Finite Group Databasebeta

← #30600047

Subgroups — #30600047

Order 15616.

660 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 102 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#21462299C211yesmaximal, characteristic
2#21461722C212yesmaximal
3D61⋊#324C211yesmaximal, characteristic
4#21460693C212yesmaximal
5#398×D61C211yesmaximal, characteristic
6C2×#14519856—21no—
7D61⋊(C22⋊C8)C2212yes—
8#16840713—24no—
9#16840704—21no—
10#16840704C2212yes—
11#16840635—24no—
12#16840608—21no—
13#16840607—21no—
14C2×C4⋊C4×D61—21no—
15C2×C4⋊C4×D61C2211yescharacteristic
16C2×C4⋊C4×D61C2211yescharacteristic
17#16839696—21no—
18#16839695—21no—
19#16839694C411yescharacteristic, complemented
20#16839693C411yescharacteristic, complemented
21#16839692C411yescharacteristic, complemented
22#95×D61—21no—
23#95×D61C2211yescharacteristic
24C61×#398C411yescharacteristic, complemented
25C2×D61⋊C8—22no—
26#14519856—42no—
27#14519856—42no—
28C2×C4⋊#6923—22no—
29C2×C4⋊#6923—22no—
30C2×C4⋊#6923D414yes—
31C2×C22⋊#6923—22no—
32C2×C4×#6923—22no—
33#14519834—22no—
34C22⋊#92420—22no—
35(C2×#6923)⋊C4—22no—
36(C2×#6923)⋊C4—22no—
37C22×C4×D61—21no—
38C22×C4×D61—21no—
39C22×C4×D61D411yescharacteristic
40C22×C4×D61D411yescharacteristic
41C22×C4×D61C2311yescharacteristic
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61C2×C411yescharacteristic
44C2×C4⋊Dic61D411yescharacteristic
45C2×#11758196—21no—
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758196C2×C411yescharacteristic
49C2×#11758196D411yescharacteristic
50C4⋊C4×D61—22no—
51C4⋊C4×D61—22no—
52C4⋊C4×D61—22no—
53C4⋊C4×D61—22no—
54C4⋊C4×D61—42no—
55C4⋊C4×D61—42no—
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59#14519684—21no—
60#14519684C2×C411yescharacteristic
61C2×C4⋊C4×C61—21no—
62C2×C4⋊C4×C61C2×C411yescharacteristic
63C2×C4⋊C4×C61C2×C411yescharacteristic
64#95×C61—21no—
65#95×C61C2×C411yescharacteristic
66C23×D61C2×D411yescharacteristic
67C22×#6923—24no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22×C411yescharacteristic
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61C22⋊C411yescharacteristic
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C2×D411yescharacteristic
78C2×C4×D61—21no—
79C2×C4×D61—21no—
80C2×C4×D61—21no—
81C2×C4×D61—21no—
82C2×C4×D61—24no—
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—42no—
86C2×C4×D61—42no—
87C2×C4×D61—41no—
88C2×C4×D61—41no—
89C2×C4×D61—42no—
90C2×C4×D61—41no—
91C2×C4×D61—41no—
92C2×C4×D61—41no—
93C2×C4×D61—41no—
94C2×C4×D61—41no—
95C2×C4×D61—41no—
96C2×C4×D61D812yes—
97C2×C4×D61SD1612yes—
98C2×C4×D61C2×D412yes—
99C2×C4×D61#4112yes—
100C22⋊#6923—42no—
101C22⋊#6923—42no—
102C2×#92420—42no—
103C4⋊#6923—44no—
104C4⋊#6923—44no—
105C4⋊#6923—82no—
106C4×#6923—44no—
107D61⋊C8—44no—
108C4⋊C4×C61—22no—
109C4⋊C4×C61—22no—
110C4⋊C4×C61—42no—
111C4⋊Dic61—22no—
112C4⋊Dic61—22no—
113C4⋊Dic61—42no—
114#11758196—22no—
115#11758196—22no—
116#11758196—42no—
117#11758196—42no—
118#11758196—42no—
119C23×C61C2×C22⋊C411yescharacteristic
120C22×D61—22no—
121C22×D61—22no—
122C22×D61#12011yescharacteristic
123C22×D61#12311yescharacteristic
124C22×D61C2×D811yescharacteristic
125C22×D61C2×SD1611yescharacteristic
126C22×D61AGL(1,ℤ/8)11yescharacteristic
127C22×D61#13711yescharacteristic
128C2×#6923—44no—
129C2×#6923—44no—
130C2×#6923—44no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—22no—
134C2×C4×C61—21no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61#10212yes—
143C2×C4×C61C4×D412yes—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—22no—
147C2×Dic61—21no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61#10212yes—
156C2×Dic61#12112yes—
157C4×D61—22no—
158C4×D61—22no—
159C4×D61—24no—
160C4×D61—22no—
161C4×D61—22no—
162C4×D61—42no—
163C4×D61—44no—
164C4×D61—44no—
165C4×D61—42no—
166C4×D61—81no—
167C4×D61—81no—
168C4×D61—81no—
169C4×D61—81no—
170C4×D61—81no—
171#92420—82no—
172#9027—611nomaximal, Sylow
173C22×C61—21no—
174C22×C61—21no—
175C22×C61#38611yescharacteristic
176C22×C61C2×#10211yescharacteristic
177C22×C61#41711yescharacteristic
178C2×D61—21no—
179C2×D61—21no—
180C2×D61—22no—
181C2×D61—22no—
182C2×D61—22no—
183C2×D61—22no—
184C2×D61#44711yescharacteristic
185C2×D61#44811yescharacteristic
186C2×D61#45612yes—
187C2×D61#45812yes—
188C2×D61#48011yescharacteristic
189C2×D61#48111yescharacteristic
190#6923—84no—
191C4×C61—22no—
192C4×C61—22no—
193C4×C61—42no—
194C4×C61—41no—
195C4×C61—81no—
196Dic61—22no—
197Dic61—22no—
198Dic61—42no—
199Dic61—41no—
200Dic61—81no—
201C2×#398—611no—
202#2316—611no—
203#1823—612no—
204C2×#324—611no—
205#1294—612no—
206C2×C61—21no—
207C2×C61—21no—
208C2×C61#187411yescharacteristic
209C2×C61#188611yescharacteristic
210C2×C61#189011yescharacteristic
211D61—21no—
212D61—21no—
213D61#161012yes—
214D61#161112yes—
215C2×#117—1221no—
216C22×C4⋊C4—611no—
217C22×C4⋊C4—611no—
218C22×C4⋊C4—1221no—
219C2×C22⋊C8—612no—
220#398—611no—
221#398—611no—
222#398—611no—
223#398—1221no—
224#398—1221no—
225#398#692311yescharacteristic, complemented
226#389—1224no—
227#380—612no—
228#380—1221no—
229C2×#95—611no—
230C2×#95—1221no—
231#340—1224no—
232#324—1221no—
233#324—1221no—
234C61#902711yescharacteristic, complemented, Sylow
235C23×C4—611no—
236C23×C4—611no—
237C23×C4—611no—
238C23×C4—1221no—
239C23×C4—1221no—
240C22×C8—1222no—
241#117—2442no—
242#117—2442no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—611no—
245C2×C4⋊C4—611no—
246C2×C4⋊C4—611no—
247C2×C4⋊C4—611no—
248C2×C4⋊C4—614no—
249C2×C4⋊C4—1222no—
250C2×C4⋊C4—1222no—
251C2×C4⋊C4—1222no—
252C2×C4⋊C4—1222no—
253C2×C4⋊C4—1222no—
254C2×C4⋊C4—1221no—
255C2×C4⋊C4—1221no—
256C2×C4⋊C4—1221no—
257C2×C4⋊C4—1222no—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4—2442no—
260C2×C4⋊C4—2442no—
261C2×C4⋊C4C2×#692311yescharacteristic
262C2×C4⋊C4C2×#692311yescharacteristic
263C2×C22⋊C4—1222no—
264C2×C42—1222no—
265C22⋊C8—1222no—
266C22⋊C8—1222no—
267#95—21no—
268#95—611no—
269#95—1221no—
270#95—1221no—
271#95—1221no—
272#95—1222no—
273#95—1222no—
274#95—1221no—
275#95C2×#692311yescharacteristic
276C24—611no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—611no—
280C22×C4—611no—
281C22×C4—612no—
282C22×C4—612no—
283C22×C4—612no—
284C22×C4—612no—
285C22×C4—611no—
286C22×C4—1221no—
287C22×C4—1221no—
288C22×C4—1221no—
289C22×C4—1221no—
290C22×C4—1221no—
291C22×C4—1221no—
292C22×C4—1224no—
293C22×C4—1221no—
294C22×C4—1224no—
295C22×C4—1221no—
296C22×C4—2442no—
297C22×C4—2441no—
298C22×C4—2441no—
299C22×C4—2441no—
300C22×C4—2441no—
301C22×C4—2441no—
302C22×C4—2441no—
303C22×C4—2442no—
304C22×C4—2441no—
305C22×C4—2441no—
306C22×C4—2442no—
307C22×C4C22⋊#692311yescharacteristic
308C22×C4C22⋊#692311yescharacteristic
309C22×C4C22×#692311yescharacteristic
310C2×C8—2442no—
311C2×C8—2444no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—42no—
315C4⋊C4—1222no—
316C4⋊C4—1222no—
317C4⋊C4—1222no—
318C4⋊C4—1222no—
319C4⋊C4—4882no—
320C4⋊C4—2442no—
321C4⋊C4—2442no—
322C4⋊C4—2442no—
323C4⋊C4—2444no—
324C4⋊C4—2444no—
325C4⋊C4—2442no—
326C22⋊C4—2442no—
327C22⋊C4—2442no—
328C42—2444no—
329C23—611no—
330C23—611no—
331C23—611no—
332C23—611no—
333C23—611no—
334C23—611no—
335C23—1222no—
336C23—1222no—
337C23C2×C22⋊#692311yescharacteristic
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—612no—
350C2×C4—612no—
351C2×C4—1222no—
352C2×C4—1224no—
353C2×C4—1222no—
354C2×C4—1222no—
355C2×C4—1222no—
356C2×C4—1221no—
357C2×C4—1221no—
358C2×C4—1222no—
359C2×C4—1221no—
360C2×C4—4881no—
361C2×C4—4881no—
362C2×C4—4881no—
363C2×C4—4881no—
364C2×C4—4881no—
365C2×C4—2442no—
366C2×C4—2442no—
367C2×C4—2444no—
368C2×C4—2444no—
369C2×C4—2441no—
370C2×C4—2444no—
371C2×C4—2444no—
372C2×C4—2444no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4#1451983912yes—
380C2×C4D4×#692312yes—
381C8—4882no—
382C22—21no—
383C22—21no—
384C22—611no—
385C22—611no—
386C22—612no—
387C22—612no—
388C22—611no—
389C22—611no—
390C22—1222no—
391C22—1222no—
392C22—1222no—
393C22—1222no—
394C22—1221no—
395C22—1221no—
396C22#1684071011yescharacteristic
397C22C2×#1451983911yescharacteristic
398C22#1684075311yescharacteristic
399C4—22no—
400C4—22no—
401C4—42no—
402C4—41no—
403C4—81no—
404C4—1222no—
405C4—1222no—
406C4—4884no—
407C4—4881no—
408C4—2442no—
409C4—2441no—
410C2—21no—
411C2—21no—
412C2—612no—
413C2—612no—
414C2—1221no—
415C2—1221no—
416C2#2146180611yescharacteristic
417C2#2146182511yescharacteristic
418C2#2146182911yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#398C4
socleC22×C61#386
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#398#6923
hypocenterC61#9027
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
19. #16839694C4semidirect
20. #16839693C4semidirect
21. #16839692C4semidirect
24. C61×#398C4semidirect
225. #398#6923semidirect
234. C61#9027semidirect

Maximal subgroups

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

subgroupindex
#214622992
#214617222
D61⋊#3242
#214606932
#398×D612
#902761