Finite Group Databasebeta

← #29568547

Subgroups — #29568547

Order 15616.

532 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 107 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#21454150C211yesmaximal, characteristic
2#21445581C211yesmaximal, characteristic, complemented
3#21443641C212yesmaximal, complemented
4#21443639C212yesmaximal, complemented
5C61×#1797C211yesmaximal, characteristic, complemented
6C2×#14519760—21no—
7C2×#14519751C411yescharacteristic
8C2×C22⋊#92418—21no—
9C2×#14519688—21no—
10#16839664C411yescharacteristic
11#16839664C2211yescharacteristic
12#16839660C411yescharacteristic
13#16839660C2211yescharacteristic
14#16839653—21no—
15#16839653—21no—
16#16839641—21no—
17#16839622C411yescharacteristic
18#16839517C2212yes—
19C22×C42×C61C2211yescharacteristic
20C2×C4⋊C8×C61—21no—
21C2×C22⋊C8×C61—21no—
22C61×#336C2212yes—
23C22×D244C2×C411yescharacteristic
24C22×C22⋊D61—21no—
25C22×#92418—22no—
26#14519760—44no—
27#14519751—42no—
28C2×C4⋊Dic61C2×C411yescharacteristic
29C2×C4⋊Dic61C2×C411yescharacteristic
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×#11758196—21no—
32C2×#11758196—21no—
33C2×C22⋊Dic61—21no—
34C2×#11758198—21no—
35C2×#11758198—21no—
36C2×#11758198—21no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C22⋊#92418—24no—
40#14519688—42no—
41#14519684—21no—
42#14519684—21no—
43#14519684—21no—
44C23×C4×C61C2×C411yescharacteristic
45C23×C4×C61D411yescharacteristic
46C23×C4×C61D411yescharacteristic
47C22×C8×C61—22no—
48C2×C42×C61—42no—
49C2×C42×C61C2×C411yescharacteristic
50C2×C42×C61D411yescharacteristic
51C2×C42×C61D411yescharacteristic
52C2×C42×C61C2311yescharacteristic
53C4⋊C8×C61—42no—
54C22⋊C8×C61—42no—
55C24×C61C22⋊C411yescharacteristic
56C23×D61—21no—
57C22×C4×C61—22no—
58C22×C4×C61—22no—
59C22×C4×C61—22no—
60C22×C4×C61—21no—
61C22×C4×C61—22no—
62C22×C4×C61—21no—
63C22×C4×C61—41no—
64C22×C4×C61—41no—
65C22×C4×C61—41no—
66C22×C4×C61—41no—
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22×C411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C22×C4×C61C2×D411yescharacteristic
73C2×C22⋊D61—42no—
74C2×C22⋊D61—42no—
75C2×C22⋊D61—42no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C2×D244—22no—
80C2×D244—41no—
81C2×D244—41no—
82C2×C8×C61—42no—
83C2×C8×C61—42no—
84C2×C8×C61—42no—
85C42×C61—22no—
86C42×C61—22no—
87C42×C61—41no—
88C42×C61—41no—
89C22⋊Dic61—42no—
90#11758198—42no—
91#11758198—42no—
92#11758198—42no—
93#11758198—41no—
94#11758198—42no—
95#11758198—41no—
96C4⋊Dic61—22no—
97C4⋊Dic61—41no—
98C4⋊Dic61—41no—
99#11758196—42no—
100#11758196—42no—
101C2×#92418—24no—
102C2×#92418—42no—
103C2×#92418—42no—
104C23×C61—21no—
105C23×C61—21no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61#10412yes—
111C23×C61C2×C22⋊C411yescharacteristic
112C22×D61—41no—
113C22×D61—41no—
114C22×D61—41no—
115C22×D61—41no—
116C22×D61—41no—
117C22×D61—41no—
118C22×D61—41no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—44no—
134C2×C4×C61—44no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×C4×C61#9912yes—
138C2×C4×C61#13112yes—
139C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C61AGL(1,ℤ/8)12yes—
141C2×C4×C61#13711yescharacteristic
142C22⋊D61—84no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152D244—82no—
153C8×C61—82no—
154#92418—44no—
155#7698—611nomaximal, Sylow
156C22×C61—22no—
157C22×C61—22no—
158C22×C61—22no—
159C22×C61—41no—
160C22×C61—41no—
161C22×C61—41no—
162C22×C61—41no—
163C22×C61#40811yescharacteristic
164C22×C61C2×#9911yescharacteristic
165C22×C61#41311yescharacteristic
166C22×C61#41711yescharacteristic
167C22×C61#41811yescharacteristic
168C22×C61C2×#10411yescharacteristic
169C22×C61#42111yescharacteristic
170C2×D61—81no—
171C2×D61—81no—
172C2×D61—81no—
173C2×D61—81no—
174C2×D61—81no—
175C2×D61—81no—
176C2×D61—81no—
177C4×C61—22no—
178C4×C61—22no—
179C4×C61—22no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—41no—
183C4×C61—41no—
184Dic61—81no—
185Dic61—81no—
186Dic61—81no—
187#2854—611no—
188#1797D6111yescharacteristic, complemented
189#1749—611no—
190#1271—612no—
191#1270—612no—
192C2×C61—42no—
193C2×C61#146811yescharacteristic
194C2×C61#148311yescharacteristic
195C2×C61#149611yescharacteristic
196C2×C61#150711yescharacteristic
197C2×C61#151311yescharacteristic
198C2×C61#153211yescharacteristic
199C2×C61#155711yescharacteristic
200D61—161no—
201C2×#123—1221no—
202C2×#121—611no—
203C22×C42C2×D6111yescharacteristic
204C2×C4⋊C8—21no—
205C2×C22⋊C8—21no—
206C2×C22⋊C8—1221no—
207C2×#97—1221no—
208#390—611no—
209#390—611no—
210#389—611no—
211#389—611no—
212#388—1221no—
213#388—1221no—
214#386—1221no—
215#380—611no—
216#336—612no—
217#336C2×D6112yes—
218C61#769811yescharacteristic, complemented, Sylow
219C22×D4—611no—
220C22×D4—1221no—
221C23×C4C4×D6111yescharacteristic
222C23×C4D24411yescharacteristic
223C23×C4C22⋊D6111yescharacteristic
224C22×C8—22no—
225C22×C8—1222no—
226#123—2444no—
227#121—2442no—
228C2×C4⋊C4—611no—
229C2×C4⋊C4—611no—
230C2×C4⋊C4—611no—
231C2×C4⋊C4—1221no—
232C2×C4⋊C4—1221no—
233C2×C22⋊C4—1221no—
234C2×C22⋊C4—1221no—
235C2×C22⋊C4—1221no—
236C2×C22⋊C4—1221no—
237C2×C22⋊C4—1221no—
238C2×C22⋊C4—1221no—
239C2×C42—42no—
240C2×C42C4×D6111yescharacteristic
241C2×C42D24411yescharacteristic
242C2×C42C22⋊D6111yescharacteristic
243C2×C42C22×D6111yescharacteristic
244C4⋊C8—42no—
245C22⋊C8—42no—
246C22⋊C8—1224no—
247#97—2442no—
248#95—1221no—
249#95—1221no—
250#95—1221no—
251C24—1221no—
252C24#1175819811yescharacteristic
253C2×D4—1222no—
254C2×D4—2441no—
255C2×D4—2441no—
256C2×D4—2442no—
257C2×D4—2442no—
258C2×D4—2442no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—1221no—
270C22×C4—1221no—
271C22×C4—1221no—
272C22×C4#1175819811yescharacteristic
273C22×C4#1175819811yescharacteristic
274C22×C4#1175819811yescharacteristic
275C22×C4C2×C4×D6111yescharacteristic
276C22×C4C2×D24411yescharacteristic
277C22×C4C2×C22⋊D6111yescharacteristic
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—1224no—
282C2×C8—2442no—
283C2×C8—2442no—
284C4⋊C4—1222no—
285C4⋊C4—2441no—
286C4⋊C4—2441no—
287C4⋊C4—2442no—
288C4⋊C4—2442no—
289C22⋊C4—2442no—
290C22⋊C4—2442no—
291C22⋊C4—2442no—
292C22⋊C4—2441no—
293C22⋊C4—2442no—
294C22⋊C4—2441no—
295C22⋊C4—2442no—
296C42—22no—
297C42—22no—
298C42—41no—
299C42—41no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—2441no—
307C23—2441no—
308C23—2441no—
309C23—2441no—
310C23—2441no—
311C23—2441no—
312C23—2441no—
313C23#1451970312yes—
314C23C2×#1175819811yescharacteristic
315D4—4882no—
316D4—4884no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—2441no—
336C2×C4—2441no—
337C2×C4—2441no—
338C2×C4—2441no—
339C2×C4—2441no—
340C2×C4—2441no—
341C2×C4—2441no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4#1451969112yes—
345C2×C4#1451978312yes—
346C2×C4#1451980311yescharacteristic
347C2×C4#1451980512yes—
348C2×C4#1451980811yescharacteristic
349C8—82no—
350C8—2444no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—4881no—
359C22—4881no—
360C22—4881no—
361C22—4881no—
362C22—4881no—
363C22—4881no—
364C22—4881no—
365C22#1683972811yescharacteristic
366C22C2×#1451969111yescharacteristic
367C22#1683974411yescharacteristic
368C22#1683976111yescharacteristic
369C22#1683976811yescharacteristic
370C22C2×#1451970311yescharacteristic
371C22#1683978011yescharacteristic
372C4—21no—
373C4—21no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—4881no—
380C4—4881no—
381C4—4881no—
382C2—42no—
383C2—9761no—
384C2#2144416511yescharacteristic
385C2#2144423011yescharacteristic
386C2#2144428911yescharacteristic
387C2#2144433611yescharacteristic
388C2#2144436011yescharacteristic
389C2#2144444011yescharacteristic
390C2#2144453911yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1797C2
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519751C4
cosocleC2×C42×C61C23
hypercenter#1797D61
hypocenterC61#7698
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. #21445581C2semidirect
3. #21443641C2semidirect
4. #21443639C2semidirect
5. C61×#1797C2semidirect
188. #1797D61semidirect
218. C61#7698semidirect

Maximal subgroups

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

subgroupindex
#214541502
#214455812
#214436412
#214436392
C61×#17972
#769861