Finite Group Databasebeta

← #29577798

Subgroups — #29577798

Order 15616.

912 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 123 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#21454848C211yesmaximal, characteristic, complemented
2#21445807C211yesmaximal, characteristic, complemented
3#21445750C212yesmaximal, complemented
4#21443686C212yesmaximal, complemented
5C61×#2408C211yesmaximal, characteristic
6#16839834—24no—
7#16839819—21no—
8C2×#14519698—24no—
9C2×#14519692—21no—
10#16839726—21no—
11C2×C22⋊#92418C2212yes—
12#16839622C2212yes—
13#16839529—21no—
14C23×D4×C61C2211yescharacteristic
15C2×#121×C61—21no—
16C2×#120×C61—21no—
17C2×C4×D4×C61C2211yescharacteristic
18C61×#394C412yescomplemented
19C61×#392C412yescomplemented
20C61×#390—21no—
21C61×#380C2211yescharacteristic
22C61×#379—21no—
23C2×#11758194—22no—
24C22×#92418D414yescomplemented
25C2×C4⋊Dic61—22no—
26C2×C22⋊Dic61—22no—
27C4⋊#92418—22no—
28#14519698—48no—
29#14519692—42no—
30C22⋊#92418—22no—
31C22⋊#92418—22no—
32C22⋊#92418—22no—
33#14519688—22no—
34#14519684—22no—
35C25×C61—21no—
36C22×D4×C61—22no—
37C22×D4×C61—22no—
38C22×D4×C61—21no—
39C22×D4×C61—22no—
40C22×D4×C61—44no—
41C22×D4×C61C2×C412yes—
42C22×D4×C61D411yescharacteristic
43C23×C4×C61D411yescharacteristic
44C23×C4×C61C2311yescharacteristic
45#121×C61—44no—
46#120×C61—44no—
47C4×D4×C61—22no—
48C4×D4×C61—22no—
49C2×C4⋊C4×C61—21no—
50C2×C4⋊C4×C61C2×C411yescharacteristic
51C2×C22⋊C4×C61—21no—
52C2×C22⋊C4×C61—22no—
53C2×C22⋊C4×C61—22no—
54C2×C22⋊C4×C61—22no—
55C2×C22⋊C4×C61C2×C411yescharacteristic
56C2×C22⋊C4×C61D411yescharacteristic
57C2×C42×C61D411yescharacteristic
58#95×C61C2×C412yes—
59C24×C61—22no—
60C24×C61—44no—
61C24×C61—42no—
62C24×C61—42no—
63C24×C61—41no—
64C24×C61—41no—
65C24×C61—82no—
66C24×C61C22⋊C411yescharacteristic
67C24×C61C2×D411yescharacteristic
68C2×D4×C61—28no—
69C2×D4×C61—24no—
70C2×D4×C61—22no—
71C2×D4×C61—22no—
72C2×D4×C61—21no—
73C2×D4×C61—21no—
74C2×D4×C61—24no—
75C2×D4×C61—44no—
76C2×D4×C61—42no—
77C2×D4×C61—44no—
78C2×D4×C61—48no—
79C2×D4×C61—44no—
80C2×D4×C61—44no—
81C2×D4×C61—41no—
82C2×D4×C61—41no—
83C2×D4×C61—82no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—22no—
87C22×C4×C61—42no—
88C22×C4×C61—42no—
89C22×C4×C61C22⋊C411yescharacteristic
90C22×C4×C61C22⋊C411yescharacteristic
91C22×C4×C61C22⋊C411yescharacteristic
92C22×C4×C61C22×C411yescharacteristic
93C22×C4×C61C2×D412yes—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61#4112yes—
96C22×Dic61—42no—
97C4⋊C4×C61—22no—
98C4⋊C4×C61—42no—
99C22⋊C4×C61—22no—
100C22⋊C4×C61—22no—
101C22⋊C4×C61—44no—
102C22⋊C4×C61—44no—
103C22⋊C4×C61—42no—
104C22⋊C4×C61—82no—
105C42×C61—21no—
106C42×C61—21no—
107C22⋊Dic61—42no—
108C22⋊Dic61—42no—
109C4⋊Dic61—44no—
110#11758194—82no—
111C2×#92418—24no—
112C2×#92418—44no—
113C23×C61—24no—
114C23×C61—21no—
115C23×C61—21no—
116C23×C61—21no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61—42no—
122C23×C61—48no—
123C23×C61—42no—
124C23×C61—42no—
125C23×C61—42no—
126C23×C61—42no—
127C23×C61—42no—
128C23×C61—84no—
129C23×C61—88no—
130C23×C61—84no—
131C23×C61—84no—
132C23×C61—82no—
133C23×C61—82no—
134C23×C61C2×C22⋊C411yescharacteristic
135C23×C61#12112yes—
136C23×C61AGL(1,ℤ/8)11yescharacteristic
137C23×C61AGL(1,ℤ/8)12yes—
138C23×C61#13711yescharacteristic
139D4×C61—28no—
140D4×C61—44no—
141D4×C61—44no—
142D4×C61—42no—
143D4×C61—42no—
144D4×C61—44no—
145D4×C61—88no—
146C2×C4×C61—24no—
147C2×C4×C61—22no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—41no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—81no—
162C2×C4×C61C4×D412yes—
163C2×C4×C61#12011yescharacteristic
164C2×C4×C61#12311yescharacteristic
165C2×Dic61—82no—
166C2×Dic61—82no—
167C2×Dic61—82no—
168#92418—44no—
169#9472—611nomaximal, Sylow
170C22×C61—22no—
171C22×C61—21no—
172C22×C61—22no—
173C22×C61—21no—
174C22×C61—41no—
175C22×C61—42no—
176C22×C61—41no—
177C22×C61—42no—
178C22×C61—44no—
179C22×C61—44no—
180C22×C61—41no—
181C22×C61—44no—
182C22×C61—41no—
183C22×C61—41no—
184C22×C61—81no—
185C22×C61—82no—
186C22×C61—82no—
187C22×C61—88no—
188C22×C61—88no—
189C22×C61—82no—
190C22×C61—82no—
191C22×C61#38611yescharacteristic
192C22×C61#41711yescharacteristic
193C22×C61#41811yescharacteristic
194C22×C61#46812yes—
195C22×C61#46912yes—
196C4×C61—21no—
197C4×C61—22no—
198C4×C61—41no—
199C4×C61—41no—
200C4×C61—41no—
201C4×C61—82no—
202C4×C61B(4,𝔽2)12yes—
203Dic61—162no—
204#2973—611no—
205#2408D6111yescharacteristic
206#1799—611no—
207#1786—612no—
208#1289—612no—
209C2×C61—22no—
210C2×C61—21no—
211C2×C61—21no—
212C2×C61—22no—
213C2×C61—81no—
214C2×C61—84no—
215C2×C61#188911yescharacteristic
216C2×C61#189511yescharacteristic
217C2×C61#191011yescharacteristic
218C23×D4C2×D6111yescharacteristic
219C2×#121—21no—
220C2×#120—21no—
221C2×C4×D4C2×D6111yescharacteristic
222#435—1224no—
223#432—1221no—
224C2×#102—1224no—
225C2×#99—1221no—
226#407—1221no—
227C2×C22⋊C8—612no—
228#394Dic6112yescomplemented
229#392Dic6112yescomplemented
230#390—21no—
231#380—612no—
232#380C2×D6111yescharacteristic
233#379—21no—
234#342—1221no—
235C61#947211yescharacteristic, complemented, Sylow
236C25—21no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—44no—
242C22×D4C2×Dic6112yes—
243C22×D4C22⋊D6111yescharacteristic
244C23×C4C22⋊D6111yescharacteristic
245C23×C4C22×D6111yescharacteristic
246C2×M16—1222no—
247C22×C8—614no—
248#121—44no—
249#120—44no—
250C4×D4—22no—
251C4×D4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—1222no—
254C2×C4⋊C4C2×Dic6111yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—1222no—
260C2×C22⋊C4C2×Dic6111yescharacteristic
261C2×C22⋊C4C22⋊D6111yescharacteristic
262C2×C42C22⋊D6111yescharacteristic
263C4⋊C8—1222no—
264#102—2448no—
265#99—2442no—
266C22⋊C8—1222no—
267C22⋊C8—1222no—
268C22⋊C8—1222no—
269#97—1222no—
270#95—1222no—
271#95C2×Dic6112yes—
272C24—22no—
273C24—44no—
274C24—42no—
275C24—41no—
276C24—42no—
277C24—41no—
278C24—82no—
279C24C22⋊Dic6111yescharacteristic
280C24C2×C22⋊D6111yescharacteristic
281C2×D4—28no—
282C2×D4—24no—
283C2×D4—22no—
284C2×D4—21no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—24no—
288C2×D4—42no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—48no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—82no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—2442no—
303C22×C4C22⋊Dic6111yescharacteristic
304C22×C4C22⋊Dic6111yescharacteristic
305C22×C4C22⋊Dic6111yescharacteristic
306C22×C4D4×D6112yes—
307C22×C4#1175822412yes—
308C22×C4C22×Dic6111yescharacteristic
309C22×C4C2×C22⋊D6111yescharacteristic
310M16—4882no—
311C2×C8—1224no—
312C2×C8—2444no—
313C4⋊C4—22no—
314C4⋊C4—42no—
315C4⋊C4—2444no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—42no—
321C22⋊C4—82no—
322C22⋊C4—2442no—
323C22⋊C4—2442no—
324C42—21no—
325C42—21no—
326C23—24no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—42no—
335C23—48no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—84no—
342C23—84no—
343C23—88no—
344C23—84no—
345C23—82no—
346C23—82no—
347C23C2×C22⋊Dic6111yescharacteristic
348C23#1451974912yes—
349C23#1451980211yescharacteristic
350C23#1451980312yes—
351C23#1451980711yescharacteristic
352D4—28no—
353D4—44no—
354D4—44no—
355D4—42no—
356D4—42no—
357D4—44no—
358D4—88no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—81no—
375C2×C4—4882no—
376C2×C4—4882no—
377C2×C4—4882no—
378C2×C4D4×Dic6112yes—
379C2×C4#1451974511yescharacteristic
380C2×C4#1451975811yescharacteristic
381C8—2444no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—22no—
386C22—41no—
387C22—42no—
388C22—41no—
389C22—42no—
390C22—44no—
391C22—44no—
392C22—41no—
393C22—44no—
394C22—41no—
395C22—41no—
396C22—81no—
397C22—82no—
398C22—82no—
399C22—88no—
400C22—88no—
401C22—82no—
402C22—82no—
403C22#1683964211yescharacteristic
404C22#1683976311yescharacteristic
405C22#1683976711yescharacteristic
406C22#1684000612yes—
407C22#1684001312yes—
408C4—21no—
409C4—22no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—82no—
414C4—9762no—
415C4#1683994812yes—
416C2—21no—
417C2—22no—
418C2—21no—
419C2—22no—
420C2—81no—
421C2—84no—
422C2#2144622811yescharacteristic
423C2#2144625811yescharacteristic
424C2#2144634711yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
Fitting, prime radicalC61×#2408C2
socleC22×C61#386
cosocleC23×C4×C61C23
hypercenter#2408D61
hypocenterC61#9472
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. #21454848C2semidirect
2. #21445807C2semidirect
3. #21445750C2semidirect
4. #21443686C2semidirect
18. C61×#394C4semidirect
19. C61×#392C4semidirect
24. C22×#92418D4semidirect
228. #394Dic61semidirect
229. #392Dic61semidirect
235. C61#9472semidirect

Maximal subgroups

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

subgroupindex
#214548482
#214458072
#214457502
#214436862
C61×#24082
#947261