Finite Group Databasebeta

← #29575784

Subgroups — #29575784

Order 15616.

902 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 902 conjugacy classes of subgroups — too many to draw. Only its 143 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×#16839625C211yesmaximal, characteristic
2#21445729C211yesmaximal, characteristic
3C2×#16839499C212yesmaximal, complemented
4#21443697C212yesmaximal, complemented
5C61×#2848C211yesmaximal, characteristic
6C22×C22⋊Dic61—21no—
7C2×C22⋊#92418—22no—
8C2×C22⋊#92418C2212yes—
9#16839625—24no—
10C2×#14519684—21no—
11C2×#14519684C2212yes—
12#16839499—24no—
13C24×C4×C61C2211yescharacteristic
14C2×C22⋊Q8×C61C412yescomplemented
15C2×C22⋊Q8×C61C2211yescharacteristic
16C61×#385—21no—
17C61×#385—22no—
18C61×#380C412yescomplemented
19C61×#380C2211yescharacteristic
20C61×#379—21no—
21C23×Dic61—22no—
22C22×#92418—22no—
23C2×C22⋊Dic61—24no—
24C2×C22⋊Dic61—44no—
25C22⋊#92418—22no—
26C22⋊#92418—22no—
27C22⋊#92418—44no—
28#14519684—24no—
29#14519684—22no—
30#14519684D414yes—
31C25×C61D411yescharacteristic
32C22×Q8×C61C2×C411yescharacteristic
33C23×C4×C61—22no—
34C23×C4×C61—22no—
35C23×C4×C61—44no—
36C23×C4×C61C2×C412yes—
37C23×C4×C61D412yes—
38C23×C4×C61D411yescharacteristic
39C23×C4×C61C2311yescharacteristic
40C22⋊Q8×C61—22no—
41C22⋊Q8×C61—22no—
42C22⋊Q8×C61—44no—
43C2×C4⋊C4×C61—22no—
44C2×C4⋊C4×C61—21no—
45C2×C4⋊C4×C61C2×C411yescharacteristic
46C2×C4⋊C4×C61C2×C412yes—
47C2×C22⋊C4×C61—22no—
48C2×C22⋊C4×C61—22no—
49C2×C22⋊C4×C61—21no—
50C2×C22⋊C4×C61—21no—
51#95×C61—22no—
52#95×C61—21no—
53C24×C61—24no—
54C24×C61—21no—
55C24×C61—21no—
56C24×C61—22no—
57C24×C61—22no—
58C24×C61—42no—
59C24×C61C22⋊C412yes—
60C24×C61C2×D411yescharacteristic
61C2×Q8×C61—22no—
62C2×Q8×C61—42no—
63C22×C4×C61—24no—
64C22×C4×C61—24no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—24no—
68C22×C4×C61—24no—
69C22×C4×C61—21no—
70C22×C4×C61—22no—
71C22×C4×C61—21no—
72C22×C4×C61—42no—
73C22×C4×C61—42no—
74C22×C4×C61—42no—
75C22×C4×C61—42no—
76C22×C4×C61—48no—
77C22×C4×C61—48no—
78C22×C4×C61C22⋊C412yes—
79C22×C4×C61C22×C411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D412yes—
82C22×C4×C61#4112yes—
83C22×Dic61—24no—
84C22×Dic61—44no—
85C22×Dic61—44no—
86C22×Dic61—42no—
87C22×Dic61—42no—
88C4⋊C4×C61—22no—
89C4⋊C4×C61—44no—
90C4⋊C4×C61—42no—
91C4⋊C4×C61—42no—
92C22⋊C4×C61—44no—
93C22⋊C4×C61—44no—
94C22⋊C4×C61—41no—
95C22⋊C4×C61—41no—
96C22⋊C4×C61—42no—
97C22⋊Dic61—48no—
98C2×#92418—44no—
99C2×#92418—42no—
100C23×C61—22no—
101C23×C61—22no—
102C23×C61—24no—
103C23×C61—24no—
104C23×C61—22no—
105C23×C61—22no—
106C23×C61—22no—
107C23×C61—22no—
108C23×C61—22no—
109C23×C61—22no—
110C23×C61—48no—
111C23×C61—48no—
112C23×C61—42no—
113C23×C61—42no—
114C23×C61—42no—
115C23×C61—42no—
116C23×C61#9914yes—
117C23×C61C2×C22⋊C411yescharacteristic
118C23×C61#12011yescharacteristic
119C23×C61#12311yescharacteristic
120C23×C61#13512yes—
121C23×C61#13712yes—
122Q8×C61—82no—
123C2×C4×C61—22no—
124C2×C4×C61—22no—
125C2×C4×C61—22no—
126C2×C4×C61—22no—
127C2×C4×C61—22no—
128C2×C4×C61—42no—
129C2×C4×C61—42no—
130C2×C4×C61—48no—
131C2×C4×C61—48no—
132C2×C4×C61—42no—
133C2×C4×C61—42no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61—42no—
139C2×C4×C61—42no—
140C2×C4×C61—42no—
141C2×C4×C61C4×D412yes—
142C2×C4×C61#12112yes—
143C2×Dic61—44no—
144C2×Dic61—44no—
145C2×Dic61—44no—
146C2×Dic61—84no—
147C2×Dic61—84no—
148#92418—82no—
149#9154—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—21no—
153C22×C61—21no—
154C22×C61—24no—
155C22×C61—24no—
156C22×C61—24no—
157C22×C61—24no—
158C22×C61—24no—
159C22×C61—48no—
160C22×C61—48no—
161C22×C61—42no—
162C22×C61—42no—
163C22×C61—42no—
164C22×C61—42no—
165C22×C61#38611yescharacteristic
166C22×C61C2×#9912yes—
167C22×C61#41611yescharacteristic
168C22×C61#41911yescharacteristic
169C22×C61#42112yes—
170C22×C61#45212yes—
171C22×C61#48512yes—
172C4×C61—22no—
173C4×C61—22no—
174C4×C61—42no—
175C4×C61—81no—
176C4×C61—81no—
177C4×C61—82no—
178Dic61—84no—
179#2848D6111yescharacteristic
180C2×#381—611no—
181#1779—611no—
182C2×#328—612no—
183#1294—612no—
184C2×C61—22no—
185C2×C61—22no—
186C2×C61—44no—
187C2×C61#151112yes—
188C2×C61#151612yes—
189C2×C61#187311yescharacteristic
190C2×C61#189711yescharacteristic
191C2×C61#189811yescharacteristic
192C24×C4C2×D6111yescharacteristic
193C2×C22⋊Q8Dic6112yescomplemented
194C2×C22⋊Q8C2×D6111yescharacteristic
195C22×C22⋊C4—1221no—
196C2×C22⋊C8—612no—
197C2×C22⋊C8—1222no—
198#385—21no—
199#385—22no—
200#381—1224no—
201#380Dic6112yescomplemented
202#380C2×D6111yescharacteristic
203#379—21no—
204C2×#95—612no—
205C2×#95—1221no—
206#328—1224no—
207C61#915411yescharacteristic, complemented, Sylow
208C25C22⋊D6111yescharacteristic
209C22×Q8C2×Dic6111yescharacteristic
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—44no—
213C23×C4—1222no—
214C23×C4C2×Dic6112yes—
215C23×C4C22⋊D6112yes—
216C23×C4C22⋊D6111yescharacteristic
217C23×C4C22×D6111yescharacteristic
218C22×C8—1222no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—44no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C2×Dic6111yescharacteristic
225C2×C4⋊C4C2×Dic6112yes—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—1224no—
231C2×C22⋊C4—2444no—
232C22⋊C8—1222no—
233C22⋊C8—1222no—
234C22⋊C8—2444no—
235#95—22no—
236#95—21no—
237#95—614no—
238#95—1224no—
239#95—1222no—
240C24—24no—
241C24—21no—
242C24—21no—
243C24—22no—
244C24—22no—
245C24—42no—
246C24C22⋊Dic6112yes—
247C24C2×C22⋊D6111yescharacteristic
248C2×Q8—22no—
249C2×Q8—42no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—48no—
264C22×C4—48no—
265C22×C4—1224no—
266C22×C4—2444no—
267C22×C4—2444no—
268C22×C4—2442no—
269C22×C4—2442no—
270C22×C4C22⋊Dic6112yes—
271C22×C4D4×D6112yes—
272C22×C4#1175822412yes—
273C22×C4C22×Dic6111yescharacteristic
274C22×C4C2×C22⋊D6111yescharacteristic
275C2×C8—2444no—
276C2×C8—2442no—
277C4⋊C4—22no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—42no—
286C22⋊C4—2448no—
287C23—22no—
288C23—22no—
289C23—24no—
290C23—24no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—48no—
298C23—48no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23#1451969214yes—
304C23C2×C22⋊Dic6111yescharacteristic
305C23#1451974511yescharacteristic
306C23#1451975811yescharacteristic
307C23#1451979712yes—
308C23#1451980812yes—
309Q8—82no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—48no—
318C2×C4—48no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—4884no—
329C2×C4—4884no—
330C2×C4—2444no—
331C2×C4—2444no—
332C2×C4—2444no—
333C2×C4D4×Dic6112yes—
334C2×C4#1451974912yes—
335C8—4882no—
336C22—24no—
337C22—24no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—24no—
343C22—24no—
344C22—24no—
345C22—48no—
346C22—48no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22#1683964211yescharacteristic
352C22C2×#1451969212yes—
353C22#1683975811yescharacteristic
354C22#1683977111yescharacteristic
355C22#1683977912yes—
356C22#1683992212yes—
357C22#1684010012yes—
358C4—22no—
359C4—22no—
360C4—42no—
361C4—82no—
362C4—81no—
363C4—81no—
364C4—4884no—
365C2—22no—
366C2—22no—
367C2—44no—
368C2#2144435212yes—
369C2#2144437512yes—
370C2#2144613211yescharacteristic
371C2#2144626811yescharacteristic
372C2#2144627311yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic61
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2848C2
socleC23×C61C2×C22⋊C4
prime radicalC25×C61D4
cosocleC23×C4×C61C23
hypercenter#2848D61
hypocenterC61#9154
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
3. C2×#16839499C2semidirect
4. #21443697C2semidirect
14. C2×C22⋊Q8×C61C4semidirect
18. C61×#380C4semidirect
193. C2×C22⋊Q8Dic61semidirect
201. #380Dic61semidirect
207. C61#9154semidirect

Maximal subgroups

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

subgroupindex
C2×#168396252
#214457292
C2×#168394992
#214436972
C61×#28482
#915461