Finite Group Databasebeta

← #30601253

Subgroups — #30601253

Order 15616.

544 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 82 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#21462670C211yesmaximal, characteristic
2#21461718C212yesmaximal, complemented
3#21461075C211yesmaximal, characteristic
4#21460720C212yesmaximal, complemented
5#21450124C211yesmaximal, characteristic
6#16840793—24no—
7#16840782—21no—
8#16840732C2212yes—
9#16840732C2212yes—
10C2×C22⋊#92420—21no—
11#16840642—24no—
12#16840607—21no—
13#16840605—21no—
14C2×#14519758C411yescharacteristic
15C2×D4×Dic61C2211yescharacteristic
16C22×#11758196C2211yescharacteristic
17#16839695—21no—
18#16839690—21no—
19#16839657—21no—
20#16839625C411yescharacteristic
21#16839623C411yescharacteristic
22#16839620—21no—
23#16839620C2211yescharacteristic
24C2×#123×C61C411yescharacteristic
25C2×#11758217—22no—
26C2×#11758217D414yes—
27C22×#92420—22no—
28#14519844—22no—
29C22⋊#92420—22no—
30C22⋊#92420—22no—
31C22⋊#92420—22no—
32C22⋊#92420—22no—
33C22⋊#92420—22no—
34#14519832—22no—
35C23×Dic61C2×C411yescharacteristic
36C23×Dic61D411yescharacteristic
37C23×Dic61C2311yescharacteristic
38#14519758—42no—
39D4×Dic61—22no—
40D4×Dic61—22no—
41C2×C4⋊Dic61D411yescharacteristic
42C2×#11758196—21no—
43C2×#11758196—21no—
44C2×#11758196—22no—
45C2×#11758196—21no—
46C2×#11758196—21no—
47C2×#11758196—22no—
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61C2×C411yescharacteristic
50C2×C22⋊Dic61C2×C411yescharacteristic
51C2×C22⋊Dic61D411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58C22×D4×C61C2×C411yescharacteristic
59C23×C4×C61C2×C411yescharacteristic
60#123×C61—42no—
61C2×C4⋊C4×C61—21no—
62C2×C22⋊C4×C61—21no—
63C2×C22⋊C4×C61C2×C411yescharacteristic
64C24×C61C22⋊C411yescharacteristic
65C24×C61C22×C411yescharacteristic
66C2×D4×C61—22no—
67C2×D4×C61—22no—
68C2×D4×C61—41no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—22no—
72C22×C4×C61—41no—
73C22×C4×C61—41no—
74C22×C4×C61C22⋊C411yescharacteristic
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—22no—
80C22×Dic61—42no—
81C22×Dic61—42no—
82C22×Dic61—41no—
83C22×Dic61—41no—
84C22×Dic61C22⋊C411yescharacteristic
85C22×Dic61C22⋊C411yescharacteristic
86C22×Dic61C2×D412yes—
87C22×Dic61C2×D411yescharacteristic
88C22×Dic61C2×D411yescharacteristic
89C22×Dic61#4112yes—
90#11758217—44no—
91#11758217—44no—
92C2×#92420—24no—
93C2×#92420—42no—
94C2×#92420—82no—
95C4⋊C4×C61—42no—
96C22⋊C4×C61—42no—
97C22⋊C4×C61—41no—
98C22⋊Dic61—22no—
99C22⋊Dic61—22no—
100C22⋊Dic61—42no—
101C22⋊Dic61—41no—
102C4⋊Dic61—22no—
103#11758196—42no—
104#11758196—42no—
105#11758196—41no—
106#11758196—42no—
107#11758196—41no—
108C4×Dic61—21no—
109C4×Dic61—21no—
110C23×C61—21no—
111C23×C61—21no—
112C23×C61—21no—
113C23×C61—21no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61#10012yes—
119C23×C61C2×C22⋊C411yescharacteristic
120C23×C61C4×D412yes—
121D4×C61—44no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—82no—
132C2×Dic61—22no—
133C2×Dic61—24no—
134C2×Dic61—21no—
135C2×Dic61—21no—
136C2×Dic61—41no—
137C2×Dic61—42no—
138C2×Dic61—42no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—81no—
148C2×Dic61—82no—
149C2×Dic61#12011yescharacteristic
150C2×Dic61#12112yes—
151C2×Dic61#12311yescharacteristic
152#92420—44no—
153#92420—82no—
154#9507—611nomaximal, Sylow
155C22×C61—21no—
156C22×C61—21no—
157C22×C61—22no—
158C22×C61—22no—
159C22×C61—22no—
160C22×C61—22no—
161C22×C61—42no—
162C22×C61—42no—
163C22×C61—42no—
164C22×C61—41no—
165C22×C61—41no—
166C22×C61—81no—
167C22×C61#38611yescharacteristic
168C22×C61C2×#10011yescharacteristic
169C22×C61#41311yescharacteristic
170C4×C61—41no—
171C4×C61—81no—
172C4×C61—81no—
173Dic61—22no—
174Dic61—21no—
175Dic61—41no—
176Dic61—41no—
177Dic61—81no—
178Dic61—81no—
179Dic61#45812yes—
180#2973—611no—
181#2486—611no—
182#1821—612no—
183#1465—611no—
184#1305—612no—
185C2×C61—21no—
186C2×C61—21no—
187C2×C61—42no—
188C2×C61—42no—
189C2×C61#189011yescharacteristic
190C2×C61#190011yescharacteristic
191C2×C61#191511yescharacteristic
192C2×#123—611no—
193C2×#123#692311yescharacteristic
194C2×C4×D4—611no—
195C22×C4⋊C4—611no—
196#435—1224no—
197#432—1221no—
198#407—612no—
199#407—612no—
200C2×C22⋊C8—1221no—
201#398—1221no—
202#397—1221no—
203#389—1221no—
204#381—611no—
205#381—611no—
206#380—611no—
207#380—1221no—
208#343—1224no—
209#324—1221no—
210#323—1221no—
211C61#950711yescharacteristic, complemented, Sylow
212C22×D4C2×#692311yescharacteristic
213C23×C4—611no—
214C23×C4—611no—
215C23×C4—611no—
216C23×C4C2×#692311yescharacteristic
217C2×M16—614no—
218C2×M16—1222no—
219C22×C8—1222no—
220#123—42no—
221#123—2442no—
222C4×D4—1222no—
223C4×D4—1222no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—611no—
226C2×C4⋊C4—1222no—
227C2×C4⋊C4—1221no—
228C2×C4⋊C4—1221no—
229C2×C4⋊C4—1222no—
230C2×C4⋊C4—1221no—
231C2×C4⋊C4—1221no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—611no—
234C2×C22⋊C4—611no—
235C2×C22⋊C4—611no—
236C2×C22⋊C4—1221no—
237C2×C22⋊C4C2×#692311yescharacteristic
238C2×C42—611no—
239C4⋊C8—1222no—
240C22⋊C8—1222no—
241C22⋊C8—1222no—
242C22⋊C8—1222no—
243C22⋊C8—1222no—
244C22⋊C8—1222no—
245#97—1222no—
246#95—1221no—
247#95—1221no—
248#95—1221no—
249#95—1221no—
250#95—1221no—
251C24C22⋊#692311yescharacteristic
252C24C22×#692311yescharacteristic
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—41no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—611no—
262C22×C4—612no—
263C22×C4—611no—
264C22×C4—612no—
265C22×C4—611no—
266C22×C4—611no—
267C22×C4—1221no—
268C22×C4—1221no—
269C22×C4—1222no—
270C22×C4—1221no—
271C22×C4—1221no—
272C22×C4—2442no—
273C22×C4—2442no—
274C22×C4—2441no—
275C22×C4—2441no—
276C22×C4C22⋊#692311yescharacteristic
277M16—2444no—
278M16—2444no—
279C2×C8—1224no—
280C2×C8—4882no—
281C2×C8—2442no—
282C4⋊C4—42no—
283C4⋊C4—1222no—
284C4⋊C4—2442no—
285C4⋊C4—2441no—
286C4⋊C4—2442no—
287C4⋊C4—2441no—
288C4⋊C4—2442no—
289C22⋊C4—42no—
290C22⋊C4—41no—
291C22⋊C4—1222no—
292C22⋊C4—1222no—
293C22⋊C4—2442no—
294C22⋊C4—2441no—
295C42—1221no—
296C42—1221no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23#1451983712yes—
306C23C2×C22⋊#692311yescharacteristic
307C23D4×#692312yes—
308D4—44no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—82no—
319C2×C4—611no—
320C2×C4—612no—
321C2×C4—611no—
322C2×C4—1222no—
323C2×C4—1224no—
324C2×C4—1221no—
325C2×C4—1221no—
326C2×C4—4881no—
327C2×C4—4882no—
328C2×C4—2442no—
329C2×C4—2441no—
330C2×C4—2442no—
331C2×C4—2441no—
332C2×C4—2441no—
333C2×C4—2441no—
334C2×C4—2441no—
335C2×C4—2441no—
336C2×C4—2441no—
337C2×C4—2441no—
338C2×C4—2441no—
339C8—4882no—
340C8—2444no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—41no—
351C22—41no—
352C22—81no—
353C22#1684071011yescharacteristic
354C22C2×#1451983711yescharacteristic
355C22#1684074911yescharacteristic
356C4—41no—
357C4—81no—
358C4—81no—
359C4—612no—
360C4—1222no—
361C4—1221no—
362C4—4881no—
363C4—4881no—
364C4—2441no—
365C4—2441no—
366C2—21no—
367C2—21no—
368C2—42no—
369C2—42no—
370C2#2146182911yescharacteristic
371C2#2146184411yescharacteristic
372C2#2146187111yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC2×#123×C61C4
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC23×Dic61C23
hypercenterC2×#123#6923
hypocenterC61#9507
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. #21461718C2semidirect
4. #21460720C2semidirect
211. C61#9507semidirect

Maximal subgroups

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

subgroupindex
#214626702
#214617182
#214610752
#214607202
#214501242
#950761