Finite Group Databasebeta

← #29577343

Subgroups — #29577343

Order 15616.

618 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2973C211yesmaximal, 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
21C61×#435—24no—
22C61×#432—21no—
23C61×#407—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—44no—
32C2×C4⋊Dic61—22no—
33C2×C4⋊Dic61C2×C411yescharacteristic
34C2×#11758196—21no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—22no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41#14519692—42no—
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—22no—
46#14519684C2×C411yescharacteristic
47#14519684C2×C411yescharacteristic
48C22×D4×C61D411yescharacteristic
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×C61C2×C411yescharacteristic
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×C61C2×D411yescharacteristic
66C23×D61—21no—
67C2×D4×C61—22no—
68C2×D4×C61—21no—
69C2×D4×C61—22no—
70C2×D4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—81no—
74C22×C4×C61—81no—
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22×C411yescharacteristic
79C22×C4×C61C2×D412yes—
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61#4112yes—
82C2×C22⋊D61—42no—
83C2×C22⋊D61—42no—
84C2×C22⋊D61—42no—
85C2×C22⋊D61—41no—
86C2×C22⋊D61—42no—
87C2×C22⋊D61—41no—
88C2×C22⋊D61—81no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61—42no—
93C22×Dic61—42no—
94C22×Dic61—42no—
95C22×Dic61—81no—
96C2×D244—22no—
97C2×D244—81no—
98M16×C61—82no—
99C2×C8×C61—24no—
100C2×C8×C61—82no—
101C4⋊C4×C61—22no—
102C22⋊C4×C61—22no—
103C22⋊C4×C61—22no—
104C42×C61—21no—
105C42×C61—21no—
106C22⋊Dic61—42no—
107C22⋊Dic61—42no—
108C22⋊Dic61—42no—
109#11758198—42no—
110#11758198—42no—
111#11758198—41no—
112#11758198—41no—
113#11758198—81no—
114C4⋊Dic61—22no—
115C4⋊Dic61—82no—
116#11758196—42no—
117C23×C61—21no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61C2×C22⋊C411yescharacteristic
125C23×C61#12112yes—
126C23×C61AGL(1,ℤ/8)11yescharacteristic
127C23×C61AGL(1,ℤ/8)12yes—
128C23×C61#13711yescharacteristic
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135C22×D61—41no—
136D4×C61—42no—
137D4×C61—42no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—42no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—81no—
151C2×C4×C61C4×D412yes—
152C2×C4×C61#12011yescharacteristic
153C2×C4×C61#12311yescharacteristic
154C22⋊D61—84no—
155C22⋊D61—82no—
156C22⋊D61—82no—
157C2×Dic61—22no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—84no—
167C2×Dic61—82no—
168C2×Dic61—82no—
169C2×Dic61—82no—
170D244—82no—
171C8×C61—44no—
172#9386—611nomaximal, Sylow
173C22×C61—21no—
174C22×C61—22no—
175C22×C61—21no—
176C22×C61—22no—
177C22×C61—42no—
178C22×C61—41no—
179C22×C61—41no—
180C22×C61—42no—
181C22×C61—41no—
182C22×C61—41no—
183C22×C61—41no—
184C22×C61—81no—
185C22×C61#38611yescharacteristic
186C22×C61#41711yescharacteristic
187C22×C61#41811yescharacteristic
188C22×C61#46812yes—
189C22×C61#46912yes—
190C2×D61—81no—
191C2×D61—81no—
192C2×D61—81no—
193C2×D61—81no—
194C2×D61—81no—
195C2×D61—81no—
196C2×D61—81no—
197C4×C61—21no—
198C4×C61—22no—
199C4×C61—21no—
200C4×C61—41no—
201C4×C61—41no—
202C4×C61—41no—
203Dic61—42no—
204Dic61—81no—
205Dic61—81no—
206Dic61—162no—
207#2973D6111yescharacteristic, complemented
208#2543—611no—
209#1799—611no—
210#1519—612no—
211#1294—612no—
212C2×C61—21no—
213C2×C61—22no—
214C2×C61—21no—
215C2×C61—22no—
216C2×C61—81no—
217C2×C61#188411yescharacteristic
218C2×C61#189511yescharacteristic
219C2×C61#191311yescharacteristic
220D61—161no—
221C2×#121—611no—
222C2×#121—1221no—
223C2×C4×D4C2×D6111yescharacteristic
224#435—24no—
225#432—21no—
226C2×#99—1221no—
227#407—21no—
228C2×C22⋊C8C2×D6112yes—
229#398—611no—
230#397—611no—
231#394—611no—
232#392—611no—
233#388—1221no—
234#386—1221no—
235#386—1221no—
236#380—612no—
237C2×#95—611no—
238#342—1221no—
239#327—1224no—
240C61#938611yescharacteristic, complemented, Sylow
241C22×D4—611no—
242C22×D4—1221no—
243C22×D4—1221no—
244C22×D4D24411yescharacteristic
245C23×C4—1221no—
246C23×C4C22⋊D6111yescharacteristic
247C23×C4C22×D6111yescharacteristic
248C2×M16—22no—
249C22×C8—22no—
250#121—1222no—
251#121—1222no—
252#121—2444no—
253C4×D4—22no—
254C4×D4—22no—
255C2×C4⋊C4—611no—
256C2×C4⋊C4—1222no—
257C2×C4⋊C4—1221no—
258C2×C4⋊C4C4×D6111yescharacteristic
259C2×C22⋊C4—1221no—
260C2×C22⋊C4—1221no—
261C2×C22⋊C4—1221no—
262C2×C22⋊C4—1221no—
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—1222no—
265C2×C22⋊C4C4×D6111yescharacteristic
266C2×C22⋊C4D24411yescharacteristic
267C2×C42C22⋊D6111yescharacteristic
268C4⋊C8—22no—
269#99—2442no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8C22⋊D6114yescomplemented
273#97—22no—
274#95—611no—
275#95—611no—
276#95—1221no—
277#95—1221no—
278#95—1221no—
279#95—1222no—
280C24—1221no—
281C24#1175819811yescharacteristic
282C24C2×D24411yescharacteristic
283C2×D4—21no—
284C2×D4—22no—
285C2×D4—21no—
286C2×D4—22no—
287C2×D4—1222no—
288C2×D4—4881no—
289C2×D4—4881no—
290C2×D4—2442no—
291C2×D4—2442no—
292C2×D4—2441no—
293C2×D4—2442no—
294C2×D4—2442no—
295C2×D4—2441no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4—1221no—
301C22×C4—1221no—
302C22×C4—1221no—
303C22×C4—4881no—
304C22×C4—2442no—
305C22×C4—2442no—
306C22×C4—2442no—
307C22×C4#1175819811yescharacteristic
308C22×C4#1175819811yescharacteristic
309C22×C4#1175819811yescharacteristic
310C22×C4C2×C4×D6111yescharacteristic
311C22×C4#1175822212yes—
312C22×C4C2×C22⋊D6112yes—
313C22×C4C2×C22⋊D6111yescharacteristic
314M16—82no—
315C2×C8—24no—
316C2×C8—82no—
317C4⋊C4—22no—
318C4⋊C4—1222no—
319C4⋊C4—4882no—
320C4⋊C4—2442no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—4881no—
324C22⋊C4—2442no—
325C22⋊C4—2442no—
326C22⋊C4—2442no—
327C22⋊C4—2442no—
328C22⋊C4—2442no—
329C22⋊C4—2441no—
330C22⋊C4—2441no—
331C42—21no—
332C42—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—2441no—
341C23—2441no—
342C23—2441no—
343C23—2441no—
344C23—2441no—
345C23—2441no—
346C23—2441no—
347C23C2×#1175819811yescharacteristic
348C23#1451975112yes—
349C23#1451980512yes—
350C23#1451980511yescharacteristic
351C23#1451981011yescharacteristic
352D4—42no—
353D4—42no—
354D4—4882no—
355D4—4882no—
356D4—4882no—
357D4—4884no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—1222no—
372C2×C4—4882no—
373C2×C4—4882no—
374C2×C4—4882no—
375C2×C4—4884no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4—2441no—
384C2×C4C4×C22⋊D6112yes—
385C2×C4#1451974611yescharacteristic
386C2×C4#1451976011yescharacteristic
387C8—44no—
388C22—21no—
389C22—22no—
390C22—21no—
391C22—22no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—81no—
400C22—4881no—
401C22—4881no—
402C22—4881no—
403C22—4881no—
404C22—4881no—
405C22—4881no—
406C22—4881no—
407C22#1683964111yescharacteristic
408C22#1683976411yescharacteristic
409C22#1683976811yescharacteristic
410C22#1684000812yes—
411C22#1684001512yes—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—9762no—
419C4—4881no—
420C4—4881no—
421C4—2442no—
422C2—22no—
423C2—21no—
424C2—21no—
425C2—22no—
426C2—81no—
427C2—9761no—
428C2#2144620211yescharacteristic
429C2#2144625711yescharacteristic
430C2#2144636111yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2973C2
socleC22×C61#386
prime radical#21450775C2
cosocleC23×C4×C61C23
hypercenter#2973D61
hypocenterC61#9386
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #21445807C2semidirect
3. #21444388C2semidirect
4. #21443699C2semidirect
5. C61×#2973C2semidirect
9. #16839692C4semidirect
10. #16839687C4semidirect
11. #16839679C4semidirect
12. #16839671C4semidirect
62. C22⋊C8×C61D4semidirect
207. #2973D61semidirect
240. C61#9386semidirect
272. 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×#29732
#938661