Finite Group Databasebeta

← #30600425

Subgroups — #30600425

Order 15616.

586 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 100 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×#16840707C211yesmaximal, characteristic
2#21461711C211yesmaximal, characteristic
3C2×#16840616C212yesmaximal
4#21460692C212yesmaximal
5#21454160C211yesmaximal, characteristic
6C2×#14519834C2212yes—
7#16840707—24no—
8C2×(C2×#6923)⋊C4—21no—
9C2×(C2×#6923)⋊C4—21no—
10C2×(C2×#6923)⋊C4C2212yes—
11#16840628—21no—
12#16840628—21no—
13#16840616—24no—
14C2×#14519753C2211yescharacteristic
15C2×C42×D61C2211yescharacteristic
16#16839669C411yescharacteristic, complemented
17#16839661C411yescharacteristic, complemented
18#16839660C411yescharacteristic, complemented
19#16839645—21no—
20#16839644—21no—
21#16839636—21no—
22#16839624—21no—
23#16839619C2211yescharacteristic
24C61×#391C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—22no—
27#14519834—22no—
28#14519834—22no—
29(C2×#6923)⋊C4—24no—
30(C2×#6923)⋊C4—24no—
31(C2×#6923)⋊C4—22no—
32(C2×#6923)⋊C4D414yes—
33C22×C4×D61D411yescharacteristic
34C22×C4×D61D411yescharacteristic
35C22×C4×D61C2311yescharacteristic
36#14519753—22no—
37#14519753—22no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61C2×C411yescharacteristic
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×#11758196—21no—
42C2×#11758196—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×C4×Dic61C2×C411yescharacteristic
46C2×C4×Dic61D411yescharacteristic
47C2×C4×Dic61D411yescharacteristic
48C42×D61—22no—
49C42×D61—22no—
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55C22×Q8×C61C2×C411yescharacteristic
56C2×C4⋊C4×C61C2×C411yescharacteristic
57C2×C42×C61C2×C411yescharacteristic
58#95×C61—21no—
59#95×C61—21no—
60C23×D61C2×D411yescharacteristic
61C22×#6923—24no—
62C22×#6923—42no—
63C22×#6923—42no—
64C22×#6923—42no—
65C22×#6923—42no—
66C22×#6923—42no—
67C22×#6923—42no—
68C2×Q8×C61—22no—
69C2×Q8×C61—41no—
70C2×Q8×C61—41no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22×C411yescharacteristic
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61C22⋊C411yescharacteristic
79C22×Dic61C22⋊C411yescharacteristic
80C22×Dic61C2×D411yescharacteristic
81C2×C4×D61—21no—
82C2×C4×D61—21no—
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—22no—
86C2×C4×D61—22no—
87C2×C4×D61—41no—
88C2×C4×D61—41no—
89C2×C4×D61—41no—
90C2×C4×D61—41no—
91C2×C4×D61C2×D412yes—
92C2×C4×D61#4112yes—
93C2×#92420—42no—
94C2×#92420—42no—
95C2×#92420—42no—
96C4⋊C4×C61—41no—
97C42×C61—22no—
98#11758198—41no—
99#11758198—41no—
100#11758198—42no—
101C4⋊Dic61—22no—
102C4⋊Dic61—42no—
103C4⋊Dic61—41no—
104#11758196—42no—
105#11758196—42no—
106C4×Dic61—22no—
107C4×Dic61—22no—
108C4×Dic61—22no—
109C23×C61C2×C22⋊C411yescharacteristic
110C22×D61—21no—
111C22×D61—21no—
112C22×D61—21no—
113C22×D61—21no—
114C22×D61#12011yescharacteristic
115C22×D61#12311yescharacteristic
116C22×D61#13512yes—
117C22×D61#13712yes—
118C2×#6923—44no—
119C2×#6923—44no—
120C2×#6923—44no—
121C2×#6923—82no—
122C2×#6923—82no—
123C2×#6923—82no—
124C2×#6923—82no—
125Q8×C61—82no—
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—21no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61C4×D412yes—
141C2×Dic61—21no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61#12112yes—
156C4×D61—42no—
157C4×D61—42no—
158C4×D61—42no—
159C4×D61—42no—
160C4×D61—42no—
161C4×D61—42no—
162#92420—82no—
163#9153—611nomaximal, Sylow
164C22×C61#38611yescharacteristic
165C22×C61#41011yescharacteristic
166C22×C61#41011yescharacteristic
167C22×C61#41611yescharacteristic
168C22×C61#41911yescharacteristic
169C22×C61#42111yescharacteristic
170C22×C61#42111yescharacteristic
171C2×D61—21no—
172C2×D61—21no—
173C2×D61—21no—
174C2×D61—21no—
175C2×D61—22no—
176C2×D61—22no—
177C2×D61—22no—
178C2×D61—22no—
179C2×D61#45212yes—
180C2×D61#48512yes—
181#6923—84no—
182C4×C61—22no—
183C4×C61—41no—
184C4×C61—41no—
185C4×C61—81no—
186C4×C61—81no—
187Dic61—22no—
188Dic61—41no—
189Dic61—41no—
190Dic61—81no—
191Dic61—81no—
192#2855—611no—
193C2×#383—611no—
194#1817—611no—
195C2×#328—612no—
196#1294—612no—
197C2×C61#151511yescharacteristic
198C2×C61#151511yescharacteristic
199C2×C61#152111yescharacteristic
200C2×C61#152111yescharacteristic
201C2×C61#187311yescharacteristic
202C2×C61#189711yescharacteristic
203C2×C61#189911yescharacteristic
204D61—22no—
205D61—22no—
206C2×C22⋊Q8—611no—
207C22×C42—611no—
208C2×C22⋊C8—612no—
209#391—611no—
210#391#692311yescharacteristic, complemented
211#389—611no—
212#389—611no—
213#387—1221no—
214#387—1221no—
215#385—1221no—
216#383—1224no—
217#381—1221no—
218#380—611no—
219C2×#95—612no—
220C2×#95—1221no—
221C2×#95—1221no—
222#336—1221no—
223#336—1221no—
224#328—1224no—
225C61#915311yescharacteristic, complemented, Sylow
226C22×Q8C2×#692311yescharacteristic
227C23×C4—611no—
228C23×C4—611no—
229C23×C4—611no—
230C23×C4—1222no—
231C22×C8—1222no—
232C22⋊Q8—1222no—
233C22⋊Q8—1222no—
234C2×C4⋊C4—611no—
235C2×C4⋊C4—611no—
236C2×C4⋊C4—1221no—
237C2×C4⋊C4—1221no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4C2×#692311yescharacteristic
240C2×C22⋊C4—1221no—
241C2×C22⋊C4—1221no—
242C2×C42—611no—
243C2×C42—611no—
244C2×C42—611no—
245C2×C42—1222no—
246C2×C42—1222no—
247C2×C42C2×#692311yescharacteristic
248C22⋊C8—1222no—
249C22⋊C8—1222no—
250#95—21no—
251#95—21no—
252#95—614no—
253#95—1221no—
254#95—1221no—
255#95—1221no—
256#95—1221no—
257#95—1222no—
258#95—1224no—
259#95—1224no—
260#95—1221no—
261C24—611no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—41no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—611no—
268C22×C4—612no—
269C22×C4—612no—
270C22×C4—611no—
271C22×C4—611no—
272C22×C4—1221no—
273C22×C4—1221no—
274C22×C4—1221no—
275C22×C4—1221no—
276C22×C4—1222no—
277C22×C4—1222no—
278C22×C4—1224no—
279C22×C4—1221no—
280C22×C4—1221no—
281C22×C4—2441no—
282C22×C4—2441no—
283C22×C4—2441no—
284C22×C4—2441no—
285C22×C4—2442no—
286C22×C4—2442no—
287C22×C4—2442no—
288C22×C4—2442no—
289C22×C4—2442no—
290C22×C4—2442no—
291C22×C4C22⋊#692311yescharacteristic
292C22×C4C22⋊#692311yescharacteristic
293C22×C4C22×#692311yescharacteristic
294C2×C8—2442no—
295C2×C8—2442no—
296C2×C8—2442no—
297C4⋊C4—41no—
298C4⋊C4—1222no—
299C4⋊C4—2441no—
300C4⋊C4—2442no—
301C4⋊C4—2442no—
302C4⋊C4—2442no—
303C22⋊C4—2442no—
304C22⋊C4—2441no—
305C22⋊C4—2441no—
306C42—22no—
307C42—1222no—
308C42—1222no—
309C42—1222no—
310C23—611no—
311C23—611no—
312C23—612no—
313C23—612no—
314C23—1221no—
315C23—1221no—
316C23—1221no—
317C23—1221no—
318C23C2×C22⋊#692311yescharacteristic
319Q8—82no—
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—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—612no—
335C2×C4—1221no—
336C2×C4—1221no—
337C2×C4—1221no—
338C2×C4—1221no—
339C2×C4—1221no—
340C2×C4—1221no—
341C2×C4—1221no—
342C2×C4—1221no—
343C2×C4—4882no—
344C2×C4—4882no—
345C2×C4—4882no—
346C2×C4—4882no—
347C2×C4—2442no—
348C2×C4—2442no—
349C2×C4—2442no—
350C2×C4—2442no—
351C2×C4—2442no—
352C2×C4—2442no—
353C2×C4—2444no—
354C2×C4—2444no—
355C2×C4—2444no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4—2441no—
359C2×C4—2441no—
360C2×C4—2441no—
361C2×C4—2441no—
362C2×C4D4×#692312yes—
363C8—4882no—
364C22—612no—
365C22—612no—
366C22—1222no—
367C22—1222no—
368C22—1221no—
369C22—1221no—
370C22—1221no—
371C22—1221no—
372C22—1222no—
373C22—1222no—
374C22#1684071011yescharacteristic
375C22#1684074111yescharacteristic
376C22#1684074211yescharacteristic
377C22#1684075211yescharacteristic
378C22#1684075511yescharacteristic
379C22#1684075911yescharacteristic
380C22#1684076111yescharacteristic
381C4—22no—
382C4—41no—
383C4—41no—
384C4—81no—
385C4—81no—
386C4—1222no—
387C4—4884no—
388C4—4881no—
389C4—4881no—
390C4—2441no—
391C4—2441no—
392C2—1222no—
393C2—1222no—
394C2#2146124111yescharacteristic
395C2#2146124311yescharacteristic
396C2#2146126011yescharacteristic
397C2#2146126311yescharacteristic
398C2#2146180511yescharacteristic
399C2#2146183811yescharacteristic
400C2#2146184211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#391#6923
hypocenterC61#9153
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
16. #16839669C4semidirect
17. #16839661C4semidirect
18. #16839660C4semidirect
24. C61×#391C4semidirect
210. #391#6923semidirect
225. C61#9153semidirect

Maximal subgroups

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

subgroupindex
C2×#168407072
#214617112
C2×#168406162
#214606922
#214541602
#915361