Finite Group Databasebeta

← D61⋊#1351 #30604807

Subgroups — D61⋊#1351 #30604807

Order 15616.

776 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 88 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#21461516C211yesmaximal, characteristic
2D61⋊#323C211yesmaximal, characteristic
3#21460814C212yesmaximal, complemented
4#21460813C212yesmaximal, complemented
5#393×D61C211yesmaximal, characteristic
6C22×C22⋊#6923—21no—
7D61⋊(C22⋊C8)—21no—
8#16840640—21no—
9#16840640C2212yes—
10#16840606C2212yes—
11#16840605—21no—
12C22×Q8×D61C411yescharacteristic, complemented
13C2×C22⋊C4×D61—21no—
14C2×C22⋊C4×D61C2211yescharacteristic
15#16839675—21no—
16#16839675C2211yescharacteristic
17#16839674—21no—
18#16839674C411yescharacteristic, complemented
19#16839673C2211yescharacteristic
20#95×D61—21no—
21#95×D61C411yescharacteristic, complemented
22C61×#393C411yescharacteristic, complemented
23C23×#6923—41no—
24C2×D61⋊C8—41no—
25C2×C22⋊#6923—24no—
26C2×C22⋊#6923—22no—
27C2×C22⋊#6923—41no—
28C2×C22⋊#6923—42no—
29#14519834—41no—
30C22⋊#92420—22no—
31C24×D61D411yescharacteristic
32C22×Dic122—21no—
33C22×Dic122C2×C411yescharacteristic, complemented
34C2×Q8×D61—44no—
35C2×Q8×D61—41no—
36C2×Q8×D61—41no—
37C22×C4×D61—21no—
38C22×C4×D61—21no—
39C22×C4×D61C2×C411yescharacteristic
40C22×C4×D61D411yescharacteristic
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61D411yescharacteristic
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198D411yescharacteristic
46C2×#11758198C2311yescharacteristic
47C22⋊C4×D61—41no—
48C22⋊C4×D61—41no—
49C22⋊C4×D61—44no—
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54#14519684C2×C411yescharacteristic, complemented
55C22×Q8×C61C2×C411yescharacteristic, complemented
56C2×C22⋊C4×C61—21no—
57C2×C22⋊C4×C61C2×C411yescharacteristic
58#95×C61—21no—
59#95×C61C2×C411yescharacteristic, complemented
60C24×C61C22⋊C411yescharacteristic
61C23×D61—21no—
62C23×D61—21no—
63C23×D61—22no—
64C23×D61—42no—
65C23×D61—42no—
66C23×D61C22⋊C411yescharacteristic
67C23×D61SD1612yes—
68C23×D61Q1612yes—
69C23×D61C2×D411yescharacteristic
70C22×#6923—42no—
71C22×#6923—44no—
72C22×#6923—82no—
73C22×#6923—82no—
74C2×Q8×C61—42no—
75C2×Q8×C61—41no—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22×C411yescharacteristic
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61C22⋊C411yescharacteristic
83C22×Dic61C2×D411yescharacteristic
84Q8×D61—44no—
85Q8×D61—44no—
86Q8×D61—44no—
87Q8×D61—44no—
88C2×C4×D61—24no—
89C2×C4×D61—21no—
90C2×C4×D61—21no—
91C2×C4×D61—21no—
92C2×C4×D61—21no—
93C2×C4×D61—42no—
94C2×C4×D61—41no—
95C2×C4×D61—41no—
96C2×C4×D61—42no—
97C2×C4×D61—42no—
98C2×C4×D61—42no—
99C2×C4×D61—42no—
100C2×C4×D61—42no—
101C2×Dic122—42no—
102C2×Dic122—42no—
103C2×Dic122—42no—
104C2×Dic122—41no—
105C2×Dic122—42no—
106C22⋊#6923—24no—
107C22⋊#6923—44no—
108C22⋊#6923—42no—
109C2×#92420—42no—
110D61⋊C8—82no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—42no—
113C22⋊Dic61—41no—
114C22⋊Dic61—42no—
115#11758198—41no—
116#11758198—41no—
117#11758198—42no—
118#11758198—42no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61#10312yes—
124C23×C61C2×C22⋊C411yescharacteristic
125C22×D61—22no—
126C22×D61—22no—
127C22×D61—21no—
128C22×D61—21no—
129C22×D61—22no—
130C22×D61—22no—
131C22×D61—22no—
132C22×D61—22no—
133C22×D61—44no—
134C22×D61—44no—
135C22×D61—42no—
136C22×D61—42no—
137C22×D61—41no—
138C22×D61—41no—
139C22×D61—84no—
140C22×D61—81no—
141C22×D61—81no—
142C22×D61#9911yescharacteristic
143C22×D61#9911yescharacteristic
144C22×D61#10312yes—
145C22×D61#10412yes—
146C22×D61C2×SD1611yescharacteristic
147C22×D61C2×Q1611yescharacteristic
148C2×#6923—44no—
149C2×#6923—82no—
150C2×#6923—82no—
151C2×#6923—88no—
152Q8×C61—42no—
153Q8×C61—42no—
154C2×C4×C61—22no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×Dic61—22no—
165C2×Dic61—21no—
166C2×Dic61—21no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C4×D61—42no—
175C4×D61—44no—
176C4×D61—44no—
177C4×D61—42no—
178C4×D61—42no—
179C4×D61—81no—
180C4×D61—84no—
181Dic122—48no—
182Dic122—42no—
183Dic122—42no—
184#92420—82no—
185C2×#1351—611nomaximal, Sylow
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—22no—
189C22×C61—42no—
190C22×C61—41no—
191C22×C61—41no—
192C22×C61—81no—
193C22×C61C2×#9911yescharacteristic
194C22×C61C2×#10311yescharacteristic
195C22×C61C2×#10411yescharacteristic
196C2×D61—21no—
197C2×D61—21no—
198C2×D61—24no—
199C2×D61—24no—
200C2×D61—21no—
201C2×D61—21no—
202C2×D61—41no—
203C2×D61—41no—
204C2×D61—44no—
205C2×D61—44no—
206C2×D61—41no—
207C2×D61—41no—
208C2×D61—82no—
209C2×D61—82no—
210C2×D61—81no—
211C2×D61#32812yes—
212C2×D61#32812yes—
213C2×D61AΣL(1,𝔽16)12yes—
214C2×D61#35412yes—
215#6923—84no—
216C4×C61—42no—
217C4×C61—41no—
218C4×C61—41no—
219C4×C61—81no—
220Dic61—41no—
221Dic61—42no—
222Dic61—41no—
223Dic61—81no—
224C2×#393—611no—
225C2×#342—611no—
226C2×#323—611no—
227#1351—612no—
228#1351—612no—
229C2×C61—21no—
230C2×C61—21no—
231C2×C61—42no—
232C2×C61C2×#32811yescharacteristic
233C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
234C2×C61C2×#35411yescharacteristic
235D61—22no—
236D61—41no—
237D61—41no—
238D61#135114yescomplemented
239C23×Q8—611no—
240C22×C22⋊C4—611no—
241C22×C22⋊C4—1221no—
242C22×C22⋊C4—1221no—
243C2×C22⋊C8—1221no—
244#393—611no—
245#393—611no—
246#393—611no—
247#393—1221no—
248#393—1221no—
249#393#692311yescharacteristic, complemented
250C2×#95—611no—
251C2×#95—1221no—
252#342—612no—
253#342—1221no—
254#323—612no—
255#323—1221no—
256C61C2×#135111yescharacteristic, complemented, Sylow
257C25—611no—
258C22×Q8—611no—
259C22×Q8—1221no—
260C22×Q8—2444no—
261C22×Q8—2441no—
262C22×Q8—2441no—
263C22×Q8C2×#692311yescharacteristic, complemented
264C23×C4—611no—
265C23×C4—611no—
266C23×C4—1221no—
267C23×C4—1221no—
268C23×C4—2441no—
269C22×C8—2441no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—611no—
272C2×C22⋊C4—611no—
273C2×C22⋊C4—611no—
274C2×C22⋊C4—1222no—
275C2×C22⋊C4—1224no—
276C2×C22⋊C4—1221no—
277C2×C22⋊C4—1221no—
278C2×C22⋊C4—1221no—
279C2×C22⋊C4—2441no—
280C2×C22⋊C4—2441no—
281C2×C22⋊C4—2444no—
282C2×C22⋊C4—2442no—
283C2×C22⋊C4—2441no—
284C2×C22⋊C4C2×#692311yescharacteristic
285C22⋊C8—1222no—
286C22⋊C8—2441no—
287#95—21no—
288#95—611no—
289#95—1221no—
290#95—1221no—
291#95—1221no—
292#95—1221no—
293#95C2×#692311yescharacteristic, complemented
294C24—611no—
295C24—612no—
296C24—612no—
297C24—611no—
298C24—1222no—
299C24—1221no—
300C24—1221no—
301C24—2442no—
302C24—2442no—
303C24C22⋊#692311yescharacteristic
304C2×Q8—42no—
305C2×Q8—41no—
306C2×Q8—2442no—
307C2×Q8—2441no—
308C2×Q8—2442no—
309C2×Q8—2442no—
310C2×Q8—2442no—
311C2×Q8—2444no—
312C2×Q8—2444no—
313C2×Q8—2444no—
314C2×Q8—2444no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—611no—
318C22×C4—611no—
319C22×C4—1221no—
320C22×C4—1221no—
321C22×C4—1221no—
322C22×C4—1221no—
323C22×C4—1224no—
324C22×C4—1221no—
325C22×C4—1221no—
326C22×C4—4882no—
327C22×C4—4882no—
328C22×C4—2442no—
329C22×C4—2442no—
330C22×C4—2442no—
331C22×C4—2442no—
332C22×C4—2441no—
333C22×C4—2441no—
334C22×C4—2442no—
335C22×C4—2442no—
336C22×C4—2444no—
337C22×C4—2442no—
338C22×C4C22⋊#692311yescharacteristic
339C22×C4C22×#692311yescharacteristic
340C2×C8—4882no—
341C2×C8—2442no—
342C22⋊C4—41no—
343C22⋊C4—42no—
344C22⋊C4—1224no—
345C22⋊C4—2441no—
346C22⋊C4—2441no—
347C22⋊C4—2442no—
348C22⋊C4—2442no—
349C22⋊C4—2441no—
350C22⋊C4—2442no—
351C22⋊C4—2442no—
352C22⋊C4—2444no—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23—611no—
358C23—611no—
359C23—612no—
360C23—612no—
361C23—611no—
362C23—611no—
363C23—1222no—
364C23—1222no—
365C23—1222no—
366C23—1222no—
367C23—1222no—
368C23—1222no—
369C23—1221no—
370C23—1221no—
371C23—4881no—
372C23—4881no—
373C23—4884no—
374C23—2441no—
375C23—2441no—
376C23—2442no—
377C23—2442no—
378C23—2444no—
379C23—2444no—
380C23#1451984012yes—
381C23C2×C22⋊#692311yescharacteristic
382Q8—42no—
383Q8—42no—
384Q8—2442no—
385Q8—2448no—
386Q8—2442no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—1222no—
398C2×C4—1221no—
399C2×C4—1221no—
400C2×C4—4881no—
401C2×C4—4884no—
402C2×C4—4888no—
403C2×C4—4882no—
404C2×C4—4882no—
405C2×C4—2442no—
406C2×C4—2444no—
407C2×C4—2442no—
408C2×C4—2444no—
409C2×C4—2442no—
410C2×C4—2441no—
411C2×C4—2441no—
412C2×C4—2444no—
413C2×C4—2441no—
414C2×C4—2441no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4—2441no—
418C8—4882no—
419C22—21no—
420C22—21no—
421C22—22no—
422C22—42no—
423C22—41no—
424C22—41no—
425C22—81no—
426C22—612no—
427C22—612no—
428C22—612no—
429C22—612no—
430C22—1224no—
431C22—1224no—
432C22—1221no—
433C22—1221no—
434C22—1221no—
435C22—1221no—
436C22—4881no—
437C22—4882no—
438C22—4882no—
439C22—2441no—
440C22—2441no—
441C22—2444no—
442C22—2444no—
443C22—2441no—
444C22—2441no—
445C22D61⋊#9911yescharacteristic
446C22C2×#1451984011yescharacteristic
447C22D61⋊#10411yescharacteristic
448C4—42no—
449C4—41no—
450C4—41no—
451C4—81no—
452C4—4881no—
453C4—4884no—
454C4—2442no—
455C4—2441no—
456C4—2441no—
457C2—21no—
458C2—21no—
459C2—42no—
460C2—614no—
461C2—1222no—
462C2—2441no—
463C2—2441no—
464C2D61⋊#32811yescharacteristic
465C2D61⋊AΣL(1,𝔽16)11yescharacteristic
466C2D61⋊#35411yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#393C4
socleC22×C61C2×#99
prime radicalC24×D61D4
cosocleC2×#11758198C23
hypercenter#393#6923
hypocenterC61C2×#1351
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #393

Complements

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

NH ≅ G/Ntype
3. #21460814C2semidirect
4. #21460813C2semidirect
12. C22×Q8×D61C4semidirect
18. #16839674C4semidirect
21. #95×D61C4semidirect
22. C61×#393C4semidirect
33. C22×Dic122C2×C4semidirect
54. #14519684C2×C4semidirect
55. C22×Q8×C61C2×C4semidirect
59. #95×C61C2×C4semidirect
238. D61#1351semidirect
249. #393#6923semidirect
256. C61C2×#1351semidirect
263. C22×Q8C2×#6923semidirect
293. #95C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214615162
D61⋊#3232
#214608142
#214608132
#393×D612
C2×#135161