Finite Group Databasebeta

← #30606295

Subgroups — #30606295

Order 15616.

1154 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 152 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×D61⋊#99C212yesmaximal
2#21460914C212yesmaximal, complemented
3#21460913C212yesmaximal, complemented
4#390×D61C211yesmaximal, characteristic
5C22×C22⋊#6923—22no—
6C2×#14519836C2214yes—
7D61⋊#99—24no—
8D61⋊#99C418yescomplemented
9C2×#14519835—22no—
10C22×D4×D61C2211yescharacteristic
11C2×C4⋊C4×D61C411yescharacteristic, complemented
12C2×C22⋊C4×D61—22no—
13C2×C42×D61C411yescharacteristic, complemented
14#16839665C411yescharacteristic, complemented
15#16839664C2211yescharacteristic
16#16839663C2211yescharacteristic
17#16839662—22no—
18C61×#390C411yescharacteristic, complemented
19C23×#6923—42no—
20C2×C22⋊#6923—28no—
21C2×C22⋊#6923—24no—
22C2×C22⋊#6923—44no—
23C2×C22⋊#6923—42no—
24#14519836—24no—
25#14519836C2×C418yescomplemented
26#14519835—28no—
27C24×D61D412yes—
28C22×D244C2311yescharacteristic
29C22×C22⋊D61D412yes—
30C2×D4×D61—42no—
31C2×D4×D61—41no—
32C2×D4×D61—41no—
33C2×D4×D61C2×C414yes—
34C2×D4×D61D412yes—
35C2×D4×D61Q812yes—
36C22×C4×D61—21no—
37C22×C4×D61—21no—
38C22×C4×D61C2×C411yescharacteristic
39C2×C4⋊Dic61C2×C411yescharacteristic, complemented
40C2×#11758196—21no—
41C4⋊C4×D61—41no—
42C4⋊C4×D61—41no—
43C2×C22⋊Dic61—22no—
44C2×#11758198—22no—
45C2×#11758198—22no—
46C22⋊C4×D61—48no—
47C2×C4×Dic61—21no—
48C2×C4×Dic61C2×C411yescharacteristic, complemented
49C42×D61—42no—
50C22×D4×C61C2×C411yescharacteristic
51C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
52C2×C22⋊C4×C61—22no—
53C2×C42×C61C2×C411yescharacteristic, complemented
54C24×C61C22⋊C412yes—
55C23×D61—22no—
56C23×D61—22no—
57C23×D61—24no—
58C23×D61—44no—
59C23×D61—44no—
60C23×D61C22⋊C411yescharacteristic
61C23×D61C22⋊C414yes—
62C23×D61C4⋊C414yes—
63C23×D61C2×D412yes—
64C22×#6923—44no—
65C22×#6923—48no—
66C22×#6923—84no—
67C22×#6923—84no—
68C2×D4×C61—41no—
69C2×D4×C61—41no—
70C2×D4×C61C4212yes—
71C2×D4×C61C4⋊C412yes—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61C22×C411yescharacteristic
75C2×C22⋊D61—42no—
76C2×C22⋊D61—42no—
77C2×C22⋊D61C22⋊C414yes—
78C2×C22⋊D61C4⋊C414yes—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61C22⋊C411yescharacteristic
82D4×D61—44no—
83D4×D61—44no—
84D4×D61—42no—
85D4×D61—42no—
86D4×D61—42no—
87D4×D61—42no—
88C2×D244—41no—
89C2×D244—41no—
90C2×D244C22×C412yes—
91C2×D244C2×D411yescharacteristic
92C2×D244C2×Q811yescharacteristic
93C2×C4×D61—22no—
94C2×C4×D61—21no—
95C2×C4×D61—21no—
96C2×C4×D61—24no—
97C2×C4×D61—42no—
98C2×C4×D61—42no—
99C2×C4×D61—44no—
100C2×C4×D61—41no—
101C2×C4×D61—41no—
102C2×C4×D61C4212yes—
103C2×C4×D61C4⋊C412yes—
104C22⋊#6923—28no—
105C22⋊#6923—44no—
106C22⋊#6923—48no—
107C4⋊C4×C61—41no—
108C22⋊C4×C61—44no—
109C42×C61—41no—
110C22⋊Dic61—44no—
111#11758198—44no—
112#11758198—44no—
113C4⋊Dic61—41no—
114#11758196—42no—
115C4×Dic61—42no—
116C4×Dic61—41no—
117C23×C61—22no—
118C23×C61—22no—
119C23×C61—42no—
120C23×C61—42no—
121C23×C61#9514yes—
122C23×C61C2×C22⋊C411yescharacteristic
123C22×D61—22no—
124C22×D61—22no—
125C22×D61—22no—
126C22×D61—22no—
127C22×D61—24no—
128C22×D61—24no—
129C22×D61—24no—
130C22×D61—24no—
131C22×D61—48no—
132C22×D61—48no—
133C22×D61—44no—
134C22×D61—44no—
135C22×D61—42no—
136C22×D61—42no—
137C22×D61—88no—
138C22×D61—82no—
139C22×D61—82no—
140C22×D61#9512yes—
141C22×D61#9911yescharacteristic
142C22×D61#9912yes—
143C22×D61#9911yescharacteristic
144C22×D61C2×C22⋊C412yes—
145C22×D61C2×C4⋊C412yes—
146C2×#6923—48no—
147C2×#6923—84no—
148C2×#6923—84no—
149C2×#6923—816no—
150D4×C61—42no—
151D4×C61—42no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—22no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—41no—
159C2×C4×C61C2×C4211yescharacteristic
160C2×C4×C61C2×C4⋊C411yescharacteristic
161C22⋊D61—44no—
162C22⋊D61—44no—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—22no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—42no—
169C2×Dic61—41no—
170C2×Dic61#9512yes—
171D244—42no—
172D244—41no—
173D244—41no—
174C4×D61—22no—
175C4×D61—22no—
176C4×D61—22no—
177C4×D61—22no—
178C4×D61—48no—
179C4×D61—82no—
180C4×D61—82no—
181C2×#1393—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—24no—
185C22×C61—44no—
186C22×C61—42no—
187C22×C61—42no—
188C22×C61—82no—
189C22×C61C2×#9511yescharacteristic
190C22×C61C2×#9911yescharacteristic
191C22×C61C2×#9911yescharacteristic
192C2×D61—21no—
193C2×D61—21no—
194C2×D61—24no—
195C2×D61—24no—
196C2×D61—22no—
197C2×D61—22no—
198C2×D61—42no—
199C2×D61—42no—
200C2×D61—48no—
201C2×D61—48no—
202C2×D61—42no—
203C2×D61—42no—
204C2×D61—84no—
205C2×D61—84no—
206C2×D61—82no—
207C2×D61#34212yes—
208C2×D61#34212yes—
209C2×D61#35312yes—
210C2×D61#35412yes—
211#6923—88no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—42no—
215C4×C61—81no—
216Dic61—22no—
217Dic61—42no—
218Dic61—81no—
219C22×#99—612no—
220C2×#390—611no—
221#1393—612no—
222#1393—612no—
223C2×C61—21no—
224C2×C61—21no—
225C2×C61—44no—
226C2×C61C2×#34211yescharacteristic
227C2×C61C2×#35311yescharacteristic
228C2×C61C2×#35411yescharacteristic
229D61—22no—
230D61—42no—
231D61—42no—
232D61#139314yescomplemented
233C23×D4—611no—
234C22×C4⋊C4—611no—
235C22×C22⋊C4—1222no—
236C22×C22⋊C4—1222no—
237C22×C42—611no—
238C2×#99—618no—
239C2×#99—614no—
240C2×#99—1222no—
241C2×#99—1224no—
242#390—611no—
243#390—611no—
244#390—611no—
245#390—1222no—
246#390#692311yescharacteristic, complemented
247C61C2×#139311yescharacteristic, complemented, Sylow
248C25—612no—
249C22×D4—612no—
250C22×D4—614no—
251C22×D4—612no—
252C22×D4—612no—
253C22×D4—611no—
254C22×D4—2442no—
255C22×D4—2441no—
256C22×D4—2441no—
257C22×D4C2×#692311yescharacteristic
258C23×C4—611no—
259C23×C4—1221no—
260C23×C4—1221no—
261C23×C4—2442no—
262C2×C4⋊C4—611no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—2441no—
265C2×C4⋊C4—2441no—
266C2×C4⋊C4C2×#692311yescharacteristic, complemented
267C2×C22⋊C4—22no—
268C2×C22⋊C4—1228no—
269C2×C22⋊C4—1224no—
270C2×C22⋊C4—1222no—
271C2×C22⋊C4—1222no—
272C2×C22⋊C4—1222no—
273C2×C22⋊C4—2448no—
274C2×C22⋊C4—2442no—
275C2×C22⋊C4—2444no—
276C2×C42—611no—
277C2×C42—1221no—
278C2×C42—2442no—
279C2×C42C2×#692311yescharacteristic, complemented
280#99—618no—
281#99—1228no—
282#99—1224no—
283C24—611no—
284C24—614no—
285C24—614no—
286C24—612no—
287C24—1222no—
288C24—1224no—
289C24—1222no—
290C24—2444no—
291C24—2444no—
292C24C22⋊#692312yes—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—614no—
296C2×D4—614no—
297C2×D4—612no—
298C2×D4—611no—
299C2×D4—611no—
300C2×D4—2441no—
301C2×D4—2441no—
302C2×D4—2444no—
303C2×D4—2444no—
304C2×D4—2442no—
305C2×D4—2442no—
306C2×D4—2442no—
307C2×D4—2442no—
308C2×D4—2442no—
309C2×D4—2442no—
310C2×D4C4×#692312yes—
311C2×D4C4⋊#692312yes—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—611no—
315C22×C4—612no—
316C22×C4—612no—
317C22×C4—1222no—
318C22×C4—1221no—
319C22×C4—1221no—
320C22×C4—1224no—
321C22×C4—1221no—
322C22×C4—1221no—
323C22×C4—4884no—
324C22×C4—4884no—
325C22×C4—2441no—
326C22×C4—2441no—
327C22×C4—2444no—
328C22×C4—2442no—
329C22×C4—2442no—
330C22×C4—2448no—
331C22×C4—2444no—
332C22×C4C22×#692311yescharacteristic
333C4⋊C4—41no—
334C4⋊C4—2441no—
335C4⋊C4—2442no—
336C22⋊C4—44no—
337C22⋊C4—1228no—
338C22⋊C4—2444no—
339C22⋊C4—2444no—
340C22⋊C4—2444no—
341C22⋊C4—2448no—
342C22⋊C4—2444no—
343C42—41no—
344C42—2441no—
345C42—2442no—
346C23—22no—
347C23—22no—
348C23—42no—
349C23—42no—
350C23—611no—
351C23—612no—
352C23—611no—
353C23—612no—
354C23—612no—
355C23—612no—
356C23—1222no—
357C23—1222no—
358C23—1224no—
359C23—1224no—
360C23—1224no—
361C23—1224no—
362C23—1222no—
363C23—1222no—
364C23—4882no—
365C23—4882no—
366C23—4888no—
367C23—2442no—
368C23—2442no—
369C23—2444no—
370C23—2444no—
371C23—2448no—
372C23—2448no—
373C23(C2×#6923)⋊C414yes—
374C23C2×C22⋊#692311yescharacteristic
375D4—42no—
376D4—42no—
377D4—2442no—
378D4—2441no—
379D4—2441no—
380D4—2444no—
381D4—2444no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—612no—
390C2×C4—1222no—
391C2×C4—1222no—
392C2×C4—1222no—
393C2×C4—1222no—
394C2×C4—1221no—
395C2×C4—1221no—
396C2×C4—1222no—
397C2×C4—4882no—
398C2×C4—4882no—
399C2×C4—48816no—
400C2×C4—4884no—
401C2×C4—4884no—
402C2×C4—2448no—
403C2×C4—2442no—
404C2×C4—2448no—
405C2×C4—2441no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4C2×C4×#692311yescharacteristic
409C2×C4C2×C4⋊#692311yescharacteristic
410C22—21no—
411C22—21no—
412C22—24no—
413C22—44no—
414C22—42no—
415C22—42no—
416C22—82no—
417C22—612no—
418C22—612no—
419C22—612no—
420C22—612no—
421C22—1224no—
422C22—1224no—
423C22—1221no—
424C22—1221no—
425C22—1222no—
426C22—1222no—
427C22—4882no—
428C22—4884no—
429C22—4884no—
430C22—2442no—
431C22—2442no—
432C22—2448no—
433C22—2448no—
434C22—2442no—
435C22—2442no—
436C22C2×(C2×#6923)⋊C411yescharacteristic
437C22D61⋊#9911yescharacteristic
438C22D61⋊#9911yescharacteristic
439C4—21no—
440C4—21no—
441C4—42no—
442C4—81no—
443C4—1222no—
444C4—4888no—
445C4—4881no—
446C4—2442no—
447C2—21no—
448C2—21no—
449C2—44no—
450C2—614no—
451C2—1222no—
452C2—2442no—
453C2—2442no—
454C2#2146151611yescharacteristic
455C2D61⋊#35311yescharacteristic
456C2D61⋊#35411yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC22×C61C2×#99
prime radicalC22×D4×D61C22
cosocleC22×D244C23
hypercenter#390#6923
hypocenterC61C2×#1393
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #21460914C2semidirect
3. #21460913C2semidirect
8. D61⋊#99C4semidirect
11. C2×C4⋊C4×D61C4semidirect
13. C2×C42×D61C4semidirect
14. #16839665C4semidirect
18. C61×#390C4semidirect
25. #14519836C2×C4semidirect
39. C2×C4⋊Dic61C2×C4semidirect
48. C2×C4×Dic61C2×C4semidirect
51. C2×C4⋊C4×C61C2×C4semidirect
53. C2×C42×C61C2×C4semidirect
232. D61#1393semidirect
246. #390#6923semidirect
247. C61C2×#1393semidirect
266. C2×C4⋊C4C2×#6923semidirect
279. C2×C42C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×D61⋊#992
#214609142
#214609132
#390×D612
C2×#139361