Finite Group Databasebeta

← #29652390

Subgroups — #29652390

Order 15616.

1310 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1310 conjugacy classes of subgroups — too many to draw. Only its 183 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#21455077C212yesmaximal, complemented
2#21449138C211yesmaximal, characteristic
3#21444776C214yesmaximal, complemented
4#21444742C214yesmaximal, complemented
5#21444352C212yesmaximal, complemented
6#21444146C211yesmaximal, characteristic, complemented
7C2×C61×#543C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C2×#14519808—22no—
10C2×#14519746—21no—
11#16839922—24no—
12#16839917C2214yescomplemented
13#16839912C2214yescomplemented
14#16839761—22no—
15#16839726C2212yes—
16#16839674C2212yes—
17#16839671C2212yes—
18#16839641—21no—
19#16839638—22no—
20#16839625C2211yescharacteristic
21#16839499C2218yescomplemented
22#16839486C2214yes—
23C61×#543—24no—
24C61×#543—22no—
25C61×#543C2214yes—
26C2×C22⋊Q8×C61—22no—
27C2×C22⋊Q8×C61—21no—
28C2×C22⋊Q8×C61—22no—
29C2×C22⋊Q8×C61C2212yes—
30C22×C22⋊C4×C61—21no—
31C22×C22⋊C4×C61C2211yescharacteristic
32C2×#11758222—22no—
33C2×#11758222—22no—
34C2×#11758222—44no—
35C22×Dic122C2311yescharacteristic
36C22×D244C2311yescharacteristic
37C22×C22⋊D61—21no—
38C22×C22⋊D61—22no—
39C22×C4×D61—21no—
40#14519808—48no—
41C2×#11758202—24no—
42C2×#11758201—24no—
43C2×#11758194—22no—
44#14519746—44no—
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61—22no—
47C2×#11758198—22no—
48#14519700—24no—
49#14519697—24no—
50C22⋊#92418—22no—
51C22⋊#92418C2314yes—
52#14519684—21no—
53#14519684—22no—
54#14519684C2312yes—
55C25×C61D411yescharacteristic
56C22×Q8×C61—21no—
57C22×Q8×C61D412yescomplemented
58C23×C4×C61—21no—
59C23×C4×C61—21no—
60C23×C4×C61D411yescharacteristic
61C23×C4×C61C2311yescharacteristic
62C22⋊Q8×C61—28no—
63C22⋊Q8×C61—28no—
64C22⋊Q8×C61—24no—
65C22⋊Q8×C61—28no—
66C22⋊Q8×C61—22no—
67C22⋊Q8×C61—22no—
68C22⋊Q8×C61—24no—
69C22⋊Q8×C61—28no—
70C22⋊Q8×C61D414yescomplemented
71C22⋊Q8×C61C2314yes—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61—22no—
74C2×C4⋊C4×C61—22no—
75C2×C4⋊C4×C61D412yescomplemented
76C2×C22⋊C4×C61—24no—
77C2×C22⋊C4×C61—22no—
78C2×C22⋊C4×C61—24no—
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61—22no—
81C2×C22⋊C4×C61—22no—
82C2×C22⋊C4×C61D412yes—
83C2×C22⋊C4×C61C2312yes—
84C24×C61—21no—
85C24×C61—21no—
86C24×C61—24no—
87C24×C61—42no—
88C24×C61—81no—
89C24×C61C2×D411yescharacteristic
90C24×C61C2×D412yes—
91C23×D61—21no—
92C2×Q8×C61—24no—
93C2×Q8×C61—22no—
94C2×Q8×C61—44no—
95C2×Q8×C61—82no—
96C2×Q8×C61C2×D414yes—
97C22×C4×C61—22no—
98C22×C4×C61—21no—
99C22×C4×C61—22no—
100C22×C4×C61—24no—
101C22×C4×C61—24no—
102C22×C4×C61—21no—
103C22×C4×C61—42no—
104C22×C4×C61—81no—
105C22×C4×C61—82no—
106C22×C4×C61—82no—
107C22×C4×C61C2×D412yes—
108C22×C4×C61C2×D412yes—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D412yes—
111C22×C4×C61C2411yescharacteristic
112C2×C22⋊D61—24no—
113C2×C22⋊D61—42no—
114C2×C22⋊D61—41no—
115C2×C22⋊D61—41no—
116C2×C22⋊D61—44no—
117C2×C22⋊D61—44no—
118C2×C22⋊D61—82no—
119C22×Dic61—21no—
120C22×Dic61—22no—
121#11758222—28no—
122#11758222—44no—
123#11758222—44no—
124#11758222—44no—
125C2×D244—22no—
126C2×D244—42no—
127C2×C4×D61—42no—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×Dic122—22no—
131C2×Dic122—42no—
132C4⋊C4×C61—24no—
133C4⋊C4×C61—28no—
134C4⋊C4×C61—24no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—24no—
137C4⋊C4×C61C2×D414yes—
138C22⋊C4×C61—28no—
139C22⋊C4×C61—24no—
140C22⋊C4×C61—24no—
141C22⋊C4×C61—24no—
142C22⋊C4×C61—24no—
143C22⋊Dic61—44no—
144C22⋊Dic61—42no—
145#11758202—48no—
146#11758201—48no—
147#11758198—82no—
148#11758194—44no—
149C2×#92418—24no—
150C23×C61—24no—
151C23×C61—24no—
152C23×C61—22no—
153C23×C61—22no—
154C23×C61—22no—
155C23×C61—22no—
156C23×C61—41no—
157C23×C61—42no—
158C23×C61—42no—
159C23×C61—42no—
160C23×C61—42no—
161C23×C61—81no—
162C23×C61—81no—
163C23×C61—88no—
164C23×C61#12011yescharacteristic
165C23×C61#12011yescharacteristic
166C23×C61#13714yescomplemented
167C23×C61C22×D411yescharacteristic
168C22×D61—22no—
169C22×D61—42no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—42no—
173Q8×C61—44no—
174Q8×C61—82no—
175Q8×C61—84no—
176C2×C4×C61—24no—
177C2×C4×C61—24no—
178C2×C4×C61—22no—
179C2×C4×C61—24no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—44no—
185C2×C4×C61—44no—
186C2×C4×C61—42no—
187C2×C4×C61—81no—
188C2×C4×C61—84no—
189C2×C4×C61—84no—
190C2×C4×C61#12012yes—
191C2×C4×C61C22×D412yes—
192C22⋊D61—44no—
193C22⋊D61—44no—
194C22⋊D61—88no—
195C2×Dic61—22no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61—42no—
201D244—24no—
202D244—42no—
203C4×D61—48no—
204Dic122—24no—
205Dic122—42no—
206#92418—44no—
207#18133—611nomaximal, Sylow
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—24no—
214C22×C61—21no—
215C22×C61—21no—
216C22×C61—44no—
217C22×C61—44no—
218C22×C61—48no—
219C22×C61—42no—
220C22×C61—42no—
221C22×C61—82no—
222C22×C61—84no—
223C22×C61B(4,𝔽2)12yes—
224C22×C61C2×#12011yescharacteristic
225C22×C61C2×#13712yes—
226C2×D61—44no—
227C2×D61—42no—
228C2×D61—82no—
229C2×D61—81no—
230C2×D61—81no—
231C4×C61—22no—
232C4×C61—22no—
233C4×C61—42no—
234C4×C61—44no—
235C4×C61—44no—
236Dic61—42no—
237Dic61—82no—
238C2×#543D6111yescharacteristic, complemented
239#3007—612no—
240#2404—611no—
241#1602—614no—
242#1596—614no—
243#1511—612no—
244#1462—611no—
245C2×C61—22no—
246C2×C61—22no—
247C2×C61—41no—
248C2×C61—44no—
249C2×C61#300411yescharacteristic
250C2×C61#301911yescharacteristic
251C2×C61C2×B(4,𝔽2)11yescharacteristic
252D61—82no—
253C22×#41—611no—
254C2×#137—1222no—
255#543—24no—
256#543—22no—
257#543C2×D6114yes—
258C2×C22⋊Q8—22no—
259C2×C22⋊Q8—21no—
260C2×C22⋊Q8—22no—
261C2×C22⋊Q8C2×D6112yes—
262C2×#120—1221no—
263C22×C22⋊C4—21no—
264C22×C22⋊C4C2×D6111yescharacteristic
265#452—1224no—
266#451—614no—
267#450—614no—
268#417—1222no—
269#407—612no—
270#393—612no—
271#392—612no—
272#386—1221no—
273#386—1222no—
274#381—611no—
275#328—618no—
276#323—614no—
277C61#1813311yescharacteristic, complemented, Sylow
278C25C22⋊D6111yescharacteristic
279C2×#41—1222no—
280C2×#41—1222no—
281C2×#41—2444no—
282C22×Q8—21no—
283C22×Q8—611no—
284C22×Q8C22⋊D6112yescomplemented
285C22×D4—611no—
286C22×D4—1221no—
287C22×D4—1222no—
288C23×C4—21no—
289C23×C4—21no—
290C23×C4—1221no—
291C23×C4C22⋊D6111yescharacteristic
292C23×C4C22×D6111yescharacteristic
293#137—2448no—
294C2×Q16—1224no—
295C2×SD16—1224no—
296C2×M16—1222no—
297C22⋊Q8—28no—
298C22⋊Q8—28no—
299C22⋊Q8—24no—
300C22⋊Q8—28no—
301C22⋊Q8—24no—
302C22⋊Q8—22no—
303C22⋊Q8—22no—
304C22⋊Q8—28no—
305C22⋊Q8C22⋊D6114yescomplemented
306C22⋊Q8C22×D6114yes—
307#120—2444no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4C22⋊D6112yescomplemented
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—1222no—
319C2×C22⋊C4—1221no—
320C2×C22⋊C4—1222no—
321C2×C22⋊C4C22⋊D6112yes—
322C2×C22⋊C4C22×D6112yes—
323#103—1224no—
324#102—1224no—
325C22⋊C8—614no—
326C22⋊C8—1222no—
327#95—612no—
328#95—1221no—
329#95—1222no—
330C24—21no—
331C24—24no—
332C24—21no—
333C24—42no—
334C24—81no—
335C24—1221no—
336C24C2×C22⋊D6111yescharacteristic
337C24C2×C22⋊D6112yes—
338#41—1228no—
339#41—2444no—
340#41—2444no—
341#41—2444no—
342C2×Q8—24no—
343C2×Q8—22no—
344C2×Q8—44no—
345C2×Q8—82no—
346C2×Q8—1222no—
347C2×Q8—2442no—
348C2×Q8C2×C22⋊D6114yes—
349C2×D4—1224no—
350C2×D4—1222no—
351C2×D4—4882no—
352C2×D4—2442no—
353C2×D4—2444no—
354C2×D4—2444no—
355C2×D4—2442no—
356C2×D4—2441no—
357C2×D4—2441no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—42no—
365C22×C4—81no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—1221no—
369C22×C4—1222no—
370C22×C4—2442no—
371C22×C4—2442no—
372C22×C4—2442no—
373C22×C4C2×C22⋊D6112yes—
374C22×C4C2×C22⋊D6112yes—
375C22×C4C2×C22⋊D6112yes—
376C22×C4C2×C22⋊D6111yescharacteristic
377C22×C4C23×D6111yescharacteristic
378Q16—2448no—
379SD16—2448no—
380M16—2444no—
381C2×C8—1224no—
382C4⋊C4—24no—
383C4⋊C4—28no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4C2×C22⋊D6114yes—
388C22⋊C4—28no—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—4882no—
394C22⋊C4—2444no—
395C22⋊C4—2442no—
396C23—24no—
397C23—24no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—41no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—81no—
408C23—81no—
409C23—88no—
410C23—1222no—
411C23—2442no—
412C23—2441no—
413C23—2441no—
414C23—2442no—
415C23#1451974611yescharacteristic
416C23#1451974611yescharacteristic
417C23#1451980814yescomplemented
418C23C22×C22⋊D6111yescharacteristic
419Q8—44no—
420Q8—82no—
421Q8—84no—
422Q8—1224no—
423Q8—2442no—
424D4—1224no—
425D4—4888no—
426D4—2442no—
427D4—2444no—
428D4—2444no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—81no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—1222no—
444C2×C4—2448no—
445C2×C4—2442no—
446C2×C4—2442no—
447C2×C4—2442no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4#1451974612yes—
451C2×C4C22×C22⋊D6112yes—
452C8—2444no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—24no—
460C22—21no—
461C22—44no—
462C22—44no—
463C22—48no—
464C22—42no—
465C22—42no—
466C22—82no—
467C22—84no—
468C22—4882no—
469C22—4881no—
470C22—4881no—
471C22—2444no—
472C22—2442no—
473C22#1683994912yes—
474C22C2×#1451974611yescharacteristic
475C22C2×#1451980812yes—
476C4—22no—
477C4—22no—
478C4—42no—
479C4—44no—
480C4—44no—
481C4—4882no—
482C4—2442no—
483C2—22no—
484C2—22no—
485C2—41no—
486C2—44no—
487C2—4882no—
488C2#2145505311yescharacteristic
489C2#2145517611yescharacteristic
490C2C2×#1683994911yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#543C2
socleC22×C61C2×#120
prime radical#21449138C2
hypercenterC2×#543D61
hypocenterC61#18133
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
1. #21455077C2semidirect
3. #21444776C2semidirect
4. #21444742C2semidirect
5. #21444352C2semidirect
6. #21444146C2semidirect
7. C2×C61×#543C2semidirect
12. #16839917C22semidirect
13. #16839912C22semidirect
21. #16839499C22semidirect
57. C22×Q8×C61D4semidirect
70. C22⋊Q8×C61D4semidirect
75. C2×C4⋊C4×C61D4semidirect
166. C23×C61#137semidirect
238. C2×#543D61semidirect
277. C61#18133semidirect
284. C22×Q8C22⋊D61semidirect
305. C22⋊Q8C22⋊D61semidirect
311. C2×C4⋊C4C22⋊D61semidirect
417. C23#14519808semidirect

Maximal subgroups

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

subgroupindex
#214550772
#214491382
#214447762
#214447422
#214443522
#214441462
C2×C61×#5432
#1813361