Finite Group Databasebeta

← #29577181

Subgroups — #29577181

Order 15616.

500 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 103 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#21455767C211yesmaximal, characteristic
2#21446803C211yesmaximal, characteristic
3#21445918C211yesmaximal, characteristic
4#21444338C211yesmaximal, characteristic
5#21443725C211yesmaximal, characteristic
6#21443705C211yesmaximal, characteristic
7C61×#2347C211yesmaximal, characteristic
8C2×#14519756C2211yescharacteristic
9#16840103—22no—
10#16840097—21no—
11#16840092—21no—
12#16839805C412yescomplemented
13#16839770C412yescomplemented
14#16839726C2211yescharacteristic
15#16839726C2211yescharacteristic
16C4×C22⋊Dic61C2211yescharacteristic
17#16839528—22no—
18#16839521—22no—
19#16839498—21no—
20#16839498—21no—
21C2×#117×C61C2211yescharacteristic
22C61×#389—21no—
23C61×#388—22no—
24C61×#383—21no—
25C61×#380C2211yescharacteristic
26C4×C4⋊C4×C61—21no—
27C4×C4⋊C4×C61—21no—
28C4×C22⋊C4×C61C2211yescharacteristic
29C22×Dic122C2×C411yescharacteristic
30C2×#11758194—21no—
31C2×#11758194C2×C412yes—
32#14519756—21no—
33#14519756—21no—
34#14519756—21no—
35#14519756—21no—
36C2×C4⋊Dic61C2×C411yescharacteristic
37C2×#11758196—21no—
38C2×C22⋊Dic61—21no—
39C2×C22⋊Dic61—21no—
40C2×C4×Dic61—21no—
41#14519710—22no—
42#14519708—22no—
43#14519700—22no—
44#14519700—22no—
45C22⋊#92418—21no—
46C22⋊#92418—21no—
47C22⋊#92418—21no—
48C22⋊#92418—21no—
49#14519684—21no—
50C23×C4×C61C2311yescharacteristic
51#117×C61—21no—
52#117×C61—21no—
53#117×C61C2×C412yes—
54#117×C61D412yes—
55C2×C4⋊C4×C61—22no—
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61D411yescharacteristic
58C2×C4⋊C4×C61D411yescharacteristic
59C2×C22⋊C4×C61—21no—
60C2×C22⋊C4×C61—21no—
61C2×C22⋊C4×C61D412yes—
62C2×C42×C61—21no—
63C2×C42×C61—21no—
64C2×C42×C61—22no—
65#95×C61—22no—
66#95×C61—22no—
67#95×C61—21no—
68#95×C61D411yescharacteristic
69#95×C61D411yescharacteristic
70C24×C61C2×D411yescharacteristic
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—22no—
76C22×C4×C61—42no—
77C22×C4×C61C22×C411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61#4111yescharacteristic
82C22×C4×C61#4111yescharacteristic
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61—41no—
86C2×Dic122—21no—
87C2×Dic122—21no—
88C2×Dic122—42no—
89C4⋊C4×C61—21no—
90C4⋊C4×C61—21no—
91C4⋊C4×C61—42no—
92C4⋊C4×C61—44no—
93C4⋊C4×C61C22⋊C412yes—
94C4⋊C4×C61C22⋊C412yes—
95C22⋊C4×C61—22no—
96C22⋊C4×C61—22no—
97C22⋊C4×C61—42no—
98C22⋊C4×C61—42no—
99C42×C61—22no—
100C42×C61—22no—
101C42×C61—42no—
102C42×C61—82no—
103C22⋊Dic61—22no—
104C22⋊Dic61—41no—
105C22⋊Dic61—41no—
106C22⋊Dic61—41no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109#11758196—41no—
110#11758196—41no—
111C4×Dic61—42no—
112#11758194—42no—
113#11758194—42no—
114C2×#92418—22no—
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—22no—
128C2×C4×C61—22no—
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—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—42no—
140C2×C4×C61—42no—
141C2×C4×C61—42no—
142C2×C4×C61—41no—
143C2×C4×C61C2×C22⋊C411yescharacteristic
144C2×C4×C61C4×D411yescharacteristic
145C2×C4×C61#12111yescharacteristic
146C2×C4×C61#12311yescharacteristic
147C2×Dic61—42no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—81no—
151C2×Dic61—81no—
152C2×Dic61—81no—
153C2×Dic61—81no—
154Dic122—42no—
155Dic122—81no—
156#92418—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—42no—
177C4×C61—42no—
178C4×C61—82no—
179C4×C61—81no—
180C4×C61#41012yes—
181Dic61—82no—
182Dic61—81no—
183Dic61—81no—
184#3095—611no—
185#2347D6111yescharacteristic
186#1996—611no—
187#1826—611no—
188#1508—611no—
189#1305—611no—
190#1297—611no—
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×#117C2×D6111yescharacteristic
200#485—1222no—
201#484—1221no—
202#483—1221no—
203#428—612no—
204#419—612no—
205#407—611no—
206#407—611no—
207#389—21no—
208#388—22no—
209#383—21no—
210#380C2×D6111yescharacteristic
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C22⋊C4—611no—
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—612no—
223C2×M16—1221no—
224C22⋊Q8—1221no—
225C22⋊Q8—1221no—
226C22⋊Q8—1221no—
227C22⋊Q8—1221no—
228#117—21no—
229#117—21no—
230#117C4×D6112yes—
231#117D24412yes—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—611no—
235C2×C4⋊C4—1221no—
236C2×C4⋊C4C22⋊D6111yescharacteristic
237C2×C4⋊C4C22⋊D6111yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—1221no—
241C2×C22⋊C4—1221no—
242C2×C22⋊C4C22⋊D6112yes—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—1221no—
247#107—1222no—
248#106—1222no—
249#103—1222no—
250#103—1222no—
251C22⋊C8—1221no—
252C22⋊C8—1221no—
253C22⋊C8—1221no—
254C22⋊C8—1221no—
255#95—22no—
256#95—22no—
257#95—21no—
258#95—1221no—
259#95C22⋊D6111yescharacteristic
260#95C22⋊D6111yescharacteristic
261C24C2×C22⋊D6111yescharacteristic
262C2×Q8—1221no—
263C2×Q8—1221no—
264C2×Q8—2442no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—1221no—
272C22×C4—1221no—
273C22×C4—2441no—
274C22×C4C2×C4×D6111yescharacteristic
275C22×C4C2×D24411yescharacteristic
276C22×C4#1175822211yescharacteristic
277C22×C4#1175822211yescharacteristic
278C22×C4C2×C22⋊D6111yescharacteristic
279C22×C4C2×C22⋊D6111yescharacteristic
280M16—2442no—
281M16—2442no—
282C2×C8—1222no—
283C2×C8—2441no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—1221no—
289C4⋊C4—1221no—
290C4⋊C4—2441no—
291C4⋊C4—2441no—
292C4⋊C4#1175819812yes—
293C4⋊C4#1175819812yes—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—1222no—
299C22⋊C4—2441no—
300C22⋊C4—2441no—
301C22⋊C4—2441no—
302C42—22no—
303C42—22no—
304C42—42no—
305C42—82no—
306C42—2442no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23C4×C22⋊D6111yescharacteristic
312C23#1451974611yescharacteristic
313C23#1451975111yescharacteristic
314Q8—4881no—
315Q8—2442no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—4881no—
337C2×C4—4881no—
338C2×C4—4881no—
339C2×C4—4881no—
340C2×C4—2442no—
341C2×C4—2441no—
342C2×C4—2441no—
343C2×C4C2×#1175819811yescharacteristic
344C2×C4C4×C22⋊D6111yescharacteristic
345C2×C4#1451975111yescharacteristic
346C2×C4#1451976011yescharacteristic
347C8—4881no—
348C8—2442no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—81no—
358C22—81no—
359C22#1683964111yescharacteristic
360C22#1683994011yescharacteristic
361C22#1683994511yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—42no—
367C4—42no—
368C4—82no—
369C4—81no—
370C4—4882no—
371C4—4881no—
372C4—4881no—
373C4#1683973512yes—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#2144618811yescharacteristic
379C2#2144627711yescharacteristic
380C2#2144632011yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2347C2
socleC22×C61#386
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#2347D61
hypocenterC61#9362
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2347

Complements

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

NH ≅ G/Ntype
12. #16839805C4semidirect
13. #16839770C4semidirect
219. C61#9362semidirect

Maximal subgroups

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

subgroupindex
#214557672
#214468032
#214459182
#214443382
#214437252
#214437052
C61×#23472
#936261