Finite Group Databasebeta

← #29577333

Subgroups — #29577333

Order 15616.

640 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 131 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#21450775C211yesmaximal, characteristic
2#21445807C211yesmaximal, characteristic, complemented
3#21444388C212yesmaximal, complemented
4#21443699C212yesmaximal, complemented
5C61×#2972C211yesmaximal, characteristic, complemented
6C2×#14519751C2211yescharacteristic
7C2×#14519749—21no—
8C2×#14519692—21no—
9#16839692C411yescharacteristic, complemented
10#16839687C411yescharacteristic, complemented
11#16839679C411yescharacteristic, complemented
12#16839671C411yescharacteristic, complemented
13#16839653—21no—
14#16839641—21no—
15#16839639—21no—
16#16839622C2212yes—
17C2×#14519684C2211yescharacteristic
18#16839529—21no—
19#16839497—24no—
20C2×C4×D4×C61C2211yescharacteristic
21C8×D4×C61—24no—
22C61×#431—21no—
23C2×C22⋊C8×C61—21no—
24C2×C22⋊C8×C61C2212yes—
25C22×D244C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27C22×C22⋊D61—21no—
28C23×Dic61—21no—
29#14519751—22no—
30#14519751—22no—
31#14519749—42no—
32#14519749—42no—
33C2×C4⋊Dic61—22no—
34C2×C4⋊Dic61C2×C411yescharacteristic
35C2×#11758196—21no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—22no—
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42#14519692—42no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—22no—
47#14519684C2×C411yescharacteristic
48#14519684C2×C411yescharacteristic
49C22×D4×C61D411yescharacteristic
50C23×C4×C61D411yescharacteristic
51C23×C4×C61C2311yescharacteristic
52C22×C8×C61—22no—
53C22×C8×C61—22no—
54C4×D4×C61—22no—
55C4×D4×C61—22no—
56C2×C4⋊C4×C61C2×C411yescharacteristic
57C2×C22⋊C4×C61C2×C411yescharacteristic
58C2×C22⋊C4×C61D411yescharacteristic
59C2×C42×C61D411yescharacteristic
60C4⋊C8×C61—22no—
61C22⋊C8×C61—22no—
62C22⋊C8×C61—22no—
63C22⋊C8×C61D414yescomplemented
64C4×C8×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—81no—
75C22×C4×C61—81no—
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22×C411yescharacteristic
80C22×C4×C61C2×D412yes—
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61#4112yes—
83C2×C22⋊D61—42no—
84C2×C22⋊D61—42no—
85C2×C22⋊D61—42no—
86C2×C22⋊D61—41no—
87C2×C22⋊D61—41no—
88C2×C22⋊D61—42no—
89C2×C22⋊D61—81no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61—21no—
93C22×Dic61—42no—
94C22×Dic61—41no—
95C22×Dic61—41no—
96C22×Dic61—42no—
97C22×Dic61—81no—
98C2×D244—22no—
99C2×D244—81no—
100C2×C8×C61—24no—
101C2×C8×C61—82no—
102C2×C8×C61—82no—
103C4⋊C4×C61—22no—
104C22⋊C4×C61—22no—
105C22⋊C4×C61—22no—
106C42×C61—21no—
107C42×C61—21no—
108C22⋊Dic61—42no—
109C22⋊Dic61—42no—
110C22⋊Dic61—42no—
111#11758198—42no—
112#11758198—42no—
113#11758198—41no—
114#11758198—41no—
115#11758198—81no—
116C4⋊Dic61—22no—
117C4⋊Dic61—82no—
118#11758196—42no—
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#10212yes—
153C2×C4×C61C4×D412yes—
154C2×C4×C61#12011yescharacteristic
155C2×C4×C61#12311yescharacteristic
156C22⋊D61—84no—
157C22⋊D61—82no—
158C22⋊D61—82no—
159C2×Dic61—22no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—82no—
169C2×Dic61—82no—
170C2×Dic61—82no—
171C2×Dic61—82no—
172C2×Dic61—82no—
173D244—82no—
174C8×C61—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—
206Dic61—42no—
207Dic61—81no—
208Dic61—81no—
209Dic61—162no—
210#2972D6111yescharacteristic, complemented
211#2543—611no—
212#1799—611no—
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—24no—
228#431—21no—
229C2×#99—1221no—
230C2×C22⋊C8—21no—
231C2×C22⋊C8C2×D6112yes—
232#398—611no—
233#397—611no—
234#394—611no—
235#392—611no—
236#388—1221no—
237#386—1221no—
238#386—1221no—
239#380—612no—
240C2×#95—611no—
241#342—1221no—
242#327—1224no—
243C61#938411yescharacteristic, complemented, Sylow
244C22×D4—611no—
245C22×D4—1221no—
246C22×D4—1221no—
247C22×D4D24411yescharacteristic
248C23×C4—1221no—
249C23×C4C22⋊D6111yescharacteristic
250C23×C4C22×D6111yescharacteristic
251C22×C8—22no—
252C22×C8—22no—
253#121—1222no—
254#121—1222no—
255#121—2442no—
256#121—2442no—
257C4×D4—22no—
258C4×D4—22no—
259C2×C4⋊C4—611no—
260C2×C4⋊C4—1222no—
261C2×C4⋊C4—1221no—
262C2×C4⋊C4C4×D6111yescharacteristic
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1221no—
268C2×C22⋊C4—1222no—
269C2×C22⋊C4C4×D6111yescharacteristic
270C2×C22⋊C4D24411yescharacteristic
271C2×C42C22⋊D6111yescharacteristic
272C4⋊C8—22no—
273#99—2442no—
274C22⋊C8—22no—
275C22⋊C8—22no—
276C22⋊C8C22⋊D6114yescomplemented
277C4×C8—22no—
278#95—611no—
279#95—611no—
280#95—1221no—
281#95—1221no—
282#95—1221no—
283#95—1222no—
284C24—1221no—
285C24#1175819811yescharacteristic
286C24C2×D24411yescharacteristic
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#1175819012yes—
299C2×D4D48812yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—81no—
303C22×C4—81no—
304C22×C4—1221no—
305C22×C4—1221no—
306C22×C4—1221no—
307C22×C4—4881no—
308C22×C4—2442no—
309C22×C4—2442no—
310C22×C4—2441no—
311C22×C4—2441no—
312C22×C4#1175819811yescharacteristic
313C22×C4#1175819811yescharacteristic
314C22×C4#1175819811yescharacteristic
315C22×C4C2×C4×D6111yescharacteristic
316C22×C4#1175822212yes—
317C22×C4C2×C22⋊D6112yes—
318C22×C4C2×C22⋊D6111yescharacteristic
319C2×C8—24no—
320C2×C8—82no—
321C2×C8—82no—
322C4⋊C4—22no—
323C4⋊C4—1222no—
324C4⋊C4—4882no—
325C4⋊C4—2442no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—4881no—
329C22⋊C4—2442no—
330C22⋊C4—2442no—
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#1451969912yes—
352C23C2×#1175819811yescharacteristic
353C23#1451975112yes—
354C23C2×D48811yescharacteristic
355C23C2×#1175819011yescharacteristic
356C23#1451979912yes—
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—1222no—
376C2×C4—4882no—
377C2×C4—4882no—
378C2×C4—4882no—
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#1451969912yes—
390C2×C4C4×C22⋊D6112yes—
391C2×C4#1451974611yescharacteristic
392C2×C4#1451976011yescharacteristic
393C8—44no—
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#1683964111yescharacteristic
414C22C2×#1451969911yescharacteristic
415C22#1683975911yescharacteristic
416C22#1683999112yes—
417C22#1683999812yes—
418C4—22no—
419C4—21no—
420C4—21no—
421C4—22no—
422C4—41no—
423C4—41no—
424C4—9762no—
425C4—4881no—
426C4—4881no—
427C4—2442no—
428C2—22no—
429C2—21no—
430C2—21no—
431C2—22no—
432C2—42no—
433C2—9761no—
434C2#2144620211yescharacteristic
435C2#2144624711yescharacteristic
436C2#2144636111yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2972C2
socleC22×C61#386
prime radical#21450775C2
cosocleC23×C4×C61C23
hypercenter#2972D61
hypocenterC61#9384
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #21445807C2semidirect
3. #21444388C2semidirect
4. #21443699C2semidirect
5. C61×#2972C2semidirect
9. #16839692C4semidirect
10. #16839687C4semidirect
11. #16839679C4semidirect
12. #16839671C4semidirect
63. C22⋊C8×C61D4semidirect
210. #2972D61semidirect
243. C61#9384semidirect
276. C22⋊C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214507752
#214458072
#214443882
#214436992
C61×#29722
#938461