Finite Group Databasebeta

← #29576054

Subgroups — #29576054

Order 15616.

684 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 127 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#21451624C211yesmaximal, characteristic
2#21448433C211yesmaximal, characteristic
3#21444392C212yesmaximal, complemented
4#21443687C212yesmaximal, complemented
5C2×C61×#323C211yesmaximal, characteristic
6C2×#14519756C2211yescharacteristic
7C22×C22⋊Dic61—21no—
8#16839698C412yescomplemented
9#16839685—22no—
10#16839674C412yescomplemented
11#16839637—24no—
12#16839635—21no—
13#16839622—21no—
14#16839622C2211yescharacteristic
15#16839622C2212yes—
16#16839618—21no—
17#16839500—24no—
18C22×C22⋊C4×C61C2211yescharacteristic
19C2×C22⋊C8×C61C2212yes—
20C61×#323—22no—
21C22×Dic122C2×C411yescharacteristic, complemented
22C23×Dic61—22no—
23#14519756—22no—
24#14519756—22no—
25C2×C4⋊Dic61—22no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61—22no—
28C2×C4⋊Dic61C2×C411yescharacteristic, complemented
29C2×#11758196—21no—
30C2×#11758196—22no—
31C2×C22⋊Dic61—22no—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61—22no—
36C2×C22⋊Dic61—22no—
37C2×C22⋊Dic61—42no—
38C2×C22⋊Dic61—42no—
39#14519684—22no—
40#14519684—22no—
41#14519684C2×C412yescomplemented
42#14519684D414yes—
43C25×C61D411yescharacteristic
44C23×C4×C61D411yescharacteristic
45C23×C4×C61C2311yescharacteristic
46C22×C8×C61—22no—
47C2×C22⋊C4×C61—22no—
48C2×C22⋊C4×C61—22no—
49C2×C22⋊C4×C61C2×C412yes—
50C2×C22⋊C4×C61D412yes—
51C22⋊C8×C61—22no—
52C22⋊C8×C61—22no—
53C24×C61—21no—
54C24×C61—21no—
55C24×C61—21no—
56C24×C61—21no—
57C24×C61—81no—
58C24×C61—81no—
59C24×C61C22⋊C412yes—
60C24×C61SD1612yes—
61C24×C61Q1612yes—
62C24×C61C2×D411yescharacteristic
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—81no—
66C22×C4×C61—81no—
67C22×C4×C61C22⋊C412yes—
68C22×C4×C61C22×C411yescharacteristic
69C22×C4×C61C2×D412yes—
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61#4112yes—
72C22×Dic61—21no—
73C22×Dic61—22no—
74C22×Dic61—21no—
75C22×Dic61—24no—
76C22×Dic61—44no—
77C22×Dic61—82no—
78C22×Dic61—82no—
79C2×Dic122—22no—
80C2×Dic122—81no—
81C2×C8×C61—42no—
82C2×C8×C61—82no—
83C22⋊C4×C61—24no—
84C22⋊C4×C61—24no—
85C22⋊Dic61—44no—
86C22⋊Dic61—41no—
87C22⋊Dic61—41no—
88C22⋊Dic61—42no—
89C22⋊Dic61—41no—
90C22⋊Dic61—42no—
91C22⋊Dic61—41no—
92C22⋊Dic61—82no—
93C22⋊Dic61—82no—
94C4⋊Dic61—22no—
95C4⋊Dic61—81no—
96C4⋊Dic61—82no—
97C4⋊Dic61—82no—
98#11758196—42no—
99#11758196—82no—
100C23×C61—22no—
101C23×C61—22no—
102C23×C61—22no—
103C23×C61—22no—
104C23×C61—22no—
105C23×C61—22no—
106C23×C61—41no—
107C23×C61—41no—
108C23×C61—42no—
109C23×C61—42no—
110C23×C61—42no—
111C23×C61—42no—
112C23×C61—81no—
113C23×C61—81no—
114C23×C61—82no—
115C23×C61—82no—
116C23×C61—82no—
117C23×C61—82no—
118C23×C61#10314yes—
119C23×C61C2×C22⋊C411yescharacteristic
120C23×C61#12011yescharacteristic
121C23×C61#12311yescharacteristic
122C23×C61C2×SD1611yescharacteristic
123C23×C61C2×Q1611yescharacteristic
124C23×C61AGL(1,ℤ/8)11yescharacteristic
125C23×C61#13711yescharacteristic
126C2×C4×C61—22no—
127C2×C4×C61—41no—
128C2×C4×C61—42no—
129C2×C4×C61—42no—
130C2×C4×C61—41no—
131C2×C4×C61—42no—
132C2×C4×C61—42no—
133C2×C4×C61—81no—
134C2×C4×C61C4×D412yes—
135C2×C4×C61#12112yes—
136C2×Dic61—42no—
137C2×Dic61—42no—
138C2×Dic61—42no—
139C2×Dic61—44no—
140C2×Dic61—44no—
141C2×Dic61—42no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—42no—
145C2×Dic61—44no—
146C2×Dic61—88no—
147Dic122—82no—
148C8×C61—82no—
149#9199—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—21no—
153C22×C61—21no—
154C22×C61—21no—
155C22×C61—21no—
156C22×C61—24no—
157C22×C61—41no—
158C22×C61—41no—
159C22×C61—41no—
160C22×C61—41no—
161C22×C61—44no—
162C22×C61—44no—
163C22×C61—44no—
164C22×C61—42no—
165C22×C61—42no—
166C22×C61—82no—
167C22×C61—82no—
168C22×C61—82no—
169C22×C61#38611yescharacteristic
170C22×C61C2×#10311yescharacteristic
171C22×C61#41711yescharacteristic
172C22×C61#45011yescharacteristic
173C22×C61#45111yescharacteristic
174C22×C61#48311yescharacteristic
175C22×C61#48411yescharacteristic
176C4×C61—22no—
177C4×C61—22no—
178C4×C61—42no—
179Dic61—82no—
180Dic61—84no—
181Dic61—81no—
182Dic61—81no—
183#2626—611no—
184#2315—611no—
185#1520—612no—
186C2×#323D6111yescharacteristic
187#1289—612no—
188C2×C61—21no—
189C2×C61—21no—
190C2×C61—21no—
191C2×C61—21no—
192C2×C61—21no—
193C2×C61—21no—
194C2×C61—44no—
195C2×C61#187211yescharacteristic
196C2×C61#191111yescharacteristic
197C2×C61#191311yescharacteristic
198C2×C22⋊Q8—611no—
199C22×C22⋊C4—1221no—
200C22×C22⋊C4C2×D6111yescharacteristic
201C2×C22⋊C8C2×D6112yes—
202#399—612no—
203#396—1222no—
204#393—612no—
205#385—1224no—
206#385—1221no—
207#380—611no—
208#380—612no—
209#380—1221no—
210#379—1221no—
211#328—1224no—
212#323—22no—
213C61#919911yescharacteristic, complemented, Sylow
214C25D24411yescharacteristic
215C22×Q8—611no—
216C23×C4—1222no—
217C23×C4D24411yescharacteristic
218C23×C4C22×D6111yescharacteristic
219C22×C8—22no—
220C22⋊Q8—1222no—
221C22⋊Q8—1222no—
222C2×C4⋊C4—611no—
223C2×C4⋊C4—1222no—
224C2×C4⋊C4—1222no—
225C2×C4⋊C4—1222no—
226C2×C4⋊C4—1221no—
227C2×C4⋊C4—1221no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—1222no—
231C2×C22⋊C4—1222no—
232C2×C22⋊C4—1222no—
233C2×C22⋊C4—1222no—
234C2×C22⋊C4—1221no—
235C2×C22⋊C4—1221no—
236C2×C22⋊C4—2442no—
237C2×C22⋊C4—2442no—
238C2×C22⋊C4C4×D6112yes—
239C2×C22⋊C4C22⋊D6112yes—
240C22⋊C8—22no—
241C22⋊C8—22no—
242#95—612no—
243#95—614no—
244#95—1222no—
245#95—1222no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—81no—
251C24—81no—
252C24#1175819012yes—
253C24Dic24412yes—
254C24#1175819812yes—
255C24C2×D24411yescharacteristic
256C2×Q8—1222no—
257C2×Q8—4881no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—1222no—
263C22×C4—1221no—
264C22×C4—1221no—
265C22×C4—1224no—
266C22×C4—4882no—
267C22×C4—4882no—
268C22×C4—2444no—
269C22×C4#1175819812yes—
270C22×C4C2×C4×D6111yescharacteristic
271C22×C4D4×D6112yes—
272C22×C4#1175822412yes—
273C22×C4C2×C22⋊D6111yescharacteristic
274C2×C8—42no—
275C2×C8—82no—
276C4⋊C4—1222no—
277C4⋊C4—4882no—
278C4⋊C4—4882no—
279C4⋊C4—4881no—
280C4⋊C4—4882no—
281C4⋊C4—2442no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—4882no—
285C22⋊C4—4882no—
286C22⋊C4—2444no—
287C22⋊C4—2441no—
288C22⋊C4—2441no—
289C22⋊C4—2441no—
290C22⋊C4—2442no—
291C22⋊C4—2442no—
292C22⋊C4—2441no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—41no—
300C23—41no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—81no—
306C23—81no—
307C23—82no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23#1451970214yes—
312C23C2×#1175819811yescharacteristic
313C23#1451974711yescharacteristic
314C23#1451976111yescharacteristic
315C23C2×#1175819011yescharacteristic
316C23C2×Dic24411yescharacteristic
317C23#1451980511yescharacteristic
318C23#1451981011yescharacteristic
319Q8—4882no—
320C2×C4—22no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—81no—
328C2×C4—4888no—
329C2×C4—2442no—
330C2×C4—2442no—
331C2×C4—2442no—
332C2×C4—2442no—
333C2×C4—2444no—
334C2×C4—2444no—
335C2×C4—2442no—
336C2×C4—2441no—
337C2×C4—2441no—
338C2×C4—2444no—
339C2×C4#1451973712yes—
340C2×C4#1451974912yes—
341C8—82no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—24no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22#1683963911yescharacteristic
362C22C2×#1451970211yescharacteristic
363C22#1683976411yescharacteristic
364C22#1683991511yescharacteristic
365C22#1683992011yescharacteristic
366C22#1684009311yescharacteristic
367C22#1684009811yescharacteristic
368C4—22no—
369C4—22no—
370C4—42no—
371C4—4884no—
372C4—4882no—
373C4—4881no—
374C4—4881no—
375C2—21no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#2144612411yescharacteristic
383C2#2144634911yescharacteristic
384C2#2144635911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#323C2
socleC22×C61#386
prime radicalC25×C61D4
cosocleC23×C4×C61C23
hypercenterC2×#323D61
hypocenterC61#9199
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #21444392C2semidirect
4. #21443687C2semidirect
8. #16839698C4semidirect
10. #16839674C4semidirect
21. C22×Dic122C2×C4semidirect
28. C2×C4⋊Dic61C2×C4semidirect
41. #14519684C2×C4semidirect
213. C61#9199semidirect

Maximal subgroups

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

subgroupindex
#214516242
#214484332
#214443922
#214436872
C2×C61×#3232
#919961