Finite Group Databasebeta

← #30600384

Subgroups — #30600384

Order 15616.

588 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 122 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×#16840779C211yesmaximal, characteristic
2#21461675C211yesmaximal, characteristic
3C2×#16840621C212yesmaximal, complemented
4#21460697C212yesmaximal, complemented
5#21454169C211yesmaximal, characteristic
6#16840779—24no—
7C2×#14519844C2212yes—
8C2×#14519844C2212yes—
9C2×C22⋊#92420—21no—
10C2×C22⋊#92420—21no—
11C2×C4×#92420—21no—
12#16840628—21no—
13#16840621—24no—
14C2×#14519776C2211yescharacteristic
15C2×#14519754—21no—
16C22×C4×Dic61C2211yescharacteristic
17#16839666—21no—
18#16839661C411yescharacteristic
19#16839659—21no—
20#16839658C411yescharacteristic
21#16839631C2211yescharacteristic
22#16839622C411yescharacteristic
23#16839620—21no—
24C61×#380C411yescharacteristic
25C22×#92420—22no—
26C22×#92420—22no—
27#14519844—22no—
28#14519844—22no—
29#14519844—22no—
30#14519844D414yes—
31C22⋊#92420—24no—
32C22⋊#92420—42no—
33C4×#92420—24no—
34C22×Dic122—21no—
35C23×Dic61D411yescharacteristic
36C23×Dic61D411yescharacteristic
37#14519776—22no—
38#14519776—22no—
39#14519754—44no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×#11758196—21no—
44C2×#11758196—21no—
45C2×#11758196—21no—
46C2×#11758196—21no—
47C2×C22⋊Dic61—21no—
48C2×C4×Dic61—22no—
49C2×C4×Dic61—22no—
50C2×C4×Dic61C2×C411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53C2×C4×Dic61C2311yescharacteristic
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57C23×C4×C61C2×C411yescharacteristic
58C2×C4⋊C4×C61C2×C411yescharacteristic
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C22⋊C4×C61—21no—
61#95×C61—21no—
62C24×C61C22⋊C411yescharacteristic
63C22×C4×C61—22no—
64C22×C4×C61—22no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—41no—
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22×C411yescharacteristic
70C22×Dic61—22no—
71C22×Dic61—22no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—22no—
75C22×Dic61—22no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61C22⋊C411yescharacteristic
79C22×Dic61C22⋊C411yescharacteristic
80C22×Dic61SD1612yes—
81C22×Dic61Q1612yes—
82C22×Dic61C2×D411yescharacteristic
83C22×Dic61C2×D411yescharacteristic
84C2×Dic122—41no—
85C2×Dic122—41no—
86C2×Dic122—42no—
87C2×Dic122—42no—
88C2×#92420—24no—
89C2×#92420—42no—
90C2×#92420—42no—
91C2×#92420—42no—
92C2×#92420—42no—
93C2×#92420—42no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—41no—
96C22⋊C4×C61—42no—
97C22⋊Dic61—42no—
98C4⋊Dic61—22no—
99C4⋊Dic61—42no—
100C4⋊Dic61—41no—
101#11758196—42no—
102#11758196—42no—
103#11758196—42no—
104#11758196—42no—
105C4×Dic61—22no—
106C4×Dic61—21no—
107C4×Dic61—22no—
108C4×Dic61—21no—
109C4×Dic61C2×D412yes—
110C4×Dic61#4112yes—
111C23×C61—21no—
112C23×C61—21no—
113C23×C61—21no—
114C23×C61—21no—
115C23×C61—21no—
116C23×C61—21no—
117C23×C61#10312yes—
118C23×C61C2×C22⋊C411yescharacteristic
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—44no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61#10112yes—
131C2×C4×C61C4×D412yes—
132C2×Dic61—21no—
133C2×Dic61—21no—
134C2×Dic61—24no—
135C2×Dic61—24no—
136C2×Dic61—21no—
137C2×Dic61—21no—
138C2×Dic61—21no—
139C2×Dic61—42no—
140C2×Dic61—42no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61#10312yes—
148C2×Dic61#12011yescharacteristic
149C2×Dic61#12112yes—
150C2×Dic61#12311yescharacteristic
151C2×Dic61C2×SD1611yescharacteristic
152C2×Dic61C2×Q1611yescharacteristic
153C2×Dic61#13512yes—
154C2×Dic61#13512yes—
155C2×Dic61AGL(1,ℤ/8)12yes—
156Dic122—82no—
157Dic122—82no—
158#92420—44no—
159#92420—82no—
160#9132—611nomaximal, Sylow
161C22×C61—22no—
162C22×C61—22no—
163C22×C61—22no—
164C22×C61—41no—
165C22×C61—41no—
166C22×C61—41no—
167C22×C61—41no—
168C22×C61#38611yescharacteristic
169C22×C61C2×#10111yescharacteristic
170C22×C61#41311yescharacteristic
171C22×C61C2×#10311yescharacteristic
172C22×C61#41611yescharacteristic
173C22×C61#41611yescharacteristic
174C22×C61#41811yescharacteristic
175C4×C61—22no—
176C4×C61—22no—
177C4×C61—81no—
178C4×C61—81no—
179Dic61—22no—
180Dic61—41no—
181Dic61—81no—
182Dic61—81no—
183Dic61#44912yes—
184Dic61#46512yes—
185Dic61#46712yes—
186Dic61#48212yes—
187C2×#431—611no—
188#2857—611no—
189#1797—611no—
190C2×#332—612no—
191#1296—612no—
192C2×C61—42no—
193C2×C61#154211yescharacteristic
194C2×C61#154411yescharacteristic
195C2×C61#154911yescharacteristic
196C2×C61#155311yescharacteristic
197C2×C61#187511yescharacteristic
198C2×C61#189211yescharacteristic
199C2×C61#190111yescharacteristic
200C2×C4⋊Q8—611no—
201C2×C22⋊Q8—1221no—
202C22×C42—611no—
203#431—1224no—
204C2×C4⋊C8—612no—
205C2×C4⋊C8—612no—
206C2×C22⋊C8—1221no—
207C2×C22⋊C8—1221no—
208C2×C4×C8—1221no—
209#391—1221no—
210#389—611no—
211#389—611no—
212#389—1221no—
213#384—611no—
214#380—611no—
215#380—1221no—
216#380#692311yescharacteristic
217#336—1221no—
218#332—1224no—
219C61#913211yescharacteristic, complemented, Sylow
220C22×Q8—1221no—
221C23×C4—611no—
222C23×C4—611no—
223C23×C4C2×#692311yescharacteristic
224C22×C8—1222no—
225C22×C8—1222no—
226C4⋊Q8—1222no—
227C4⋊Q8—1222no—
228C22⋊Q8—2444no—
229C2×C4⋊C4—611no—
230C2×C4⋊C4—611no—
231C2×C4⋊C4—1221no—
232C2×C4⋊C4—1221no—
233C2×C4⋊C4—1221no—
234C2×C4⋊C4—1221no—
235C2×C4⋊C4—1221no—
236C2×C4⋊C4C2×#692311yescharacteristic
237C2×C4⋊C4C2×#692311yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—1221no—
240C2×C42—611no—
241C2×C42—611no—
242C2×C42—611no—
243C2×C42—611no—
244C2×C42—1222no—
245C2×C42—1222no—
246C4⋊C8—614no—
247C4⋊C8—1222no—
248C4⋊C8—1222no—
249C4⋊C8—1222no—
250C22⋊C8—1224no—
251C22⋊C8—2442no—
252C4×C8—1224no—
253#95—21no—
254#95—1221no—
255#95—1221no—
256#95—1221no—
257C24C22⋊#692311yescharacteristic
258C2×Q8—2442no—
259C2×Q8—2441no—
260C2×Q8—2441no—
261C2×Q8—2442no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—41no—
267C22×C4—611no—
268C22×C4—611no—
269C22×C4—612no—
270C22×C4—612no—
271C22×C4—611no—
272C22×C4—611no—
273C22×C4—1222no—
274C22×C4—1222no—
275C22×C4—1221no—
276C22×C4—1221no—
277C22×C4—1222no—
278C22×C4—1222no—
279C22×C4—1221no—
280C22×C4—1221no—
281C22×C4C22⋊#692311yescharacteristic
282C22×C4C22×#692311yescharacteristic
283C2×C8—1224no—
284C2×C8—2442no—
285C2×C8—2442no—
286C2×C8—2442no—
287C2×C8—2442no—
288C2×C8—2442no—
289C4⋊C4—22no—
290C4⋊C4—41no—
291C4⋊C4—1222no—
292C4⋊C4—2441no—
293C4⋊C4—2442no—
294C4⋊C4—2442no—
295C4⋊C4—2442no—
296C4⋊C4—2442no—
297C4⋊C4—2442no—
298C22⋊C4—42no—
299C22⋊C4—2442no—
300C42—612no—
301C42—612no—
302C42—1221no—
303C42—1222no—
304C42—1221no—
305C42—1222no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23#1451984012yes—
313C23C2×C22⋊#692311yescharacteristic
314Q8—4882no—
315Q8—4882no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—44no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—611no—
328C2×C4—612no—
329C2×C4—611no—
330C2×C4—611no—
331C2×C4—611no—
332C2×C4—612no—
333C2×C4—612no—
334C2×C4—612no—
335C2×C4—612no—
336C2×C4—1224no—
337C2×C4—1224no—
338C2×C4—1221no—
339C2×C4—1221no—
340C2×C4—1221no—
341C2×C4—1221no—
342C2×C4—1221no—
343C2×C4—2442no—
344C2×C4—2442no—
345C2×C4—2441no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4—2441no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4#1451983812yes—
352C2×C4D4×#692312yes—
353C8—4882no—
354C8—2444no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22#1684071011yescharacteristic
363C22C2×#1451983811yescharacteristic
364C22#1684074911yescharacteristic
365C22C2×#1451984011yescharacteristic
366C22#1684075211yescharacteristic
367C22#1684075211yescharacteristic
368C22#1684075411yescharacteristic
369C4—22no—
370C4—22no—
371C4—81no—
372C4—81no—
373C4—612no—
374C4—612no—
375C4—612no—
376C4—612no—
377C4—1222no—
378C4—4881no—
379C4—4881no—
380C4—2441no—
381C2—42no—
382C2#2146132111yescharacteristic
383C2#2146132411yescharacteristic
384C2#2146133411yescharacteristic
385C2#2146134311yescharacteristic
386C2#2146180711yescharacteristic
387C2#2146183211yescharacteristic
388C2#2146184611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#380C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#380#6923
hypocenterC61#9132
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. C2×#16840621C2semidirect
4. #21460697C2semidirect
219. C61#9132semidirect

Maximal subgroups

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

subgroupindex
C2×#168407792
#214616752
C2×#168406212
#214606972
#214541692
#913261