
216#339—1222no—
217#328—1221no—
218#328—1221no—
219C61#936211yescharacteristic, complemented, Sylow
220C22×Q8—611no—
221C23×C4C22×D6111yescharacteristic
222C2×M16—1221no—
223C2×M16C4×D6112yes—
224C22⋊Q8—1221no—
225C22⋊Q8—1221no—
226C22⋊Q8—1221no—
227C22⋊Q8—1221no—
228#117—612no—
229#117—612no—
230#117—1221no—
231#117—1221no—
232C2×C4⋊C4—611no—
233C2×C4⋊C4—611no—
234C2×C4⋊C4—611no—
235C2×C4⋊C4—1222no—
236C2×C4⋊C4—1221no—
237C2×C4⋊C4—1221no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—612no—
241C2×C22⋊C4—1221no—
242C2×C22⋊C4—1221no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—1221no—
246C2×C42—1222no—
247#107—1222no—
248#106—1222no—
249#103—1222no—
250#103—1222no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—1221no—
254C22⋊C8—1221no—
255#95—21no—
256#95—1222no—
257#95—1222no—
258#95—1221no—
259#95D24411yescharacteristic
260#95D24411yescharacteristic
261C24D4×D6111yescharacteristic
262C2×Q8—1221no—
263C2×Q8—1221no—
264C2×Q8—2442no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—1221no—
270C22×C4—1221no—
271C22×C4—1221no—
272C22×C4—1221no—
273C22×C4—1222no—
274C22×C4C2×C4×D6111yescharacteristic
275C22×C4C2×D24411yescharacteristic
276C22×C4#1175822211yescharacteristic
277C22×C4D4×D6111yescharacteristic
278C22×C4D4×D6111yescharacteristic
279C22×C4#1175822611yescharacteristic
280M16—42no—
281M16—2442no—
282C2×C8—22no—
283C2×C8—2441no—
284C4⋊C4—612no—
285C4⋊C4—612no—
286C4⋊C4—1221no—
287C4⋊C4—1221no—
288C4⋊C4—1221no—
289C4⋊C4—1221no—
290C4⋊C4—2444no—
291C4⋊C4—2442no—
292C4⋊C4—2441no—
293C4⋊C4—2441no—
294C22⋊C4—22no—
295C22⋊C4—42no—
296C22⋊C4—41no—
297C22⋊C4—1222no—
298C22⋊C4—1222no—
299C22⋊C4—2441no—
300C22⋊C4—2441no—
301C22⋊C4—2442no—
302C42—42no—
303C42—42no—
304C42—1222no—
305C42—1222no—
306C42—4882no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23#1451973911yescharacteristic
312C23#1451974711yescharacteristic
313C23#1451975011yescharacteristic
314Q8—4881no—
315Q8—2442no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—1222no—
332C2×C4—1222no—
333C2×C4—4881no—
334C2×C4—4881no—
335C2×C4—2442no—
336C2×C4—2442no—
337C2×C4—2442no—
338C2×C4—2441no—
339C2×C4—2441no—
340C2×C4—2442no—
341C2×C4—2441no—
342C2×C4—2441no—
343C2×C4C22⋊C4×D6111yescharacteristic
344C2×C4C4×D24411yescharacteristic
345C2×C4#1451975211yescharacteristic
346C2×C4#1451976211yescharacteristic
347C8—42no—
348C8—4881no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—81no—
358C22—81no—
359C22#1683964011yescharacteristic
360C22#1683994111yescharacteristic
361C22#1683994611yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—81no—
367C4—82no—
368C4—4882no—
369C4—4881no—
370C4—4881no—
371C4—2442no—
372C4—2442no—
373C4#1683973412yes—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#2144618911yescharacteristic
379C2#2144627611yescharacteristic
380C2#2144631911yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1297C2
socleC22×C61#386
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#1297D61
hypocenterC61#9362
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
13. #16839804C4semidirect
14. #16839772C4semidirect
219. C61#9362semidirect

Maximal subgroups

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

subgroupindex
#214557682
#214487102
#214468042
#214459192
#214443382
#214437242
C61×#12972
#936261