Finite Group Databasebeta

← #29577455

Subgroups — #29577455

Order 15616.

840 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 97 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#21450751C211yesmaximal, characteristic
2#21448171C211yesmaximal, characteristic, complemented
3#21444335C211yesmaximal, characteristic, complemented
4#21444145C211yesmaximal, characteristic, complemented
5#21443705C211yesmaximal, characteristic, complemented
6#21443704C211yesmaximal, characteristic, complemented
7C61×#1507C211yesmaximal, characteristic, complemented
8C2×#14519758—21no—
9C2×#14519745—21no—
10C22×#11758198C2211yescharacteristic
11C2×C22⋊C4×D61—21no—
12#16839726C2211yescharacteristic
13#16839697C412yescomplemented
14#16839670C412yescomplemented
15#16839662—22no—
16C22⋊C4×Dic61—21no—
17C4×#11758198—22no—
18C4×C22⋊Dic61C2211yescharacteristic
19C4×C22⋊Dic61C2211yescharacteristic
20#16839609—22no—
21#16839496—22no—
22#16839485—22no—
23C2×#121×C61C2211yescharacteristic
24C61×#407C2211yescharacteristic
25C4×C22⋊C4×C61C2211yescharacteristic
26C61×#327—22no—
27C24×D61D411yescharacteristic
28C22×C22⋊D61—22no—
29C23×Dic61D411yescharacteristic
30C22×C4×D61—22no—
31C2×#11758194—21no—
32#14519758—44no—
33#14519745—44no—
34C2×#11758196—22no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—22no—
38C2×#11758198—22no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—22no—
44C2×#11758198C2×C412yes—
45C2×#11758198D412yes—
46C22⋊C4×D61—28no—
47C2×C4×Dic61—22no—
48C2×C4×Dic61—21no—
49C2×C4×Dic61—21no—
50C22⋊#92418—21no—
51C22⋊#92418—21no—
52#14519684—22no—
53#14519684C2×C412yescomplemented
54#14519684D412yescomplemented
55C22×D4×C61—21no—
56C22×D4×C61C2×C411yescharacteristic, complemented
57C23×C4×C61C2311yescharacteristic
58C2×M16×C61—21no—
59#121×C61—21no—
60#121×C61—21no—
61#121×C61—21no—
62#121×C61—21no—
63C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
64C2×C22⋊C4×C61—21no—
65C2×C22⋊C4×C61—21no—
66C2×C42×C61—21no—
67C22⋊C8×C61—21no—
68C22⋊C8×C61—21no—
69#95×C61D412yescomplemented
70C24×C61—21no—
71C24×C61C2×D411yescharacteristic
72C23×D61—21no—
73C23×D61—21no—
74C23×D61—24no—
75C23×D61—42no—
76C23×D61—81no—
77C23×D61C22⋊C412yes—
78C2×D4×C61—21no—
79C2×D4×C61—21no—
80C2×D4×C61—42no—
81C2×D4×C61—42no—
82C2×D4×C61—41no—
83C2×D4×C61—41no—
84C2×D4×C61—81no—
85C22×C4×C61—22no—
86C22×C4×C61—21no—
87C22×C4×C61—81no—
88C22×C4×C61C22×C411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61#4111yescharacteristic
93C22×C4×C61#4111yescharacteristic
94C2×C22⋊D61—44no—
95C2×C22⋊D61—44no—
96C2×C22⋊D61—82no—
97C22×Dic61—22no—
98C22×Dic61—22no—
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61—42no—
102C22×Dic61C22⋊C412yes—
103C2×C4×D61—28no—
104C2×C4×D61—82no—
105C2×C4×D61—82no—
106M16×C61—81no—
107C2×C8×C61—41no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C22⋊C4×C61—24no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113C42×C61—81no—
114C22⋊Dic61—24no—
115C22⋊Dic61—24no—
116C22⋊Dic61—44no—
117#11758198—28no—
118#11758198—22no—
119#11758198—22no—
120#11758198—22no—
121#11758198—22no—
122#11758198—44no—
123#11758196—44no—
124C4×Dic61—81no—
125C4×Dic61—81no—
126C4×Dic61—82no—
127#11758194—81no—
128C2×#92418—41no—
129C23×C61—22no—
130C23×C61—22no—
131C23×C61—41no—
132C23×C61—41no—
133C23×C61—41no—
134C23×C61—42no—
135C23×C61—41no—
136C23×C61—42no—
137C23×C61C2×C22⋊C411yescharacteristic
138C23×C61#12011yescharacteristic
139C23×C61#12311yescharacteristic
140C22×D61—22no—
141C22×D61—22no—
142C22×D61—22no—
143C22×D61—24no—
144C22×D61—22no—
145C22×D61—42no—
146C22×D61—42no—
147C22×D61—42no—
148C22×D61—42no—
149C22×D61—88no—
150C22×D61—81no—
151C22×D61—81no—
152C22×D61#10414yes—
153D4×C61—81no—
154D4×C61—81no—
155D4×C61—82no—
156D4×C61—82no—
157C2×C4×C61—24no—
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×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61C4×D411yescharacteristic
170C2×C4×C61C4×D411yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12111yescharacteristic
173C22⋊D61—88no—
174C2×Dic61—24no—
175C2×Dic61—24no—
176C2×Dic61—22no—
177C2×Dic61—42no—
178C2×Dic61—42no—
179C2×Dic61—42no—
180C2×Dic61—42no—
181C2×Dic61—42no—
182C2×Dic61—42no—
183C2×Dic61—81no—
184C4×D61—88no—
185C8×C61—81no—
186#92418—81no—
187#9409—611nomaximal, Sylow
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—44no—
196C22×C61—82no—
197C22×C61—82no—
198C22×C61—81no—
199C22×C61—81no—
200C22×C61—81no—
201C22×C61#38611yescharacteristic
202C22×C61C2×#10412yes—
203C22×C61#45312yes—
204C2×D61—24no—
205C2×D61—44no—
206C2×D61—44no—
207C2×D61—42no—
208C2×D61—42no—
209C2×D61—44no—
210C2×D61—82no—
211C2×D61—84no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—44no—
217C4×C61—81no—
218Dic61—44no—
219Dic61—44no—
220Dic61—42no—
221#2539—611no—
222#2273—611no—
223#1507—611no—
224#1507D6111yescharacteristic, complemented
225#1462—611no—
226#1297—611no—
227#1297—611no—
228C2×C61—22no—
229C2×C61—22no—
230C2×C61—41no—
231C2×C61—161no—
232C2×C61#188711yescharacteristic
233C2×C61#189611yescharacteristic
234C2×C61#191311yescharacteristic
235D61—44no—
236C2×#123—1221no—
237C2×#121C2×D6111yescharacteristic
238C2×#120—1221no—
239C22×C22⋊C4—611no—
240C22×C22⋊C4—1221no—
241#407—611no—
242#407C2×D6111yescharacteristic
243#399—612no—
244#392—612no—
245#390—1222no—
246C4×C22⋊C4—611no—
247C4×C22⋊C4—611no—
248C4×C22⋊C4—1222no—
249C4×C22⋊C4—1221no—
250C4×C22⋊C4—1222no—
251C4×C22⋊C4C2×D6111yescharacteristic
252#327—22no—
253#327—1222no—
254#323—1222no—
255C61#940911yescharacteristic, complemented, Sylow
256C25—611no—
257C22×D4—21no—
258C22×D4—1222no—
259C22×D4C4×D6111yescharacteristic, complemented
260C23×C4—611no—
261C23×C4—1222no—
262C23×C4C22×D6111yescharacteristic
263C2×M16—21no—
264C2×M16—1221no—
265#123—2444no—
266#121—21no—
267#121—21no—
268#121—21no—
269#121—21no—
270#120—2444no—
271C2×C4⋊C4—1222no—
272C2×C4⋊C4C4×D6111yescharacteristic, complemented
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—612no—
276C2×C22⋊C4—612no—
277C2×C22⋊C4—1228no—
278C2×C22⋊C4—1222no—
279C2×C22⋊C4—1221no—
280C2×C22⋊C4—1221no—
281C2×C22⋊C4—1221no—
282C2×C22⋊C4—1221no—
283C2×C22⋊C4—1221no—
284C2×C22⋊C4—1221no—
285C2×C22⋊C4—1222no—
286C2×C22⋊C4—1222no—
287C2×C42—21no—
288C2×C42—1222no—
289C2×C42—1221no—
290C2×C42—1221no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—1221no—
294C22⋊C8—1221no—
295#95—612no—
296#95—612no—
297#95—1222no—
298#95C22⋊D6112yescomplemented
299C24—21no—
300C24—612no—
301C24—1224no—
302C24—1221no—
303C24—1221no—
304C24—4881no—
305C24—2442no—
306C24D4×D6111yescharacteristic
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—81no—
314C2×D4—4882no—
315C2×D4—2444no—
316C2×D4—2444no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—81no—
320C22×C4—612no—
321C22×C4—1228no—
322C22×C4—1221no—
323C22×C4—1222no—
324C22×C4—1222no—
325C22×C4—1221no—
326C22×C4—4882no—
327C22×C4—4882no—
328C22×C4—2442no—
329C22×C4C2×C4×D6111yescharacteristic
330C22×C4#1175822211yescharacteristic
331C22×C4D4×D6111yescharacteristic
332C22×C4D4×D6111yescharacteristic
333C22×C4#1175822411yescharacteristic
334C22×C4C2×C22⋊D6111yescharacteristic
335M16—81no—
336M16—4881no—
337C2×C8—41no—
338C2×C8—2441no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—2444no—
342C22⋊C4—24no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—1228no—
346C22⋊C4—1224no—
347C22⋊C4—1224no—
348C22⋊C4—1222no—
349C22⋊C4—1222no—
350C22⋊C4—1222no—
351C22⋊C4—1222no—
352C22⋊C4—2444no—
353C22⋊C4—2444no—
354C42—81no—
355C42—4881no—
356C42—4881no—
357C42—4882no—
358C23—22no—
359C23—22no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—41no—
366C23—614no—
367C23—1224no—
368C23—1222no—
369C23—1222no—
370C23—1222no—
371C23—1222no—
372C23—4881no—
373C23—4881no—
374C23—4888no—
375C23—2442no—
376C23—2442no—
377C23—2442no—
378C23—2442no—
379C23C22⋊C4×D6111yescharacteristic
380C23#1451974511yescharacteristic
381C23#1451975911yescharacteristic
382D4—81no—
383D4—81no—
384D4—82no—
385D4—82no—
386D4—4888no—
387C2×C4—24no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—1224no—
400C2×C4—1224no—
401C2×C4—1222no—
402C2×C4—4888no—
403C2×C4—4881no—
404C2×C4—2442no—
405C2×C4—2442no—
406C2×C4—2442no—
407C2×C4—2442no—
408C2×C4—2442no—
409C2×C4—2442no—
410C2×C4#1451973711yescharacteristic
411C2×C4C4×C22⋊D6111yescharacteristic
412C2×C4#1451974811yescharacteristic
413C2×C4#1451975011yescharacteristic
414C8—81no—
415C8—4881no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—44no—
424C22—82no—
425C22—82no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—1224no—
430C22—4882no—
431C22—4884no—
432C22—2444no—
433C22—2444no—
434C22—2444no—
435C22—2442no—
436C22—2442no—
437C22#1683963811yescharacteristic
438C22#104×D6112yes—
439C22#1683992812yes—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—44no—
445C4—81no—
446C4—2444no—
447C4—2444no—
448C4—2442no—
449C2—22no—
450C2—22no—
451C2—41no—
452C2—161no—
453C2—2444no—
454C2#2144621411yescharacteristic
455C2#2144625911yescharacteristic
456C2#2144635811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1507C2
socleC22×C61#386
prime radical#21450751C2
cosocleC23×C4×C61C23
hypercenter#1507D61
hypocenterC61#9409
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1507

Complements

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

NH ≅ G/Ntype
2. #21448171C2semidirect
3. #21444335C2semidirect
4. #21444145C2semidirect
5. #21443705C2semidirect
6. #21443704C2semidirect
7. C61×#1507C2semidirect
13. #16839697C4semidirect
14. #16839670C4semidirect
53. #14519684C2×C4semidirect
54. #14519684D4semidirect
56. C22×D4×C61C2×C4semidirect
63. C2×C4⋊C4×C61C2×C4semidirect
69. #95×C61D4semidirect
224. #1507D61semidirect
255. C61#9409semidirect
259. C22×D4C4×D61semidirect
272. C2×C4⋊C4C4×D61semidirect
298. #95C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214507512
#214481712
#214443352
#214441452
#214437052
#214437042
C61×#15072
#940961