Finite Group Databasebeta

← #30595752

Subgroups — #30595752

Order 15616.

630 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 106 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#21461483C211yesmaximal, characteristic, complemented
2#21461480C211yesmaximal, characteristic, complemented
3#21460638C212yesmaximal, complemented
4#21460636C212yesmaximal, complemented
5#21454152C211yesmaximal, characteristic
6C22×#11758213—21no—
7C22×D61⋊C8—21no—
8#16840628—21no—
9#16840628—21no—
10#16840628C2212yes—
11#16840628C2212yes—
12C2×#14519762—21no—
13C2×#14519752C411yescharacteristic
14C2×C42×D61C2211yescharacteristic
15#16839664C411yescharacteristic
16#16839663C2211yescharacteristic
17#16839658C2211yescharacteristic
18#16839652—21no—
19#16839651—21no—
20#16839640—21no—
21#16839619C411yescharacteristic
22C61×#389C411yescharacteristic
23C2×#11758213—24no—
24C2×#11758213—42no—
25C2×#11758213—42no—
26C2×D61⋊C8—24no—
27C2×D61⋊C8—42no—
28C2×D61⋊C8—42no—
29C22×#92420—22no—
30C22×#92420—22no—
31C22×D244—21no—
32C22×D244C2×C411yescharacteristic
33C22×C4×D61C2×C411yescharacteristic
34C22×C4×D61D411yescharacteristic
35C22×C4×D61D411yescharacteristic
36#14519762—42no—
37#14519762—42no—
38#14519752—42no—
39C2×C4⋊Dic61C2×C411yescharacteristic
40C2×#11758196—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×C4×Dic61D411yescharacteristic
48C2×C4×Dic61D411yescharacteristic
49C2×C4×Dic61C2311yescharacteristic
50C42×D61—42no—
51#14519684—21no—
52#14519684—21no—
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61C2×C411yescharacteristic
55C2×C4⋊C4×C61C2×C411yescharacteristic
56C2×C42×C61C2×C411yescharacteristic
57#95×C61—21no—
58C23×D61—21no—
59C23×D61C22⋊C411yescharacteristic
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22×C411yescharacteristic
65C22×Dic61—21no—
66C22×Dic61C22⋊C411yescharacteristic
67C22×Dic61C2×D411yescharacteristic
68C22×Dic61C2×D411yescharacteristic
69C2×D244—22no—
70C2×D244—41no—
71C2×D244—41no—
72C2×D244—42no—
73C2×D244—42no—
74C2×D244—42no—
75C2×C4×D61—22no—
76C2×C4×D61—22no—
77C2×C4×D61—22no—
78C2×C4×D61—22no—
79C2×C4×D61—21no—
80C2×C4×D61—21no—
81C2×C4×D61—22no—
82C2×C4×D61—22no—
83C2×C4×D61—41no—
84C2×C4×D61D812yes—
85C2×C4×D61SD1612yes—
86C2×#92420—24no—
87C2×#92420—24no—
88C2×#92420—42no—
89C2×#92420—42no—
90C2×#92420—42no—
91C2×#92420—42no—
92#11758213—48no—
93D61⋊C8—48no—
94C4⋊C4×C61—41no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—42no—
97C42×C61—41no—
98#11758198—42no—
99#11758198—42no—
100#11758198—42no—
101#11758198—42no—
102#11758198—41no—
103#11758198—41no—
104#11758198—41no—
105#11758198—41no—
106C4⋊Dic61—22no—
107#11758196—41no—
108#11758196—41no—
109C4×Dic61—41no—
110C4×Dic61—41no—
111C4×Dic61—41no—
112C23×C61C2×C22⋊C411yescharacteristic
113C22×D61—21no—
114C22×D61—21no—
115C22×D61—21no—
116C22×D61—21no—
117C22×D61—41no—
118C22×D61—41no—
119C22×D61—41no—
120C22×D61—41no—
121C22×D61—41no—
122C22×D61—41no—
123C22×D61—41no—
124C22×D61#9912yes—
125C22×D61#10212yes—
126C22×D61#10412yes—
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—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61#10212yes—
140C2×C4×C61#10412yes—
141C2×C4×C61#13112yes—
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×Dic61C2×D811yescharacteristic
153C2×Dic61C2×SD1611yescharacteristic
154C2×Dic61AGL(1,ℤ/8)11yescharacteristic
155C2×Dic61#13711yescharacteristic
156D244—82no—
157D244—84no—
158C4×D61—24no—
159C4×D61—24no—
160C4×D61—24no—
161C4×D61—24no—
162C4×D61—44no—
163#92420—44no—
164#92420—44no—
165#7678—611nomaximal, Sylow
166C22×C61#40811yescharacteristic
167C22×C61C2×#9911yescharacteristic
168C22×C61#41311yescharacteristic
169C22×C61C2×#10211yescharacteristic
170C22×C61#41711yescharacteristic
171C22×C61C2×#10411yescharacteristic
172C22×C61C2×#10411yescharacteristic
173C2×D61—22no—
174C2×D61—22no—
175C2×D61—22no—
176C2×D61—22no—
177C2×D61—22no—
178C2×D61—22no—
179C2×D61—81no—
180C2×D61—81no—
181C2×D61—81no—
182C2×D61—81no—
183C2×D61—81no—
184C2×D61—81no—
185C2×D61—81no—
186C2×D61#32714yes—
187C4×C61—22no—
188C4×C61—22no—
189C4×C61—22no—
190C4×C61—81no—
191C4×C61—81no—
192Dic61—21no—
193Dic61—21no—
194Dic61—21no—
195Dic61—21no—
196Dic61—41no—
197Dic61—81no—
198#2854—611no—
199#1728—611no—
200C2×#336—611no—
201#1271—612no—
202#1270—612no—
203C2×C61#146811yescharacteristic
204C2×C61#148311yescharacteristic
205C2×C61#149211yescharacteristic
206C2×C61C2×#32711yescharacteristic
207C2×C61#150411yescharacteristic
208C2×C61#152411yescharacteristic
209C2×C61#154711yescharacteristic
210D61—24no—
211D61—161no—
212C22×M16—1221no—
213C23×C8—1221no—
214C2×#123—1221no—
215C2×#121—611no—
216C22×C42—611no—
217#390—611no—
218#390—611no—
219#389—611no—
220#389#692311yescharacteristic
221#388—1221no—
222#388—1221no—
223#386—1221no—
224#380—611no—
225#336—612no—
226#336—612no—
227#336—1221no—
228#336—1221no—
229C61#767811yescharacteristic, complemented, Sylow
230C22×D4—611no—
231C22×D4—1221no—
232C23×C4—611no—
233C23×C4—611no—
234C23×C4—611no—
235C2×M16—1224no—
236C2×M16—2442no—
237C2×M16—2442no—
238C22×C8—1222no—
239C22×C8—1222no—
240C22×C8—1224no—
241C22×C8—2442no—
242C22×C8—2442no—
243#123—2442no—
244#123—2442no—
245#121—2442no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—611no—
248C2×C4⋊C4—1221no—
249C2×C4⋊C4C2×#692311yescharacteristic
250C2×C4⋊C4C2×#692311yescharacteristic
251C2×C22⋊C4—1221no—
252C2×C22⋊C4—1221no—
253C2×C22⋊C4—1221no—
254C2×C22⋊C4—1221no—
255C2×C22⋊C4—1221no—
256C2×C22⋊C4—1221no—
257C2×C42—611no—
258C2×C42—611no—
259C2×C42—611no—
260C2×C42—2442no—
261C2×C42C2×#692311yescharacteristic
262#95—21no—
263#95—1221no—
264#95—1221no—
265C24—611no—
266C24—1221no—
267C2×D4—1222no—
268C2×D4—2441no—
269C2×D4—2441no—
270C2×D4—2442no—
271C2×D4—2442no—
272C2×D4—2442no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—611no—
276C22×C4—611no—
277C22×C4—612no—
278C22×C4—612no—
279C22×C4—611no—
280C22×C4—1221no—
281C22×C4—1221no—
282C22×C4—1222no—
283C22×C4—1222no—
284C22×C4—1222no—
285C22×C4—1222no—
286C22×C4—1222no—
287C22×C4—1222no—
288C22×C4—1221no—
289C22×C4—2441no—
290C22×C4C22⋊#692311yescharacteristic
291C22×C4C22⋊#692311yescharacteristic
292C22×C4C22×#692311yescharacteristic
293M16—2448no—
294C2×C8—1224no—
295C2×C8—1224no—
296C2×C8—2442no—
297C2×C8—2442no—
298C2×C8—2448no—
299C2×C8—2442no—
300C2×C8—2442no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—1222no—
305C4⋊C4—2441no—
306C4⋊C4—2441no—
307C22⋊C4—2441no—
308C22⋊C4—2441no—
309C22⋊C4—2441no—
310C22⋊C4—2441no—
311C22⋊C4—2442no—
312C22⋊C4—2442no—
313C22⋊C4—2442no—
314C22⋊C4—2442no—
315C42—41no—
316C42—2441no—
317C42—2441no—
318C42—2441no—
319C23—612no—
320C23—612no—
321C23—612no—
322C23—1221no—
323C23—1221no—
324C23—1221no—
325C23—1221no—
326C23—2441no—
327C23—2441no—
328C23—2441no—
329C23—2441no—
330C23—2441no—
331C23—2441no—
332C23—2441no—
333C23C2×C22⋊#692311yescharacteristic
334D4—4882no—
335D4—4884no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—611no—
349C2×C4—611no—
350C2×C4—611no—
351C2×C4—611no—
352C2×C4—1224no—
353C2×C4—1224no—
354C2×C4—1224no—
355C2×C4—1224no—
356C2×C4—1221no—
357C2×C4—1221no—
358C2×C4—1221no—
359C2×C4—1221no—
360C2×C4—1221no—
361C2×C4—1221no—
362C2×C4—1221no—
363C2×C4—2444no—
364C2×C4—2441no—
365C2×C4—2441no—
366C2×C4—2441no—
367C2×C4#1451983912yes—
368C2×C4#1451984212yes—
369C2×C4#1451986512yes—
370C8—2444no—
371C8—2444no—
372C22—614no—
373C22—1222no—
374C22—1222no—
375C22—1222no—
376C22—1222no—
377C22—1222no—
378C22—1222no—
379C22—4881no—
380C22—4881no—
381C22—4881no—
382C22—4881no—
383C22—4881no—
384C22—4881no—
385C22—4881no—
386C22#1684073711yescharacteristic
387C22D61⋊#9911yescharacteristic
388C22#1684074811yescharacteristic
389C22C2×#1451983911yescharacteristic
390C22#1684075311yescharacteristic
391C22D61⋊#10411yescharacteristic
392C22C2×#1451984211yescharacteristic
393C4—22no—
394C4—22no—
395C4—22no—
396C4—81no—
397C4—81no—
398C4—1221no—
399C4—1221no—
400C4—1221no—
401C4—1221no—
402C4—4881no—
403C4—2441no—
404C2—1224no—
405C2—9761no—
406C2#2146108711yescharacteristic
407C2#2146113711yescharacteristic
408C2#2146116611yescharacteristic
409C2D61⋊#32711yescharacteristic
410C2#2146120611yescharacteristic
411C2#2146127111yescharacteristic
412C2#2146133011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#389C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519752C4
cosocleC2×C4×Dic61C23
hypercenter#389#6923
hypocenterC61#7678
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
1. #21461483C2semidirect
2. #21461480C2semidirect
3. #21460638C2semidirect
4. #21460636C2semidirect
229. C61#7678semidirect

Maximal subgroups

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

subgroupindex
#214614832
#214614802
#214606382
#214606362
#214541522
#767861