Finite Group Databasebeta

← #30599250

Subgroups — #30599250

Order 15616.

940 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 126 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#21461642C211yesmaximal, characteristic
2#21460993C211yesmaximal, characteristic
3#21460692C214yesmaximal, complemented
4C2×#16839619C211yesmaximal, characteristic
5C22×C22⋊#6923—21no—
6C2×#14519834—21no—
7C2×#14519834C2212yes—
8C2×C22⋊#92420—21no—
9C2×(C2×#6923)⋊C4—21no—
10C2×(C2×#6923)⋊C4C2212yes—
11C23×C4×D61C2211yescharacteristic
12C22×C4⋊Dic61C411yescharacteristic, complemented
13C22×#11758198—21no—
14#16839619—21no—
15#16839619—21no—
16#16839619C412yescomplemented
17#16839619C2212yes—
18C2×#14519684—21no—
19C22×C4⋊C4×C61C411yescharacteristic, complemented
20C23×#6923—22no—
21C22×#92420—22no—
22C2×C22⋊#6923—24no—
23C2×C22⋊#6923—42no—
24C2×C22⋊#6923—42no—
25#14519834—24no—
26#14519834—42no—
27C22⋊#92420—42no—
28(C2×#6923)⋊C4—24no—
29(C2×#6923)⋊C4—42no—
30C24×D61D411yescharacteristic
31C23×Dic61—21no—
32C23×Dic61D411yescharacteristic
33C22×C4×D61—24no—
34C22×C4×D61—42no—
35C22×C4×D61—42no—
36C22×C4×D61C2×C411yescharacteristic
37C22×C4×D61D411yescharacteristic
38C22×C4×D61D411yescharacteristic
39C22×C4×D61C2311yescharacteristic
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—42no—
42C2×C4⋊Dic61C2×C412yes—
43C2×#11758198—22no—
44C2×#11758198—22no—
45C2×#11758198—44no—
46#14519684—22no—
47#14519684—22no—
48C23×C4×C61—21no—
49C23×C4×C61C2×C411yescharacteristic
50C2×C4⋊C4×C61—21no—
51C2×C4⋊C4×C61—42no—
52C2×C4⋊C4×C61C2×C412yes—
53C24×C61C22⋊C411yescharacteristic
54C23×D61—22no—
55C23×D61—22no—
56C23×D61—22no—
57C23×D61—22no—
58C23×D61—41no—
59C23×D61—41no—
60C23×D61C22⋊C411yescharacteristic
61C23×D61SD1612yes—
62C23×D61Q1612yes—
63C23×D61C2×D411yescharacteristic
64C22×#6923—24no—
65C22×#6923—42no—
66C22×#6923—42no—
67C22×#6923—42no—
68C22×#6923—42no—
69C22×#6923—42no—
70C22×#6923—42no—
71C22×C4×C61—22no—
72C22×C4×C61—22no—
73C22×C4×C61—22no—
74C22×C4×C61—22no—
75C22×C4×C61—42no—
76C22×C4×C61—42no—
77C22×C4×C61—42no—
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22×C411yescharacteristic
80C22×Dic61—22no—
81C22×Dic61—22no—
82C22×Dic61—22no—
83C22×Dic61—41no—
84C22×Dic61—42no—
85C22×Dic61—42no—
86C22×Dic61—42no—
87C22×Dic61C22⋊C411yescharacteristic
88C22×Dic61C2×D411yescharacteristic
89C2×C4×D61—22no—
90C2×C4×D61—22no—
91C2×C4×D61—22no—
92C2×C4×D61—22no—
93C2×C4×D61—22no—
94C2×C4×D61—41no—
95C2×C4×D61—41no—
96C2×C4×D61—41no—
97C2×C4×D61—41no—
98C2×C4×D61—41no—
99C2×C4×D61—41no—
100C2×C4×D61—416no—
101C2×C4×D61#4114yes—
102C22⋊#6923—48no—
103C2×#92420—42no—
104C2×#92420—42no—
105C2×#92420—42no—
106C4⋊C4×C61—42no—
107C4⋊C4×C61—42no—
108#11758198—44no—
109#11758198—44no—
110C4⋊Dic61—42no—
111C4⋊Dic61—42no—
112C23×C61—21no—
113C23×C61—21no—
114C23×C61—21no—
115C23×C61—21no—
116C23×C61#9912yes—
117C23×C61#10312yes—
118C23×C61#10412yes—
119C23×C61C2×C22⋊C411yescharacteristic
120C22×D61—21no—
121C22×D61—21no—
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—24no—
125C22×D61—24no—
126C22×D61—21no—
127C22×D61—21no—
128C22×D61—21no—
129C22×D61—21no—
130C22×D61—21no—
131C22×D61—21no—
132C22×D61—24no—
133C22×D61—24no—
134C22×D61—42no—
135C22×D61—42no—
136C22×D61—44no—
137C22×D61—44no—
138C22×D61—44no—
139C22×D61—44no—
140C22×D61#10312yes—
141C22×D61#12312yes—
142C22×D61C2×SD1611yescharacteristic
143C22×D61C2×Q1611yescharacteristic
144C22×D61AGL(1,ℤ/8)12yes—
145C2×#6923—44no—
146C2×#6923—44no—
147C2×#6923—44no—
148C2×#6923—82no—
149C2×#6923—82no—
150C2×#6923—82no—
151C2×#6923—82no—
152C2×C4×C61—28no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—84no—
161C2×C4×C61#9912yes—
162C2×C4×C61#11712yes—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—21no—
167C2×Dic61—21no—
168C2×Dic61—44no—
169C2×Dic61—42no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61—84no—
173C2×Dic61#12312yes—
174C4×D61—44no—
175C4×D61—44no—
176C4×D61—44no—
177C4×D61—44no—
178#92420—82no—
179#8713—611nomaximal, Sylow
180C22×C61—22no—
181C22×C61—22no—
182C22×C61—22no—
183C22×C61—22no—
184C22×C61—22no—
185C22×C61—22no—
186C22×C61#32814yes—
187C22×C61#38111yescharacteristic
188C22×C61C2×#9911yescharacteristic
189C22×C61C2×#9911yescharacteristic
190C22×C61C2×#10311yescharacteristic
191C22×C61#41811yescharacteristic
192C22×C61C2×#10411yescharacteristic
193C22×C61#42111yescharacteristic
194C2×D61—22no—
195C2×D61—22no—
196C2×D61—21no—
197C2×D61—21no—
198C2×D61—22no—
199C2×D61—22no—
200C2×D61—21no—
201C2×D61—21no—
202C2×D61—22no—
203C2×D61—22no—
204C2×D61—22no—
205C2×D61—48no—
206C2×D61—48no—
207C2×D61—41no—
208C2×D61—41no—
209C2×D61—41no—
210C2×D61—41no—
211C2×D61#48112yes—
212C2×D61#48412yes—
213#6923—84no—
214C4×C61—22no—
215C4×C61—22no—
216C4×C61—82no—
217Dic61—22no—
218Dic61—41no—
219Dic61—82no—
220C2×#380—611no—
221#1779—611no—
222C22⋊#95—611no—
223#1294—614no—
224C2×C61—24no—
225C2×C61C2×#32811yescharacteristic
226C2×C61#150711yescharacteristic
227C2×C61#151011yescharacteristic
228C2×C61#152011yescharacteristic
229C2×C61#181911yescharacteristic
230C2×C61#182411yescharacteristic
231C2×C61#182611yescharacteristic
232D61—22no—
233D61—22no—
234D61—42no—
235C24×C4—611no—
236C22×C4⋊C4—611no—
237C22×C4⋊C4#692311yescharacteristic, complemented
238C22×C22⋊C4—1221no—
239C22×C22⋊C4—1221no—
240C2×C22⋊C8—612no—
241C2×C22⋊C8—1221no—
242C2×C22⋊C8—1221no—
243#380—612no—
244#380—612no—
245#380—1221no—
246#380—1221no—
247C2×#95—612no—
248C2×#95—1221no—
249C2×#95—1221no—
250C61#871311yescharacteristic, complemented, Sylow
251C25—611no—
252C23×C4—21no—
253C23×C4—611no—
254C23×C4—611no—
255C23×C4—611no—
256C23×C4—611no—
257C23×C4—611no—
258C23×C4—1224no—
259C23×C4—1222no—
260C23×C4—1221no—
261C23×C4—2442no—
262C23×C4—2442no—
263C23×C4C2×#692311yescharacteristic
264C22×C8—1222no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—612no—
268C2×C4⋊C4—1221no—
269C2×C4⋊C4—2442no—
270C2×C4⋊C4C2×#692312yes—
271C2×C22⋊C4—1222no—
272C2×C22⋊C4—1222no—
273C2×C22⋊C4—1224no—
274C2×C22⋊C4—2444no—
275C2×C22⋊C4—2442no—
276C2×C22⋊C4—2442no—
277C22⋊C8—1224no—
278C22⋊C8—2442no—
279C22⋊C8—2442no—
280#95—1222no—
281#95—1222no—
282#95—1224no—
283#95—2442no—
284C24—611no—
285C24—611no—
286C24—612no—
287C24—612no—
288C24—1222no—
289C24—1222no—
290C24—1222no—
291C24—1222no—
292C24—2441no—
293C24—2441no—
294C24C22⋊#692311yescharacteristic
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—611no—
303C22×C4—614no—
304C22×C4—611no—
305C22×C4—1222no—
306C22×C4—1222no—
307C22×C4—1222no—
308C22×C4—1222no—
309C22×C4—1222no—
310C22×C4—1222no—
311C22×C4—1222no—
312C22×C4—1224no—
313C22×C4—1222no—
314C22×C4—2441no—
315C22×C4—2441no—
316C22×C4—2441no—
317C22×C4—2441no—
318C22×C4—2441no—
319C22×C4—2441no—
320C22×C4—24416no—
321C22×C4—2441no—
322C22×C4—2442no—
323C22×C4—2442no—
324C22×C4—2442no—
325C22×C4—2442no—
326C22×C4—2442no—
327C22×C4—2442no—
328C22×C4—2442no—
329C22×C4—2442no—
330C22×C4—2442no—
331C22×C4C22⋊#692311yescharacteristic
332C22×C4C22×#692311yescharacteristic
333C2×C8—2442no—
334C2×C8—2442no—
335C2×C8—2442no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—2442no—
339C4⋊C4—2442no—
340C22⋊C4—2444no—
341C22⋊C4—2444no—
342C22⋊C4—2448no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—612no—
348C23—611no—
349C23—611no—
350C23—612no—
351C23—612no—
352C23—1224no—
353C23—1224no—
354C23—1221no—
355C23—1221no—
356C23—1221no—
357C23—1221no—
358C23—1221no—
359C23—1221no—
360C23—1221no—
361C23—1221no—
362C23—1221no—
363C23—1221no—
364C23—1224no—
365C23—1224no—
366C23—2442no—
367C23—2442no—
368C23—2444no—
369C23—2444no—
370C23—2444no—
371C23—2444no—
372C23#1451983612yes—
373C23#1451984012yes—
374C23#1451984112yes—
375C23C2×C22⋊#692311yescharacteristic
376C2×C4—28no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—84no—
385C2×C4—612no—
386C2×C4—1221no—
387C2×C4—1221no—
388C2×C4—1221no—
389C2×C4—1221no—
390C2×C4—1221no—
391C2×C4—4884no—
392C2×C4—4882no—
393C2×C4—4882no—
394C2×C4—4882no—
395C2×C4—4882no—
396C2×C4—2444no—
397C2×C4—2444no—
398C2×C4—2444no—
399C2×C4—2444no—
400C2×C4—2444no—
401C2×C4—2444no—
402C2×C4—2444no—
403C2×C4—2444no—
404C2×C4—2442no—
405C2×C4—2442no—
406C2×C4—2442no—
407C2×C4#1451983512yes—
408C2×C4#1451985612yes—
409C8—4882no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—612no—
417C22—612no—
418C22—1221no—
419C22—1221no—
420C22—1221no—
421C22—1221no—
422C22—1222no—
423C22—1222no—
424C22—1222no—
425C22—1222no—
426C22—1222no—
427C22—1222no—
428C22—1222no—
429C22—2448no—
430C22—2448no—
431C22—2441no—
432C22—2441no—
433C22—2441no—
434C22—2441no—
435C22#1684061514yes—
436C22#1684070511yescharacteristic
437C22C2×#1451983511yescharacteristic
438C22C2×#1451983611yescharacteristic
439C22C2×#1451984011yescharacteristic
440C22#1684075411yescharacteristic
441C22C2×#1451984111yescharacteristic
442C22#1684076111yescharacteristic
443C4—22no—
444C4—22no—
445C4—82no—
446C4—1222no—
447C4—4884no—
448C4—4882no—
449C4—2441no—
450C2—24no—
451C2—1222no—
452C2—1222no—
453C2—2442no—
454C2C2×#1684061511yescharacteristic
455C2#2146121511yescharacteristic
456C2#2146122611yescharacteristic
457C2#2146125911yescharacteristic
458C2#2146171411yescharacteristic
459C2#2146172311yescharacteristic
460C2#2146172511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×C4⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#8713
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #21460692C2semidirect
12. C22×C4⋊Dic61C4semidirect
16. #16839619C4semidirect
19. C22×C4⋊C4×C61C4semidirect
237. C22×C4⋊C4#6923semidirect
250. C61#8713semidirect

Maximal subgroups

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

subgroupindex
#214616422
#214609932
#214606922
C2×#168396192
#871361