Finite Group Databasebeta

← #29575558

Subgroups — #29575558

Order 15616.

648 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 119 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#21448473C211yesmaximal, characteristic
2#21444137C211yesmaximal, characteristic, complemented
3C2×#16839485C212yesmaximal, complemented
4#21443665C212yesmaximal, complemented
5C61×#2973C211yesmaximal, characteristic, complemented
6C2×#14519745C411yescharacteristic
7C22×#11758198C2211yescharacteristic
8C2×C22⋊#92418C2212yes—
9#16839662—21no—
10#16839653—21no—
11#16839652—21no—
12#16839642C411yescharacteristic
13#16839641—21no—
14#16839634C411yescharacteristic
15#16839617C411yescharacteristic
16C4×C22⋊Dic61C2211yescharacteristic
17#16839488—21no—
18#16839486—21no—
19#16839485—24no—
20C2×C4×D4×C61C2211yescharacteristic
21C61×#435—24no—
22C61×#432—21no—
23C61×#407—21no—
24C2×C22⋊C8×C61C2212yes—
25C24×D61C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27C23×Dic61C2×C411yescharacteristic
28C22×#92418—22no—
29#14519745—22no—
30#14519745—22no—
31C2×#11758196—21no—
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61C2×C411yescharacteristic
34C2×C22⋊Dic61C2×C411yescharacteristic
35C2×#11758198—21no—
36C2×#11758198—22no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×#11758198—22no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×C4×Dic61—21no—
43C22⋊#92418—22no—
44C22⋊#92418—22no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48C22×D4×C61C2×C411yescharacteristic
49C23×C4×C61D411yescharacteristic
50C23×C4×C61C2311yescharacteristic
51C2×M16×C61—22no—
52C22×C8×C61—22no—
53C4×D4×C61—22no—
54C4×D4×C61—22no—
55C2×C4⋊C4×C61D411yescharacteristic
56C2×C22⋊C4×C61C2×C411yescharacteristic
57C2×C22⋊C4×C61D411yescharacteristic
58C2×C42×C61D411yescharacteristic
59C4⋊C8×C61—22no—
60C22⋊C8×C61—22no—
61C22⋊C8×C61—22no—
62C22⋊C8×C61D414yescomplemented
63#97×C61—22no—
64C24×C61C22⋊C411yescharacteristic
65C24×C61C22×C411yescharacteristic
66C23×D61—21no—
67C23×D61—22no—
68C23×D61—21no—
69C23×D61—22no—
70C23×D61—81no—
71C23×D61—81no—
72C23×D61M1612yescomplemented
73C2×D4×C61—22no—
74C2×D4×C61—21no—
75C2×D4×C61—21no—
76C2×D4×C61—22no—
77C22×C4×C61—21no—
78C22×C4×C61—21no—
79C22×C4×C61—81no—
80C22×C4×C61—81no—
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22⋊C411yescharacteristic
84C22×C4×C61C2×D412yes—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61#4112yes—
88C2×C22⋊D61—24no—
89C2×C22⋊D61—81no—
90C2×C22⋊D61—81no—
91C22×Dic61—21no—
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61—22no—
95C22×Dic61—81no—
96M16×C61—82no—
97C2×C8×C61—24no—
98C2×C8×C61—82no—
99C4⋊C4×C61—22no—
100C22⋊C4×C61—22no—
101C22⋊C4×C61—22no—
102C42×C61—21no—
103C42×C61—21no—
104C22⋊Dic61—21no—
105C22⋊Dic61—22no—
106C22⋊Dic61—81no—
107C22⋊Dic61M1612yescomplemented
108#11758198—24no—
109#11758198—41no—
110#11758198—42no—
111#11758198—41no—
112#11758198—81no—
113#11758198—81no—
114#11758196—81no—
115C4×Dic61—81no—
116C2×#92418—42no—
117C2×#92418—82no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61C2×C22⋊C411yescharacteristic
126C23×C61C4×D412yes—
127C23×C61C2×M1611yescharacteristic
128C23×C61C2×M1611yescharacteristic
129C23×C61#13112yes—
130C22×D61—22no—
131C22×D61—24no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135C22×D61—41no—
136C22×D61—41no—
137C22×D61—41no—
138C22×D61—42no—
139C22×D61—82no—
140C22×D61—82no—
141C22×D61—82no—
142C22×D61—82no—
143C22×D61—84no—
144D4×C61—42no—
145D4×C61—42no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—42no—
152C2×C4×C61—41no—
153C2×C4×C61—42no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—81no—
159C2×C4×C61#12011yescharacteristic
160C2×C4×C61#12112yes—
161C2×C4×C61#12311yescharacteristic
162C22⋊D61—84no—
163C2×Dic61—22no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—82no—
173C8×C61—44no—
174#92418—82no—
175#9109—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—22no—
178C22×C61—21no—
179C22×C61—22no—
180C22×C61—41no—
181C22×C61—41no—
182C22×C61—42no—
183C22×C61—42no—
184C22×C61—41no—
185C22×C61—41no—
186C22×C61—41no—
187C22×C61—81no—
188C22×C61#38611yescharacteristic
189C22×C61#40711yescharacteristic
190C22×C61#40811yescharacteristic
191C22×C61#43512yes—
192C22×C61#43612yes—
193C2×D61—42no—
194C2×D61—42no—
195C2×D61—42no—
196C2×D61—81no—
197C2×D61—81no—
198C2×D61—84no—
199C2×D61—81no—
200C2×D61—81no—
201C2×D61—84no—
202C4×C61—22no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—41no—
206C4×C61—41no—
207C4×C61—41no—
208Dic61—42no—
209Dic61—81no—
210Dic61—81no—
211#2973D6111yescharacteristic, complemented
212#2321—611no—
213#1459—611no—
214C2×#323—612no—
215#1282—612no—
216C2×C61—21no—
217C2×C61—21no—
218C2×C61—22no—
219C2×C61—22no—
220C2×C61—81no—
221C2×C61#187011yescharacteristic
222C2×C61#191311yescharacteristic
223C2×C61#191411yescharacteristic
224D61—82no—
225C2×#120—611no—
226C2×C4×D4C2×D6111yescharacteristic
227C22×C22⋊C4—611no—
228#435—24no—
229#432—21no—
230#407—21no—
231C2×C22⋊C8—612no—
232C2×C22⋊C8C2×D6112yes—
233#390—1221no—
234#388—1221no—
235#388—1221no—
236#386—611no—
237#386—1221no—
238#385—611no—
239#379—611no—
240C4×C22⋊C4—611no—
241#324—1221no—
242#323—1221no—
243#323—1224no—
244C61#910911yescharacteristic, complemented, Sylow
245C25—611no—
246C22×D4—1221no—
247C22×D4C4×D6111yescharacteristic
248C23×C4—611no—
249C23×C4C22⋊D6111yescharacteristic
250C23×C4C22×D6111yescharacteristic
251C2×M16—22no—
252C22×C8—22no—
253C22×C8—1222no—
254#120—1222no—
255#120—1222no—
256C4×D4—22no—
257C4×D4—22no—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4D24411yescharacteristic
260C2×C22⋊C4—611no—
261C2×C22⋊C4—611no—
262C2×C22⋊C4—1221no—
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1222no—
268C2×C22⋊C4—1222no—
269C2×C22⋊C4—1221no—
270C2×C22⋊C4C4×D6111yescharacteristic
271C2×C22⋊C4D24411yescharacteristic
272C2×C42—1221no—
273C2×C42C22⋊D6111yescharacteristic
274C4⋊C8—22no—
275C22⋊C8—22no—
276C22⋊C8—22no—
277C22⋊C8—1222no—
278C22⋊C8—1222no—
279C22⋊C8C22⋊D6114yescomplemented
280#97—22no—
281#95—1221no—
282#95—1221no—
283#95—1221no—
284C24—612no—
285C24—1222no—
286C24—1221no—
287C24—1222no—
288C24—1221no—
289C24—4881no—
290C24—4881no—
291C24#1175819811yescharacteristic
292C24C2×C4×D6111yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—22no—
297C2×D4—1224no—
298C2×D4—4881no—
299C2×D4—4881no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—81no—
303C22×C4—81no—
304C22×C4—1221no—
305C22×C4—1222no—
306C22×C4—1221no—
307C22×C4—1221no—
308C22×C4—4881no—
309C22×C4#1175819811yescharacteristic
310C22×C4#1175819811yescharacteristic
311C22×C4#1175819811yescharacteristic
312C22×C4C2×D24411yescharacteristic
313C22×C4#1175822212yes—
314C22×C4C2×C22⋊D6112yes—
315C22×C4C2×C22⋊D6111yescharacteristic
316M16—82no—
317C2×C8—24no—
318C2×C8—82no—
319C2×C8—4882no—
320C2×C8—2442no—
321C4⋊C4—22no—
322C4⋊C4—4881no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—612no—
326C22⋊C4—1224no—
327C22⋊C4—1221no—
328C22⋊C4—1222no—
329C22⋊C4—4881no—
330C22⋊C4—4881no—
331C22⋊C4—4881no—
332C22⋊C4—2441no—
333C22⋊C4—2442no—
334C22⋊C4—2441no—
335C42—21no—
336C42—21no—
337C42—4881no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—1224no—
346C23—1222no—
347C23—4882no—
348C23—4882no—
349C23—4884no—
350C23—4882no—
351C23—4882no—
352C23—2441no—
353C23—2441no—
354C23—2441no—
355C23—2441no—
356C23—2441no—
357C23—2442no—
358C23—2441no—
359C23C2×#1175819811yescharacteristic
360C23C4×C22⋊D6112yes—
361C23M16×D6111yescharacteristic
362C23M16×D6111yescharacteristic
363C23#1451978512yes—
364D4—42no—
365D4—42no—
366D4—4884no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—81no—
380C2×C4—1222no—
381C2×C4—4882no—
382C2×C4—2441no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4#1451974611yescharacteristic
391C2×C4#1451975112yes—
392C2×C4#1451976011yescharacteristic
393C8—44no—
394C8—4882no—
395C22—21no—
396C22—22no—
397C22—21no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—81no—
407C22—4884no—
408C22—4884no—
409C22—4881no—
410C22—4881no—
411C22—4881no—
412C22—4881no—
413C22—2442no—
414C22—2442no—
415C22—2442no—
416C22#1683964111yescharacteristic
417C22#1683972511yescharacteristic
418C22#1683973011yescharacteristic
419C22#1683983112yes—
420C22#1683983812yes—
421C4—21no—
422C4—22no—
423C4—21no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—4881no—
428C4—4881no—
429C4—2442no—
430C2—21no—
431C2—21no—
432C2—22no—
433C2—22no—
434C2—81no—
435C2—4882no—
436C2#2144611611yescharacteristic
437C2#2144636111yescharacteristic
438C2#2144636611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2973C2
socleC22×C61#386
prime radicalC2×#14519745C4
cosocleC23×C4×C61C23
hypercenter#2973D61
hypocenterC61#9109
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #21444137C2semidirect
3. C2×#16839485C2semidirect
4. #21443665C2semidirect
5. C61×#2973C2semidirect
62. C22⋊C8×C61D4semidirect
72. C23×D61M16semidirect
107. C22⋊Dic61M16semidirect
211. #2973D61semidirect
244. C61#9109semidirect
279. C22⋊C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214484732
#214441372
C2×#168394852
#214436652
C61×#29732
#910961