Finite Group Databasebeta

← #29577331

Subgroups — #29577331

Order 15616.

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

normal subgroups only

This group has 640 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#21454843C211yesmaximal, characteristic, complemented
2#21450775C211yesmaximal, characteristic
3#21444386C212yesmaximal, complemented
4#21443697C212yesmaximal, complemented
5C61×#1799C211yesmaximal, characteristic, complemented
6C2×#14519751C2211yescharacteristic
7C2×#14519749—21no—
8D4×#92418—24no—
9#16839815—21no—
10C2×C22⋊#92418—21no—
11C2×C22⋊#92418C2212yes—
12#16839692C411yescharacteristic, complemented
13#16839687C411yescharacteristic, complemented
14#16839679C411yescharacteristic, complemented
15#16839671C411yescharacteristic, complemented
16#16839653—21no—
17#16839641—21no—
18#16839639—21no—
19C2×#14519684C2211yescharacteristic
20#16839495—24no—
21C2×C4×D4×C61C2211yescharacteristic
22C2×#99×C61—21no—
23C61×#380C2212yes—
24C61×#342—21no—
25C22×D244C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27C22×C22⋊D61—21no—
28C23×Dic61—21no—
29C22×#92418—22no—
30C22×#92418—22no—
31#14519751—22no—
32#14519751—22no—
33#14519749—42no—
34#14519749—42no—
35C2×C4⋊Dic61C2×C411yescharacteristic
36C2×#11758196—21no—
37C2×C22⋊Dic61—21no—
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C4⋊#92418—22no—
43C22⋊#92418—22no—
44C22⋊#92418—22no—
45C22⋊#92418D414yescomplemented
46C4×#92418—22no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684C2×C411yescharacteristic
51#14519684C2×C411yescharacteristic
52C22×D4×C61D411yescharacteristic
53C23×C4×C61D411yescharacteristic
54C23×C4×C61C2311yescharacteristic
55C4×D4×C61—22no—
56C4×D4×C61—22no—
57C2×C4⋊C4×C61—22no—
58C2×C4⋊C4×C61C2×C411yescharacteristic
59C2×C22⋊C4×C61—22no—
60C2×C22⋊C4×C61C2×C411yescharacteristic
61C2×C22⋊C4×C61D411yescharacteristic
62C2×C42×C61D411yescharacteristic
63#99×C61—42no—
64#95×C61—22no—
65C24×C61C22⋊C411yescharacteristic
66C24×C61C2×D411yescharacteristic
67C23×D61—21no—
68C2×D4×C61—22no—
69C2×D4×C61—22no—
70C2×D4×C61D812yes—
71C2×D4×C61SD1612yes—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—42no—
75C22×C4×C61—81no—
76C22×C4×C61—81no—
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22×C411yescharacteristic
81C22×C4×C61C2×D412yes—
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61#4112yes—
84C2×C22⋊D61—42no—
85C2×C22⋊D61—42no—
86C2×C22⋊D61—42no—
87C2×C22⋊D61—41no—
88C2×C22⋊D61—41no—
89C2×C22⋊D61—42no—
90C2×C22⋊D61—81no—
91C22×Dic61—21no—
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61—42no—
95C22×Dic61—41no—
96C22×Dic61—41no—
97C22×Dic61—81no—
98C2×D244—22no—
99C2×D244—81no—
100C4⋊C4×C61—22no—
101C4⋊C4×C61—82no—
102C22⋊C4×C61—22no—
103C22⋊C4×C61—22no—
104C22⋊C4×C61—42no—
105C22⋊C4×C61—42no—
106C42×C61—21no—
107C42×C61—21no—
108C22⋊Dic61—42no—
109#11758198—42no—
110#11758198—42no—
111#11758198—41no—
112#11758198—41no—
113#11758198—81no—
114C4⋊Dic61—22no—
115#11758196—42no—
116C2×#92418—24no—
117C2×#92418—82no—
118C2×#92418—82no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61#10212yes—
126C23×C61C2×C22⋊C411yescharacteristic
127C23×C61#12112yes—
128C23×C61C2×D811yescharacteristic
129C23×C61C2×SD1611yescharacteristic
130C23×C61#13512yes—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135C22×D61—41no—
136C22×D61—41no—
137C22×D61—41no—
138D4×C61—24no—
139D4×C61—24no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—42no—
145C2×C4×C61—42no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—42no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—81no—
152C2×C4×C61—82no—
153C2×C4×C61—82no—
154C2×C4×C61—82no—
155C2×C4×C61#10212yes—
156C2×C4×C61C4×D412yes—
157C2×C4×C61#12011yescharacteristic
158C2×C4×C61#12311yescharacteristic
159C22⋊D61—84no—
160C22⋊D61—82no—
161C22⋊D61—82no—
162C2×Dic61—22no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—82no—
172C2×Dic61—82no—
173D244—82no—
174#92418—44no—
175#9384—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—22no—
178C22×C61—21no—
179C22×C61—22no—
180C22×C61—22no—
181C22×C61—41no—
182C22×C61—41no—
183C22×C61—42no—
184C22×C61—42no—
185C22×C61—41no—
186C22×C61—41no—
187C22×C61—81no—
188C22×C61#38611yescharacteristic
189C22×C61C2×#10211yescharacteristic
190C22×C61#41611yescharacteristic
191C22×C61#46512yes—
192C22×C61#46612yes—
193C2×D61—81no—
194C2×D61—81no—
195C2×D61—81no—
196C2×D61—81no—
197C2×D61—81no—
198C2×D61—81no—
199C2×D61—81no—
200C4×C61—22no—
201C4×C61—21no—
202C4×C61—22no—
203C4×C61—21no—
204C4×C61—41no—
205C4×C61—41no—
206C4×C61—162no—
207Dic61—42no—
208Dic61—81no—
209Dic61—81no—
210#2972—611no—
211#2543—611no—
212#1799D6111yescharacteristic, complemented
213#1519—612no—
214#1294—612no—
215C2×C61—21no—
216C2×C61—22no—
217C2×C61—21no—
218C2×C61—22no—
219C2×C61—42no—
220C2×C61#188411yescharacteristic
221C2×C61#189311yescharacteristic
222C2×C61#191311yescharacteristic
223D61—161no—
224C2×#121—611no—
225C2×#121—1221no—
226C2×C4×D4C2×D6111yescharacteristic
227C8×D4—1224no—
228#431—1221no—
229C2×#99—21no—
230C2×C22⋊C8—612no—
231C2×C22⋊C8—1221no—
232#398—611no—
233#397—611no—
234#394—611no—
235#392—611no—
236#388—1221no—
237#386—1221no—
238#386—1221no—
239#380C2×D6112yes—
240C2×#95—611no—
241#342—21no—
242#327—1224no—
243C61#938411yescharacteristic, complemented, Sylow
244C22×D4—611no—
245C22×D4—1221no—
246C22×D4—1221no—
247C22×D4C22⋊D6111yescharacteristic
248C23×C4—1221no—
249C23×C4D24411yescharacteristic
250C23×C4C22×D6111yescharacteristic
251C22×C8—1222no—
252C22×C8—1222no—
253#121—1222no—
254#121—1222no—
255#121—2442no—
256#121—2442no—
257C4×D4—22no—
258C4×D4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—611no—
261C2×C4⋊C4—1221no—
262C2×C4⋊C4C4×D6111yescharacteristic
263C2×C22⋊C4—22no—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1221no—
268C2×C22⋊C4—1221no—
269C2×C22⋊C4C4×D6111yescharacteristic
270C2×C22⋊C4C22⋊D6111yescharacteristic
271C2×C42D24411yescharacteristic
272C4⋊C8—1222no—
273#99—42no—
274C22⋊C8—614no—
275C22⋊C8—1222no—
276C22⋊C8—1222no—
277C4×C8—1222no—
278#95—22no—
279#95—611no—
280#95—611no—
281#95—1221no—
282#95—1221no—
283#95—1221no—
284C24—1221no—
285C24#1175819811yescharacteristic
286C24C2×C22⋊D6111yescharacteristic
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—1222no—
290C2×D4—4881no—
291C2×D4—4881no—
292C2×D4—2442no—
293C2×D4—2442no—
294C2×D4—2442no—
295C2×D4—2441no—
296C2×D4—2441no—
297C2×D4—2442no—
298C2×D4#1175819912yes—
299C2×D4#1175820112yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—42no—
303C22×C4—81no—
304C22×C4—81no—
305C22×C4—1221no—
306C22×C4—1221no—
307C22×C4—1221no—
308C22×C4—4881no—
309C22×C4—2442no—
310C22×C4—2441no—
311C22×C4—2441no—
312C22×C4#1175819811yescharacteristic
313C22×C4#1175819811yescharacteristic
314C22×C4#1175819811yescharacteristic
315C22×C4C2×C4×D6111yescharacteristic
316C22×C4C2×D24411yescharacteristic
317C22×C4D4×D6112yes—
318C22×C4#1175822412yes—
319C2×C8—1224no—
320C2×C8—4882no—
321C2×C8—4882no—
322C4⋊C4—22no—
323C4⋊C4—82no—
324C4⋊C4—1222no—
325C4⋊C4—2442no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—4881no—
331C22⋊C4—2442no—
332C22⋊C4—2442no—
333C22⋊C4—2442no—
334C22⋊C4—2441no—
335C22⋊C4—2441no—
336C42—21no—
337C42—21no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—2441no—
345C23—2441no—
346C23—2441no—
347C23—2441no—
348C23—2441no—
349C23—2441no—
350C23—2441no—
351C23#1451969712yes—
352C23C2×#1175819811yescharacteristic
353C23#1451974912yes—
354C23C2×#1175819911yescharacteristic
355C23C2×#1175820111yescharacteristic
356C23#1451979712yes—
357D4—24no—
358D4—24no—
359D4—4882no—
360D4—4882no—
361D4—4882no—
362D4—4884no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—1222no—
379C2×C4—4882no—
380C2×C4—4882no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4#1451969712yes—
390C2×C4#1451973712yes—
391C2×C4#1451974711yescharacteristic
392C2×C4#1451976111yescharacteristic
393C8—2444no—
394C22—21no—
395C22—22no—
396C22—21no—
397C22—22no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—4881no—
407C22—4881no—
408C22—4881no—
409C22—4881no—
410C22—4881no—
411C22—4881no—
412C22—4881no—
413C22#1683963911yescharacteristic
414C22C2×#1451969711yescharacteristic
415C22#1683975611yescharacteristic
416C22#1683999312yes—
417C22#1683999612yes—
418C4—22no—
419C4—21no—
420C4—21no—
421C4—22no—
422C4—41no—
423C4—41no—
424C4—162no—
425C4—4881no—
426C4—4881no—
427C4—2442no—
428C2—22no—
429C2—21no—
430C2—21no—
431C2—22no—
432C2—42no—
433C2—9761no—
434C2#2144620011yescharacteristic
435C2#2144624511yescharacteristic
436C2#2144635911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1799C2
socleC22×C61#386
prime radical#21450775C2
cosocleC23×C4×C61C23
hypercenter#1799D61
hypocenterC61#9384
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
1. #21454843C2semidirect
3. #21444386C2semidirect
4. #21443697C2semidirect
5. C61×#1799C2semidirect
12. #16839692C4semidirect
13. #16839687C4semidirect
14. #16839679C4semidirect
15. #16839671C4semidirect
45. C22⋊#92418D4semidirect
212. #1799D61semidirect
243. C61#9384semidirect

Maximal subgroups

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

subgroupindex
#214548432
#214507752
#214443862
#214436972
C61×#17992
#938461