Finite Group Databasebeta

← #29577358

Subgroups — #29577358

Order 15616.

496 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 99 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#21451506C211yesmaximal, characteristic
2#21444416C212yesmaximal, complemented
3#21444153C211yesmaximal, characteristic, complemented
4#21443713C212yesmaximal, complemented
5C61×#2921C211yesmaximal, characteristic, complemented
6C2×#14519765C2211yescharacteristic
7#16839819C2212yes—
8#16839682C411yescharacteristic
9#16839679C411yescharacteristic
10#16839674C411yescharacteristic
11#16839671C411yescharacteristic
12#16839662—21no—
13#16839653—21no—
14#16839652—21no—
15#16839641—21no—
16#16839608C2211yescharacteristic
17#16839501—24no—
18#16839488—21no—
19#16839486—21no—
20C2×#117×C61C2211yescharacteristic
21C61×#433—24no—
22C61×#423—21no—
23C61×#407—21no—
24C4×M16×C61C2212yes—
25C22×Dic122C2×C411yescharacteristic
26C22×D244C2×C411yescharacteristic
27C22×C22⋊D61—21no—
28#14519765—22no—
29#14519765—22no—
30C2×#11758196—21no—
31C2×C22⋊Dic61—21no—
32C2×#11758198—21no—
33C2×#11758198—21no—
34C2×#11758198—21no—
35C2×#11758198—21no—
36C2×#11758198—21no—
37C2×C4×Dic61—21no—
38C4⋊#92418—22no—
39C22⋊#92418—22no—
40#14519688—22no—
41#14519684—21no—
42#14519684—21no—
43#14519684—21no—
44#14519684C2×C411yescharacteristic
45#14519684C2×C411yescharacteristic
46C23×C4×C61D411yescharacteristic
47C2×M16×C61—22no—
48#117×C61—22no—
49#117×C61—22no—
50C2×C4⋊C4×C61C2×C411yescharacteristic
51C2×C4⋊C4×C61D411yescharacteristic
52C2×C22⋊C4×C61C2×C411yescharacteristic
53C2×C22⋊C4×C61D411yescharacteristic
54C2×C42×C61D411yescharacteristic
55C2×C42×C61C2311yescharacteristic
56C4⋊C8×C61—24no—
57C22⋊C8×C61—24no—
58#97×C61D414yescomplemented
59C4×C8×C61—22no—
60C24×C61C22⋊C411yescharacteristic
61C23×D61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—41no—
65C22×C4×C61—41no—
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22×C411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C2×C22⋊D61—42no—
73C2×C22⋊D61—42no—
74C2×C22⋊D61—81no—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C2×D244—22no—
79C2×D244—81no—
80C2×Dic122—22no—
81C2×Dic122—81no—
82M16×C61—42no—
83M16×C61—42no—
84C2×C8×C61—24no—
85C4⋊C4×C61—22no—
86C4⋊C4×C61—22no—
87C22⋊C4×C61—22no—
88C22⋊C4×C61—22no—
89C42×C61—21no—
90C42×C61—21no—
91C42×C61C2×D412yes—
92C42×C61#4112yes—
93C22⋊Dic61—81no—
94#11758198—42no—
95#11758198—42no—
96#11758198—41no—
97#11758198—42no—
98#11758198—41no—
99#11758198—81no—
100#11758196—81no—
101C4×Dic61—42no—
102C2×#92418—42no—
103C23×C61—21no—
104C23×C61—21no—
105C23×C61—21no—
106C23×C61—41no—
107C23×C61—41no—
108C23×C61C2×C22⋊C411yescharacteristic
109C22×D61—41no—
110C22×D61—41no—
111C22×D61—41no—
112C22×D61—41no—
113C22×D61—41no—
114C22×D61—41no—
115C22×D61—41no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61C4×D412yes—
130C2×C4×C61#12011yescharacteristic
131C2×C4×C61#12112yes—
132C2×C4×C61#12311yescharacteristic
133C22⋊D61—84no—
134C2×Dic61—22no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143D244—82no—
144Dic122—82no—
145C8×C61—44no—
146#92418—82no—
147#9389—611nomaximal, Sylow
148C22×C61—21no—
149C22×C61—21no—
150C22×C61—42no—
151C22×C61—41no—
152C22×C61—41no—
153C22×C61—41no—
154C22×C61—81no—
155C22×C61#38611yescharacteristic
156C22×C61#42111yescharacteristic
157C22×C61#42111yescharacteristic
158C2×D61—81no—
159C2×D61—81no—
160C2×D61—81no—
161C2×D61—81no—
162C2×D61—81no—
163C2×D61—81no—
164C2×D61—81no—
165C4×C61—22no—
166C4×C61—21no—
167C4×C61—21no—
168C4×C61—22no—
169C4×C61—41no—
170C4×C61—41no—
171C4×C61—41no—
172Dic61—42no—
173Dic61—81no—
174Dic61—81no—
175#2921D6111yescharacteristic, complemented
176#2610—611no—
177#1527—612no—
178#1464—611no—
179#1300—612no—
180C2×C61—21no—
181C2×C61—21no—
182C2×C61—81no—
183C2×C61#188411yescharacteristic
184C2×C61#189711yescharacteristic
185C2×C61#191411yescharacteristic
186D61—161no—
187C2×#124—611no—
188C2×#117C2×D6111yescharacteristic
189#433—24no—
190#432—612no—
191#423—21no—
192#407—21no—
193C4×M16C2×D6112yes—
194#395—611no—
195#394—611no—
196#393—611no—
197#392—611no—
198#390—1221no—
199#388—1221no—
200#388—1221no—
201#386—1221no—
202#376—611no—
203#329—1224no—
204#324—1221no—
205#323—1221no—
206C61#938911yescharacteristic, complemented, Sylow
207C22×Q8—611no—
208C22×D4—611no—
209C22×D4—1221no—
210C23×C4C22⋊D6111yescharacteristic
211C2×M16—22no—
212#124—1222no—
213#124—1222no—
214#117—22no—
215#117—22no—
216C2×C4⋊C4—1221no—
217C2×C4⋊C4C4×D6111yescharacteristic
218C2×C4⋊C4D24411yescharacteristic
219C2×C22⋊C4—1221no—
220C2×C22⋊C4—1221no—
221C2×C22⋊C4—1221no—
222C2×C22⋊C4—1221no—
223C2×C22⋊C4—1221no—
224C2×C22⋊C4—1221no—
225C2×C22⋊C4C4×D6111yescharacteristic
226C2×C22⋊C4D24411yescharacteristic
227C2×C42—1221no—
228C2×C42C22⋊D6111yescharacteristic
229C2×C42C22×D6111yescharacteristic
230C4⋊C8—24no—
231C4⋊C8—1222no—
232C22⋊C8—24no—
233C22⋊C8—1222no—
234#97—1222no—
235#97C22⋊D6114yescomplemented
236C4×C8—22no—
237#95—611no—
238#95—611no—
239#95—1221no—
240#95—1221no—
241#95—1221no—
242C24—1221no—
243C24#1175819811yescharacteristic
244C2×Q8—1222no—
245C2×Q8—4881no—
246C2×D4—1222no—
247C2×D4—4881no—
248C2×D4—4881no—
249C2×D4—2442no—
250C2×D4—2442no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—1221no—
256C22×C4—1221no—
257C22×C4—1221no—
258C22×C4#1175819811yescharacteristic
259C22×C4#1175819811yescharacteristic
260C22×C4#1175819811yescharacteristic
261C22×C4C2×C4×D6111yescharacteristic
262C22×C4C2×D24411yescharacteristic
263C22×C4C2×C22⋊D6111yescharacteristic
264M16—42no—
265M16—42no—
266C2×C8—24no—
267C2×C8—2442no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—4881no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—4881no—
274C22⋊C4—4881no—
275C22⋊C4—2442no—
276C22⋊C4—2442no—
277C22⋊C4—2441no—
278C22⋊C4—2442no—
279C22⋊C4—2441no—
280C42—21no—
281C42—21no—
282C42—2442no—
283C42#1175822212yes—
284C42C2×C22⋊D6112yes—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—2441no—
291C23—2441no—
292C23—2441no—
293C23—2441no—
294C23—2441no—
295C23—2441no—
296C23—2441no—
297C23C2×#1175819811yescharacteristic
298Q8—4882no—
299D4—4882no—
300D4—4884no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—1222no—
315C2×C4—2441no—
316C2×C4—2441no—
317C2×C4—2441no—
318C2×C4—2441no—
319C2×C4—2441no—
320C2×C4—2441no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4C4×C22⋊D6112yes—
324C2×C4#1451974611yescharacteristic
325C2×C4#1451975112yes—
326C2×C4#1451976011yescharacteristic
327C8—44no—
328C8—4882no—
329C22—21no—
330C22—21no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—4881no—
337C22—4881no—
338C22—4881no—
339C22—4881no—
340C22—4881no—
341C22—4881no—
342C22—4881no—
343C22#1683964111yescharacteristic
344C22#1683978011yescharacteristic
345C22#1683978011yescharacteristic
346C4—22no—
347C4—21no—
348C4—21no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—4881no—
354C4—4881no—
355C4—2442no—
356C2—21no—
357C2—21no—
358C2—81no—
359C2—9761no—
360C2#2144620211yescharacteristic
361C2#2144626711yescharacteristic
362C2#2144636611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2921C2
socleC22×C61#386
prime radical#16839671C4
cosocleC2×C42×C61C23
hypercenter#2921D61
hypocenterC61#9389
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #21444416C2semidirect
3. #21444153C2semidirect
4. #21443713C2semidirect
5. C61×#2921C2semidirect
58. #97×C61D4semidirect
175. #2921D61semidirect
206. C61#9389semidirect
235. #97C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214515062
#214444162
#214441532
#214437132
C61×#29212
#938961