Finite Group Databasebeta

← #29589133

Subgroups — #29589133

Order 15616.

504 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 89 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#21454262C211yesmaximal, characteristic
2#349×D61C212yesmaximal, complemented
3#21443755C211yesmaximal, characteristic, complemented
4#21443754C212yesmaximal, complemented
5C61×#1320C211yesmaximal, characteristic, complemented
6C22×#11758222C411yescharacteristic
7C2×M16×D61C411yescharacteristic
8C2×M16×D61C2211yescharacteristic
9M32×D61—22no—
10#16839727—22no—
11#16839727—22no—
12#16839726C411yescharacteristic
13#16839725C411yescharacteristic
14#16839725C2211yescharacteristic
15#16839544C2212yes—
16#16839543—22no—
17C61×#407C2211yescharacteristic
18C61×#349C2212yes—
19C2×#11758222—21no—
20C2×#11758222—21no—
21C2×#11758222—21no—
22C2×#11758222—21no—
23C2×#11758222—42no—
24C2×#11758222—42no—
25C22×Dic122C811yescharacteristic
26C22×D244C811yescharacteristic
27C22×C22⋊D61C811yescharacteristic
28C22×C22⋊D61C811yescharacteristic
29C22×C4×D61C2×C411yescharacteristic
30C22×C4×D61C2×C411yescharacteristic
31C2×#11758194C2×C411yescharacteristic
32C2×#11758194C2×C411yescharacteristic
33C2×#11758189—21no—
34C2×#11758189D412yes—
35M16×D61—42no—
36M16×D61—42no—
37C2×C8×D61—21no—
38C2×C8×D61D412yes—
39#14519717—22no—
40#14519716—42no—
41C16×D61—42no—
42C22⋊#92418—22no—
43#14519689—22no—
44#14519689—22no—
45C23×C4×C61C2×C411yescharacteristic
46C2×M16×C61C2×C411yescharacteristic
47C2×M16×C61C2311yescharacteristic
48M32×C61—22no—
49C22⋊C8×C61—22no—
50C24×C61C2×C811yescharacteristic
51C23×D61C2×C811yescharacteristic
52C23×D61C2×C811yescharacteristic
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—41no—
56C22×C4×C61—41no—
57C22×C4×C61C2×C811yescharacteristic
58C22×C4×C61C22×C411yescharacteristic
59C2×C22⋊D61—21no—
60C2×C22⋊D61—21no—
61C2×C22⋊D61—21no—
62C2×C22⋊D61—21no—
63C2×C22⋊D61—42no—
64C2×C22⋊D61—42no—
65C22×Dic61C2×C811yescharacteristic
66C22×Dic61C2×C811yescharacteristic
67#11758222—44no—
68#11758222—42no—
69#11758222—42no—
70#11758222—44no—
71#11758222—42no—
72#11758222—42no—
73C2×D244—21no—
74C2×D244—21no—
75C2×D244—42no—
76C2×C4×D61—22no—
77C2×C4×D61—22no—
78C2×C4×D61—21no—
79C2×C4×D61—21no—
80C2×C4×D61—41no—
81C2×C4×D61—41no—
82C2×C4×D61C22⋊C411yescharacteristic
83C2×C4×D61C22⋊C411yescharacteristic
84C2×C4×D61M1611yescharacteristic
85C2×C4×D61M1611yescharacteristic
86C2×Dic122—21no—
87C2×Dic122—21no—
88C2×Dic122—42no—
89M16×C61—41no—
90M16×C61—41no—
91C2×C8×C61—21no—
92C2×C8×C61C2×D412yes—
93#11758194—41no—
94#11758194—41no—
95C2×#92418—21no—
96C2×#92418C22⋊C412yes—
97#11758189—42no—
98#11758189—42no—
99C8×D61—22no—
100C8×D61—22no—
101C8×D61—42no—
102C16×C61—42no—
103#11758185—42no—
104C23×C61—21no—
105C23×C61—41no—
106C23×C61—81no—
107C23×C61C22×C811yescharacteristic
108C22×D61—22no—
109C22×D61—21no—
110C22×D61—41no—
111C22×D61—81no—
112C22×D61—81no—
113C22×D61C22⋊C811yescharacteristic
114C22×D61C22⋊C811yescharacteristic
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—42no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61C2×C22⋊C411yescharacteristic
123C2×C4×C61C2×M1611yescharacteristic
124C22⋊D61—42no—
125C22⋊D61—42no—
126C22⋊D61—42no—
127C22⋊D61—42no—
128C2×Dic61—22no—
129C2×Dic61—21no—
130C2×Dic61—41no—
131C2×Dic61C22⋊C811yescharacteristic
132C2×Dic61C22⋊C811yescharacteristic
133D244—42no—
134D244—42no—
135C4×D61—22no—
136C4×D61—21no—
137C4×D61—21no—
138C4×D61—22no—
139C4×D61—42no—
140C4×D61—41no—
141C4×D61—41no—
142C4×D61#9912yes—
143C4×D61#10012yes—
144Dic122—42no—
145Dic122—42no—
146C8×C61—22no—
147C8×C61—41no—
148#92418—22no—
149#92418—41no—
150#11436—611nomaximal, Sylow
151C22×C61—21no—
152C22×C61—41no—
153C22×C61—42no—
154C22×C61—41no—
155C22×C61—82no—
156C22×C61C2×C22⋊C811yescharacteristic
157C2×D61—21no—
158C2×D61—41no—
159C2×D61—41no—
160C2×D61—42no—
161C2×D61—41no—
162C2×D61—82no—
163C2×D61—82no—
164C2×D61#32312yes—
165C4×C61—21no—
166C4×C61—41no—
167C4×C61C2×#9911yescharacteristic
168C4×C61C2×#10011yescharacteristic
169Dic61—21no—
170Dic61—41no—
171Dic61#32312yes—
172#2879—611no—
173C2×#349—612no—
174#1320—611no—
175#1320—612no—
176#1320D6111yescharacteristic, complemented
177C2×C61—21no—
178C2×C61—41no—
179C2×C61—81no—
180C2×C61C2×#32311yescharacteristic
181D61—22no—
182D61—41no—
183D61—81no—
184C22×#41—611no—
185C22×M16—611no—
186C22×M16—611no—
187C2×M32—1222no—
188#408—1222no—
189#408—1222no—
190#407—611no—
191#407—611no—
192#407—611no—
193#407C2×D6111yescharacteristic
194#349—612no—
195#349—1222no—
196#349C2×D6112yes—
197C61#1143611yescharacteristic, complemented, Sylow
198C2×#41—1221no—
199C2×#41—1221no—
200C2×#41—1221no—
201C2×#41—1221no—
202C2×#41—2442no—
203C2×#41—2442no—
204C22×Q8—611no—
205C22×D4—611no—
206C22×D4—611no—
207C22×D4—611no—
208C23×C4—611no—
209C23×C4—611no—
210C23×C4C4×D6111yescharacteristic
211C2×M16—611no—
212C2×M16—612no—
213C2×M16—611no—
214C2×M16—1221no—
215C2×M16—2442no—
216C2×M16—2442no—
217C2×M16C4×D6111yescharacteristic
218C2×M16C22×D6111yescharacteristic
219C22×C8—612no—
220C22×C8—1221no—
221M32—22no—
222M32—1222no—
223M32—2442no—
224C2×C16—2442no—
225C22⋊C8—22no—
226C22⋊C8—1222no—
227C22⋊C8—1222no—
228C22⋊C8—1222no—
229C24—611no—
230C24—611no—
231C24C8×D6111yescharacteristic
232#41—2442no—
233#41—2442no—
234#41—2444no—
235#41—2444no—
236#41—2442no—
237#41—2442no—
238C2×Q8—1221no—
239C2×Q8—1221no—
240C2×Q8—2442no—
241C2×D4—1221no—
242C2×D4—1221no—
243C2×D4—1221no—
244C2×D4—1221no—
245C2×D4—1221no—
246C2×D4—1221no—
247C2×D4—2442no—
248C2×D4—2442no—
249C2×D4—2442no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—611no—
255C22×C4—611no—
256C22×C4—611no—
257C22×C4—611no—
258C22×C4—611no—
259C22×C4—611no—
260C22×C4—1222no—
261C22×C4—1222no—
262C22×C4—1221no—
263C22×C4—1221no—
264C22×C4—2441no—
265C22×C4—2441no—
266C22×C4C8×D6111yescharacteristic
267C22×C4C2×C4×D6111yescharacteristic
268M16—41no—
269M16—41no—
270M16—2442no—
271M16—2442no—
272M16—2441no—
273M16—2441no—
274C2×C8—21no—
275C2×C8—612no—
276C2×C8—1222no—
277C2×C8—1222no—
278C2×C8—1221no—
279C2×C8—2442no—
280C2×C8D4×D6112yes—
281C16—42no—
282C16—2442no—
283C23—21no—
284C23—41no—
285C23—81no—
286C23—611no—
287C23—611no—
288C23—1222no—
289C23—1221no—
290C23—4881no—
291C23—4881no—
292C23—2441no—
293C23C2×C8×D6111yescharacteristic
294Q8—2442no—
295Q8—2442no—
296D4—2442no—
297D4—2442no—
298D4—2442no—
299D4—2442no—
300D4—2442no—
301D4—2442no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—611no—
310C2×C4—612no—
311C2×C4—612no—
312C2×C4—611no—
313C2×C4—1222no—
314C2×C4—1222no—
315C2×C4—1222no—
316C2×C4—1221no—
317C2×C4—1221no—
318C2×C4—1221no—
319C2×C4—2442no—
320C2×C4—2441no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4C22⋊C4×D6111yescharacteristic
324C2×C4M16×D6111yescharacteristic
325C8—22no—
326C8—41no—
327C8—1222no—
328C8—2441no—
329C22—21no—
330C22—41no—
331C22—42no—
332C22—41no—
333C22—82no—
334C22—612no—
335C22—1221no—
336C22—4882no—
337C22—4882no—
338C22—2441no—
339C22—2442no—
340C22—2441no—
341C22—2441no—
342C22C22⋊C8×D6111yescharacteristic
343C4—21no—
344C4—41no—
345C4—612no—
346C4—1221no—
347C4—2441no—
348C4#99×D6111yescharacteristic
349C4#100×D6111yescharacteristic
350C2—21no—
351C2—41no—
352C2—81no—
353C2—1222no—
354C2—4881no—
355C2—2441no—
356C2#323×D6111yescharacteristic
subgroupG/H
centerC4#99×D61
commutatorC23×C61C22×C8
FrattiniC2×M16C22×D61
FittingC61×#1320C2
socleC2×C61C2×#323
prime radicalC22×#11758222C4
cosocleC2×M16×C61C23
hypercenter#1320D61
hypocenterC61#11436
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1320

Complements

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

NH ≅ G/Ntype
2. #349×D61C2semidirect
3. #21443755C2semidirect
4. #21443754C2semidirect
5. C61×#1320C2semidirect
176. #1320D61semidirect
197. C61#11436semidirect

Maximal subgroups

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

subgroupindex
#214542622
#349×D612
#214437552
#214437542
C61×#13202
#1143661