Finite Group Databasebeta

← #29571450

Subgroups — #29571450

Order 15616.

684 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×#16840374C211yesmaximal, characteristic
2#21444215C212yesmaximal, complemented
3#21444213C212yesmaximal, complemented
4#21444133C211yesmaximal, characteristic
5C61×#1458C211yesmaximal, characteristic
6#16840374—24no—
7#16840374—24no—
8C2×#14519777C411yescharacteristic
9C2×#14519770—21no—
10C2×#14519770—21no—
11C2×#14519769C411yescharacteristic
12C2×#14519767—21no—
13C2×#14519767—21no—
14C2×#14519756C411yescharacteristic
15C2×#14519756C411yescharacteristic
16C2×C4×Dic122C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18#16839815C2212yes—
19#16839494—24no—
20#16839492—24no—
21#16839488—21no—
22#16839486—21no—
23C2×#117×C61C2211yescharacteristic
24C61×#431C2212yes—
25C61×#324—21no—
26C61×#323—21no—
27C22×Dic122C2×C411yescharacteristic
28C22×Dic122C2×C411yescharacteristic
29#14519777—22no—
30#14519777—22no—
31#14519770—28no—
32#14519770—28no—
33#14519769—22no—
34#14519769—22no—
35#14519767—28no—
36#14519767—28no—
37#14519756—22no—
38#14519756—22no—
39#14519756—22no—
40#14519756—22no—
41C4×Dic122—22no—
42C4×Dic122—22no—
43C4×Dic122D414yes—
44C4×Dic122D414yes—
45C2×C4⋊Dic61—21no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61C2×C411yescharacteristic
48C2×C4⋊Dic61C2×C411yescharacteristic
49C2×#11758196—21no—
50C2×#11758196—21no—
51C2×#11758196—21no—
52C2×#11758196—21no—
53C2×#11758196—21no—
54C2×#11758196—21no—
55C2×C22⋊Dic61—21no—
56C2×C22⋊Dic61—21no—
57C2×C4×Dic61—21no—
58C2×C4×Dic61—21no—
59C4⋊#92418—22no—
60C22⋊#92418—22no—
61C4×#92418—22no—
62C23×C4×C61C2×C411yescharacteristic
63#117×C61—22no—
64#117×C61—22no—
65C2×C4⋊C4×C61D411yescharacteristic
66C2×C4⋊C4×C61D411yescharacteristic
67C2×C22⋊C4×C61D411yescharacteristic
68C2×C22⋊C4×C61D411yescharacteristic
69C2×C42×C61C2×C411yescharacteristic
70C2×C42×C61C2311yescharacteristic
71C4⋊C8×C61—22no—
72C22⋊C8×C61—22no—
73C4×C8×C61—22no—
74C24×C61C22⋊C411yescharacteristic
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—22no—
78C22×C4×C61—22no—
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22×C411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89C2×Dic122—22no—
90C2×Dic122—22no—
91C2×Dic122—22no—
92C2×Dic122—22no—
93C2×Dic122—21no—
94C2×Dic122—21no—
95C2×Dic122C22⋊C412yes—
96C2×Dic122C22⋊C412yes—
97C2×C8×C61—42no—
98C4⋊C4×C61—22no—
99C4⋊C4×C61—22no—
100C22⋊C4×C61—22no—
101C22⋊C4×C61—22no—
102C42×C61—21no—
103C42×C61—21no—
104C42×C61C2×D412yes—
105C42×C61C2×D412yes—
106C22⋊Dic61—24no—
107C22⋊Dic61—24no—
108C4⋊Dic61—24no—
109C4⋊Dic61—24no—
110C4⋊Dic61—21no—
111C4⋊Dic61—21no—
112C4⋊Dic61C22⋊C412yes—
113C4⋊Dic61C22⋊C412yes—
114#11758196—24no—
115#11758196—24no—
116#11758196—24no—
117#11758196—24no—
118#11758196—24no—
119#11758196—24no—
120C4×Dic61—24no—
121C4×Dic61—24no—
122C2×#92418—42no—
123C23×C61—21no—
124C23×C61—21no—
125C23×C61—21no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61C2×C22⋊C411yescharacteristic
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61C2×C22⋊C411yescharacteristic
141C2×C4×C61C2×C22⋊C411yescharacteristic
142C2×C4×C61#12012yes—
143C2×C4×C61#12012yes—
144C2×C4×C61#13112yes—
145C2×C4×C61#13112yes—
146C2×Dic61—22no—
147C2×Dic61—22no—
148C2×Dic61—22no—
149C2×Dic61—22no—
150C2×Dic61—42no—
151C2×Dic61—42no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156Dic122—24no—
157Dic122—24no—
158Dic122—42no—
159Dic122—42no—
160C8×C61—82no—
161#92418—82no—
162#8327—611nomaximal, Sylow
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—41no—
166C22×C61—41no—
167C22×C61—41no—
168C22×C61—81no—
169C22×C61—81no—
170C22×C61#37911yescharacteristic
171C22×C61#40811yescharacteristic
172C22×C61#40811yescharacteristic
173C4×C61—22no—
174C4×C61—22no—
175C4×C61—41no—
176C4×C61—41no—
177C4×C61—41no—
178C4×C61#40812yes—
179C4×C61#40812yes—
180Dic61—42no—
181Dic61—42no—
182Dic61—42no—
183Dic61—42no—
184C2×#541—611no—
185#1479—612no—
186#1479—612no—
187#1458—611no—
188#1458D6111yescharacteristic
189C2×C61—21no—
190C2×C61—21no—
191C2×C61—81no—
192C2×C61#178011yescharacteristic
193C2×C61#179611yescharacteristic
194C2×C61#179611yescharacteristic
195#541—1224no—
196#541—1224no—
197C2×C4⋊Q8—611no—
198C2×#126—1221no—
199C2×#126—1221no—
200C2×#125—611no—
201C2×#125—1221no—
202C2×#125—1221no—
203C2×C22⋊Q8—611no—
204C2×C22⋊Q8—611no—
205C2×C4×Q8—611no—
206C2×C4×Q8—611no—
207C2×#117C2×D6111yescharacteristic
208#431—612no—
209#431C2×D6112yes—
210Q8⋊C8—1224no—
211Q8⋊C8—1224no—
212#324—21no—
213#324—1221no—
214#323—21no—
215#323—1221no—
216C61#832711yescharacteristic, complemented, Sylow
217C22×Q8—611no—
218C22×Q8—611no—
219C23×C4C4×D6111yescharacteristic
220C4⋊Q8—1222no—
221C4⋊Q8—1222no—
222#126—1228no—
223#126—1228no—
224#125—1222no—
225#125—1222no—
226#125—1228no—
227#125—1228no—
228C22⋊Q8—1222no—
229C22⋊Q8—1222no—
230C22⋊Q8—1222no—
231C22⋊Q8—1222no—
232C4×Q8—614no—
233C4×Q8—614no—
234C4×Q8—1222no—
235C4×Q8—1222no—
236#117—22no—
237#117—22no—
238C2×C4⋊C4—611no—
239C2×C4⋊C4—611no—
240C2×C4⋊C4—1221no—
241C2×C4⋊C4—1221no—
242C2×C4⋊C4—1221no—
243C2×C4⋊C4—1221no—
244C2×C4⋊C4—1221no—
245C2×C4⋊C4—1221no—
246C2×C4⋊C4—1221no—
247C2×C4⋊C4—1221no—
248C2×C4⋊C4D24411yescharacteristic
249C2×C4⋊C4C22⋊D6111yescharacteristic
250C2×C22⋊C4—1221no—
251C2×C22⋊C4—1221no—
252C2×C22⋊C4D24411yescharacteristic
253C2×C22⋊C4C22⋊D6111yescharacteristic
254C2×C42—1221no—
255C2×C42—1221no—
256C2×C42C4×D6111yescharacteristic
257C2×C42C22×D6111yescharacteristic
258C4⋊C8—22no—
259C4⋊C8—1222no—
260C22⋊C8—22no—
261C22⋊C8—1222no—
262C4×C8—22no—
263C4×C8—1222no—
264C24#1175819811yescharacteristic
265C2×Q8—612no—
266C2×Q8—612no—
267C2×Q8—1221no—
268C2×Q8—1221no—
269C2×Q8—1222no—
270C2×Q8—1222no—
271C2×Q8—1222no—
272C2×Q8—1222no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—1221no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—1221no—
281C22×C4#1175819811yescharacteristic
282C22×C4#1175819811yescharacteristic
283C22×C4#1175819811yescharacteristic
284C22×C4C2×C4×D6111yescharacteristic
285C22×C4C2×D24411yescharacteristic
286C22×C4C2×C22⋊D6111yescharacteristic
287C2×C8—42no—
288C2×C8—2442no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—612no—
292C4⋊C4—612no—
293C4⋊C4—1224no—
294C4⋊C4—1224no—
295C4⋊C4—1224no—
296C4⋊C4—1224no—
297C4⋊C4—1224no—
298C4⋊C4—1224no—
299C4⋊C4—1221no—
300C4⋊C4—1221no—
301C4⋊C4—1224no—
302C4⋊C4—1224no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—1224no—
306C22⋊C4—1224no—
307C42—21no—
308C42—21no—
309C42—1224no—
310C42—1224no—
311C42D4×D6112yes—
312C42D4×D6112yes—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23C2×#1175819811yescharacteristic
319Q8—1224no—
320Q8—1224no—
321Q8—2442no—
322Q8—2442no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—1222no—
335C2×C4—1222no—
336C2×C4—1222no—
337C2×C4—1222no—
338C2×C4—2442no—
339C2×C4—2442no—
340C2×C4—2441no—
341C2×C4—2441no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4C22⋊C4×D6111yescharacteristic
345C2×C4C22⋊C4×D6111yescharacteristic
346C2×C4#1451974512yes—
347C2×C4#1451974712yes—
348C2×C4#1451978312yes—
349C2×C4#1451978512yes—
350C8—82no—
351C8—4882no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—81no—
358C22—81no—
359C22#1683961711yescharacteristic
360C22#1683972811yescharacteristic
361C22#1683973011yescharacteristic
362C4—22no—
363C4—22no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—2442no—
368C4—2442no—
369C4—2442no—
370C4—2442no—
371C4#1683972712yes—
372C4#1683972712yes—
373C2—21no—
374C2—21no—
375C2—81no—
376C2#2144573011yescharacteristic
377C2#2144579211yescharacteristic
378C2#2144579511yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1458C2
socleC22×C61#379
prime radicalC24×C61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1458D61
hypocenterC61#8327
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #21444215C2semidirect
3. #21444213C2semidirect
216. C61#8327semidirect

Maximal subgroups

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

subgroupindex
C2×#168403742
#214442152
#214442132
#214441332
C61×#14582
#832761