Finite Group Databasebeta

← #30602683

Subgroups — #30602683

Order 15616.

854 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519831C211yesmaximal, characteristic
2C2×#16840636C211yesmaximal, characteristic
3C2×#16840635C212yesmaximal
4#21460996C212yesmaximal
5#340×D61C211yesmaximal, characteristic
6C22×D61⋊C8C2211yescharacteristic
7C22×C4⋊#6923C2212yes—
8C22×C4×#6923C2211yescharacteristic
9C2×#14519831—24no—
10C2×#14519831C414yescomplemented
11C2×(C2×#6923)⋊C4—22no—
12C2×(C2×#6923)⋊C4—22no—
13#16840636C414yescomplemented
14#16840635—24no—
15C22×C8×D61C2211yescharacteristic
16C2×C4⋊C4×D61C2212yes—
17#16839525C411yescharacteristic, complemented
18#16839524C412yescomplemented
19C61×#340C411yescharacteristic, complemented
20C23×#6923—22no—
21C23×#6923D412yes—
22C2×D61⋊C8—22no—
23C2×D61⋊C8—21no—
24C2×D61⋊C8C2×C412yes—
25C2×D61⋊C8D414yes—
26C22×#92420—21no—
27C2×C4⋊#6923—22no—
28C2×C4⋊#6923—22no—
29C2×C4⋊#6923—22no—
30C2×C4⋊#6923—22no—
31C2×C4⋊#6923—22no—
32C2×C4⋊#6923C2×C414yes—
33C2×C4×#6923—22no—
34C2×C4×#6923—22no—
35C2×C4×#6923C2×C412yes—
36C2×C4×#6923Q812yes—
37#14519831—28no—
38(C2×#6923)⋊C4—44no—
39(C2×#6923)⋊C4—44no—
40C22×C4×D61—22no—
41C22×C4×D61C2311yescharacteristic
42C22×#92418C2×C411yescharacteristic
43C2×C8×D61—22no—
44C2×C8×D61—22no—
45C2×C8×D61C2×C412yes—
46C2×C8×D61C2×C412yes—
47C2×C4⋊Dic61C2×C412yes—
48C2×#11758196—22no—
49C4⋊C4×D61—22no—
50C4⋊C4×D61—22no—
51C4⋊C4×D61—22no—
52C4⋊C4×D61—22no—
53C22×C8×C61C2×C411yescharacteristic
54C2×C4⋊C4×C61C2×C412yes—
55C23×D61C2×D411yescharacteristic
56C22×#6923—22no—
57C22×#6923—22no—
58C22×#6923—22no—
59C22×#6923—22no—
60C22×#6923—22no—
61C22×#6923—22no—
62C22×#6923—24no—
63C22×#6923—42no—
64C22×#6923—42no—
65C22×#6923—42no—
66C22×#6923—42no—
67C22×#6923—42no—
68C22×#6923—42no—
69C22×#6923C4⋊C414yes—
70C22×C4×C61—22no—
71C22×C4×C61C22×C411yescharacteristic
72C22×Dic61—22no—
73C22×Dic61#4111yescharacteristic
74C2×C4×D61—42no—
75C2×C4×D61—42no—
76C2×C4×D61—42no—
77C2×C4×D61—42no—
78C2×C4×D61—42no—
79C2×C4×D61—42no—
80C2×C4×D61C22×C411yescharacteristic
81C2×C4×D61C22×C411yescharacteristic
82C2×C4×D61C2×D412yes—
83C2×C4×D61C2×Q811yescharacteristic
84C2×C4×D61#4112yes—
85C2×C4×D61#4112yes—
86C2×C4×D61#4112yes—
87C2×C4×D61#4111yescharacteristic
88C2×#92420—22no—
89C2×#92420—24no—
90C4⋊#6923—24no—
91C4⋊#6923—24no—
92C4⋊#6923—42no—
93C4⋊#6923—42no—
94C4×#6923—22no—
95C4×#6923—22no—
96C4×#6923—24no—
97D61⋊C8—22no—
98D61⋊C8—24no—
99D61⋊C8C22⋊C414yes—
100C2×C8×C61—22no—
101C2×C8×C61C4212yes—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—22no—
104C4⋊Dic61—22no—
105C4⋊Dic61—22no—
106#11758196—42no—
107#11758196—42no—
108C2×#92418—22no—
109C2×#92418C4212yes—
110C8×D61—22no—
111C8×D61—22no—
112C8×D61—24no—
113C23×C61C4×D411yescharacteristic
114C22×D61C2×C4⋊C411yescharacteristic
115C22×D61C4×D411yescharacteristic
116C22×D61#12112yes—
117C22×D61#12412yes—
118C22×D61AGL(1,ℤ/8)12yes—
119C22×D61AGL(1,ℤ/8)12yes—
120C22×D61#13712yes—
121C22×D61#13712yes—
122C2×#6923—24no—
123C2×#6923—24no—
124C2×#6923—24no—
125C2×#6923—42no—
126C2×#6923—42no—
127C2×#6923—42no—
128C2×#6923—42no—
129C2×#6923—44no—
130C2×#6923—44no—
131C2×#6923—44no—
132C2×#6923—82no—
133C2×#6923—82no—
134C2×#6923—82no—
135C2×#6923—82no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61—42no—
139C2×C4×C61C2×C4211yescharacteristic
140C2×C4×C61#11712yes—
141C2×C4×C61C4×D412yes—
142C2×C4×C61C4×Q811yescharacteristic
143C2×Dic61—42no—
144C2×Dic61—42no—
145C2×Dic61—42no—
146C2×Dic61#11711yescharacteristic
147C2×Dic61C4×Q811yescharacteristic
148C2×Dic61#12312yes—
149C2×Dic61#12612yes—
150C4×D61—82no—
151C4×D61—82no—
152C4×D61—82no—
153C4×D61—82no—
154C4×D61C2×C22⋊C411yescharacteristic
155C4×D61#11711yescharacteristic
156C4×D61#11711yescharacteristic
157C4×D61#11711yescharacteristic
158C4×D61#11712yes—
159C4×D61C4×D412yes—
160C4×D61C22⋊Q812yes—
161C4×D61#12312yes—
162C4×D61#12512yes—
163C4×D61#12612yes—
164#92420—24no—
165C8×C61—22no—
166#92418—22no—
167#10821—611nomaximal, Sylow
168C22×C61C4×C4⋊C411yescharacteristic
169C22×C61#38812yes—
170C22×C61#44012yes—
171C22×C61#44111yescharacteristic
172C22×C61#44211yescharacteristic
173C2×D61#38011yescharacteristic
174C2×D61#38211yescharacteristic
175C2×D61#38812yes—
176C2×D61#42812yes—
177C2×D61#42812yes—
178C2×D61#44012yes—
179C2×D61#44111yescharacteristic
180C2×D61#44211yescharacteristic
181C2×D61#46812yes—
182C2×D61#46812yes—
183C2×D61#46912yes—
184C2×D61#47012yes—
185C2×D61#48912yes—
186C2×D61#48912yes—
187C2×D61#49012yes—
188C2×D61#49112yes—
189#6923—44no—
190#6923—84no—
191C4×C61—82no—
192C4×C61#37611yescharacteristic
193C4×C61C4×C22⋊C411yescharacteristic
194C4×C61#38712yes—
195Dic61—82no—
196Dic61#38111yescharacteristic
197Dic61#38311yescharacteristic
198Dic61#38712yes—
199C22×#97—611no—
200C2×#340—611no—
201C2×#340—611no—
202C2×#340—612no—
203C4⋊#95—612no—
204C2×C61#143411yescharacteristic
205C2×C61#177611yescharacteristic
206C2×C61#177611yescharacteristic
207C2×C61#193512yes—
208C2×C61#193612yes—
209D61#180912yes—
210D61#183612yes—
211D61#193512yes—
212D61#193612yes—
213C23×C8—611no—
214C23×C8—611no—
215C22×C4⋊C4—612no—
216C22×C4⋊C4—612no—
217C22×C42—611no—
218C2×#97—614no—
219C2×#97—1224no—
220C2×#95—1222no—
221C2×#95—1222no—
222#340—614no—
223#340—611no—
224#340—612no—
225#340—1224no—
226#340#692311yescharacteristic, complemented
227C61#1082111yescharacteristic, complemented, Sylow
228C23×C4—611no—
229C23×C4—612no—
230C23×C4—1222no—
231C23×C4—1222no—
232C22×C8—612no—
233C22×C8—614no—
234C22×C8—611no—
235C22×C8—612no—
236C22×C8—612no—
237C22×C8—1222no—
238C22×C8—1222no—
239C22×C8—1221no—
240C22×C8—1222no—
241C22×C8—1221no—
242C22×C8C2×#692311yescharacteristic
243C2×C4⋊C4—614no—
244C2×C4⋊C4—612no—
245C2×C4⋊C4—1222no—
246C2×C4⋊C4—1222no—
247C2×C4⋊C4—1222no—
248C2×C4⋊C4—1222no—
249C2×C4⋊C4—1222no—
250C2×C4⋊C4—1222no—
251C2×C4⋊C4—1222no—
252C2×C4⋊C4—1222no—
253C2×C4⋊C4—1222no—
254C2×C4⋊C4—1222no—
255C2×C4⋊C4C2×#692312yes—
256C2×C42—612no—
257C2×C42—612no—
258C2×C42—1222no—
259C2×C42—1222no—
260#97—1228no—
261#95—2444no—
262#95—2444no—
263C24—611no—
264C22×C4—22no—
265C22×C4—611no—
266C22×C4—611no—
267C22×C4—611no—
268C22×C4—612no—
269C22×C4—612no—
270C22×C4—612no—
271C22×C4—612no—
272C22×C4—611no—
273C22×C4—611no—
274C22×C4—614no—
275C22×C4—1222no—
276C22×C4—1222no—
277C22×C4—1222no—
278C22×C4—1222no—
279C22×C4—1222no—
280C22×C4—1222no—
281C22×C4—1224no—
282C22×C4—1222no—
283C22×C4—2442no—
284C22×C4—2442no—
285C22×C4—2442no—
286C22×C4—2442no—
287C22×C4—2442no—
288C22×C4—2442no—
289C22×C4—2442no—
290C22×C4—2442no—
291C22×C4—2442no—
292C22×C4—2442no—
293C22×C4—2442no—
294C22×C4—2442no—
295C22×C4C22×#692311yescharacteristic
296C2×C8—22no—
297C2×C8—614no—
298C2×C8—612no—
299C2×C8—1222no—
300C2×C8—1222no—
301C2×C8—1224no—
302C2×C8—1224no—
303C2×C8—1222no—
304C2×C8—1224no—
305C2×C8—1222no—
306C2×C8—1222no—
307C2×C8C4×#692312yes—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—1224no—
311C4⋊C4—1224no—
312C4⋊C4—1222no—
313C4⋊C4—1222no—
314C4⋊C4—2442no—
315C4⋊C4—2442no—
316C4⋊C4—2442no—
317C4⋊C4—2442no—
318C42—1222no—
319C42—1224no—
320C42—1222no—
321C23—612no—
322C23—612no—
323C23—612no—
324C23—612no—
325C23—612no—
326C23—612no—
327C23—611no—
328C23—611no—
329C23D4×#692311yescharacteristic
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—612no—
334C2×C4—612no—
335C2×C4—611no—
336C2×C4—611no—
337C2×C4—612no—
338C2×C4—612no—
339C2×C4—612no—
340C2×C4—612no—
341C2×C4—611no—
342C2×C4—612no—
343C2×C4—611no—
344C2×C4—611no—
345C2×C4—611no—
346C2×C4—612no—
347C2×C4—1224no—
348C2×C4—1224no—
349C2×C4—1224no—
350C2×C4—4882no—
351C2×C4—4882no—
352C2×C4—4882no—
353C2×C4—4882no—
354C2×C4—4882no—
355C2×C4—4882no—
356C2×C4—4882no—
357C2×C4—4882no—
358C2×C4—2444no—
359C2×C4—2444no—
360C2×C4—2444no—
361C2×C4—2442no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4—2442no—
365C2×C4—2442no—
366C2×C4—2442no—
367C2×C4—2442no—
368C2×C4C2×C4×#692311yescharacteristic
369C2×C4#1451985612yes—
370C2×C4D4×#692312yes—
371C2×C4Q8×#692311yescharacteristic
372C8—22no—
373C8—1224no—
374C8—1222no—
375C22—611no—
376C22—612no—
377C22—611no—
378C22—612no—
379C22—612no—
380C22—612no—
381C22—611no—
382C22—611no—
383C22—612no—
384C22—612no—
385C22—612no—
386C22—612no—
387C22—612no—
388C22—612no—
389C22—612no—
390C22—612no—
391C22C4⋊C4×#692311yescharacteristic
392C22#1684071212yes—
393C22#1684080112yes—
394C22#1684080211yescharacteristic
395C22#1684080311yescharacteristic
396C4—82no—
397C4—611no—
398C4—611no—
399C4—612no—
400C4—4884no—
401C4—4882no—
402C4—2444no—
403C4#1684069811yescharacteristic
404C4C22⋊C4×#692311yescharacteristic
405C4#1684071112yes—
406C2—612no—
407C2—612no—
408C2—612no—
409C2—612no—
410C2#2146099411yescharacteristic
411C2#2146163411yescharacteristic
412C2#2146163511yescharacteristic
413C2#2146191012yes—
414C2#2146191112yes—
subgroupG/H
centerC23D4×#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#340C4
socleC23×C61C4×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#340#6923
hypocenterC61#10821
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
10. C2×#14519831C4semidirect
13. #16840636C4semidirect
17. #16839525C4semidirect
18. #16839524C4semidirect
19. C61×#340C4semidirect
226. #340#6923semidirect
227. C61#10821semidirect

Maximal subgroups

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

subgroupindex
C22×#145198312
C2×#168406362
C2×#168406352
#214609962
#340×D612
#1082161