Finite Group Databasebeta

← #29718420

Subgroups — #29718420

Order 15616.

958 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840097C211yesmaximal, characteristic
2#21449459C211yesmaximal, characteristic
3#21448696C211yesmaximal, characteristic
4#21447210C212yesmaximal, complemented
5#21447064C212yesmaximal, complemented
6#21446740C212yesmaximal, complemented
7#21446463C212yesmaximal, complemented
8#21446242C211yesmaximal, characteristic
9#21445997C211yesmaximal, characteristic
10#21445754C211yesmaximal, characteristic
11C22×#103×C61C211yesmaximal, characteristic
12C2×#14519775C2212yes—
13C2×#14519756C2211yescharacteristic
14#16840097—24no—
15C2×#14519710C2211yescharacteristic
16C2×#14519710C2211yescharacteristic
17C2×#14519706C2212yes—
18C2×#14519702C2212yes—
19C2×#14519701C2212yes—
20C2×#14519701C2212yes—
21C2×#14519700C2211yescharacteristic
22C2×#14519700C2211yescharacteristic
23C2×C22⋊#92418C2211yescharacteristic
24#16839681—22no—
25#16839669C2212yes—
26#16839661C2212yes—
27#16839648—22no—
28#16839635—21no—
29#16839635—21no—
30#16839622—21no—
31#16839622C2211yescharacteristic
32C4⋊C4×Dic61C2212yes—
33C4×C22⋊Dic61—21no—
34C4×C22⋊Dic61C2211yescharacteristic
35#16839525C2212yes—
36#16839516C2212yes—
37C23×Q8×C61C2211yescharacteristic
38C23×C8×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2211yescharacteristic
40C2×#103×C61—24no—
41C2×#103×C61C2212yes—
42C2×#103×C61C2212yes—
43C22×Dic122C2311yescharacteristic
44C22×#92418C2311yescharacteristic
45C22×#92418C2311yescharacteristic
46#14519775—24no—
47#14519775—42no—
48#14519756—22no—
49#14519756—22no—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61C2311yescharacteristic
52C2×C4⋊Dic61C2312yes—
53C2×#11758196—21no—
54C2×#11758196—22no—
55C2×#11758196—22no—
56C2×C22⋊Dic61—21no—
57C2×C22⋊Dic61—21no—
58C2×C22⋊Dic61D411yescharacteristic
59C2×C22⋊Dic61D411yescharacteristic
60C2×C4×Dic61—22no—
61C2×C4×Dic61C2312yes—
62#14519710—22no—
63#14519710—22no—
64#14519706—42no—
65#14519702—42no—
66#14519701—24no—
67#14519701—42no—
68#14519700—22no—
69#14519700—22no—
70C22⋊#92418—22no—
71#14519684—21no—
72#14519684—22no—
73#14519684—22no—
74#14519684—22no—
75#14519684—22no—
76#14519684D412yes—
77C22×Q8×C61—21no—
78C22×Q8×C61—24no—
79C22×Q8×C61—22no—
80C22×Q8×C61—22no—
81C22×Q8×C61—42no—
82C22×Q8×C61C2312yes—
83C23×C4×C61—21no—
84C23×C4×C61—21no—
85C23×C4×C61C2311yescharacteristic
86C22×C8×C61—22no—
87C22×C8×C61—42no—
88C22×C8×C61C2312yes—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61D414yes—
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C4⋊C4×C61C2311yescharacteristic
94#103×C61—24no—
95#103×C61—24no—
96C24×C61C2×D411yescharacteristic
97C2×Q8×C61—22no—
98C2×Q8×C61—28no—
99C2×Q8×C61—22no—
100C2×Q8×C61—22no—
101C2×Q8×C61—22no—
102C2×Q8×C61—22no—
103C2×Q8×C61—44no—
104C2×Q8×C61—48no—
105C2×Q8×C61—44no—
106C2×Q8×C61—42no—
107C2×Q8×C61#4114yes—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—24no—
114C22×C4×C61—44no—
115C22×C4×C61—42no—
116C22×C4×C61—82no—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D412yes—
119C22×C4×C61#4112yes—
120C22×C4×C61C2411yescharacteristic
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—22no—
124C22×Dic61C2×D412yes—
125C22×Dic61C2×D412yes—
126C2×Dic122—21no—
127C2×Dic122—21no—
128C2×Dic122—22no—
129C2×Dic122—22no—
130C2×C8×C61—22no—
131C2×C8×C61—22no—
132C2×C8×C61—22no—
133C2×C8×C61—44no—
134C4⋊C4×C61—24no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61C2×D412yes—
138C4⋊C4×C61C2×D412yes—
139C22⋊Dic61—22no—
140C22⋊Dic61—42no—
141C22⋊Dic61—42no—
142C22⋊Dic61Q1614yes—
143C4⋊Dic61—21no—
144C4⋊Dic61—22no—
145C4⋊Dic61—21no—
146C4⋊Dic61—22no—
147C4⋊Dic61—82no—
148#11758196—44no—
149#11758196—44no—
150#11758196—42no—
151C4×Dic61—22no—
152C4×Dic61—22no—
153C4×Dic61—82no—
154C2×#92418—21no—
155C2×#92418—21no—
156C2×#92418—22no—
157C2×#92418—22no—
158C23×C61—22no—
159C23×C61—22no—
160C23×C61#12112yes—
161C23×C61C2×Q1612yes—
162C23×C61#13712yes—
163C23×C61C22×D411yescharacteristic
164Q8×C61—24no—
165Q8×C61—24no—
166Q8×C61—48no—
167Q8×C61—42no—
168Q8×C61—42no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—24no—
173C2×C4×C61—22no—
174C2×C4×C61—22no—
175C2×C4×C61—24no—
176C2×C4×C61—48no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—84no—
182C2×C4×C61#12112yes—
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C2×#4112yes—
185C2×C4×C61C2×#4111yescharacteristic
186C2×C4×C61H(5,𝔽2)12yes—
187C2×Dic61—22no—
188C2×Dic61—22no—
189C2×Dic61—22no—
190C2×Dic61—22no—
191C2×Dic61—22no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—42no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61C2×Q1614yes—
200Dic122—41no—
201Dic122—41no—
202Dic122—42no—
203C8×C61—42no—
204#92418—41no—
205#92418—41no—
206#24482—611nomaximal, Sylow
207C22×C61—22no—
208C22×C61—22no—
209C22×C61—24no—
210C22×C61—24no—
211C22×C61#46214yes—
212C22×C61C2×#12111yescharacteristic
213C22×C61#54612yes—
214C22×C61C22×Q1611yescharacteristic
215C22×C61C2×#13711yescharacteristic
216C22×C61#57712yes—
217C4×C61—22no—
218C4×C61—44no—
219C4×C61—42no—
220C4×C61—42no—
221C4×C61#52511yescharacteristic
222C4×C61#53511yescharacteristic
223C4×C61#55112yes—
224Dic61—24no—
225Dic61—42no—
226Dic61—81no—
227Dic61—81no—
228Dic61—82no—
229C2×#484—611no—
230C22×#103D6111yescharacteristic
231#2434—611no—
232#2345—611no—
233#2068—612no—
234#2043—612no—
235#1981—612no—
236#1928—612no—
237#1892—611no—
238#1850—611no—
239#1787—611no—
240C2×C61—24no—
241C2×C61#241111yescharacteristic
242C2×C61C2×#46211yescharacteristic
243C2×C61#304211yescharacteristic
244C2×C61#329912yes—
245C2×C61#332012yes—
246C23×Q8C2×D6111yescharacteristic
247C23×C8C2×D6111yescharacteristic
248C2×C4⋊Q8—612no—
249C2×C22⋊Q8—611no—
250C22×C4⋊C4C2×D6111yescharacteristic
251#484—1224no—
252C2×#107—611no—
253C2×#107—611no—
254C2×C4⋊C8—612no—
255C2×#103—24no—
256C2×#103—612no—
257C2×#103—611no—
258C2×#103—611no—
259C2×#103—612no—
260C2×#103—612no—
261C2×#103C2×D6112yes—
262C2×#103C2×D6112yes—
263C2×C22⋊C8—611no—
264#395—1222no—
265#391—612no—
266#389—612no—
267#387—1222no—
268#385—1221no—
269#385—1221no—
270#380—611no—
271#380—1221no—
272C4×C4⋊C4—612no—
273C4×C22⋊C4—611no—
274C4×C22⋊C4—1221no—
275#340—612no—
276#336—612no—
277C61#2448211yescharacteristic, complemented, Sylow
278C22×Q8—21no—
279C22×Q8—24no—
280C22×Q8—22no—
281C22×Q8—22no—
282C22×Q8—42no—
283C22×Q8—611no—
284C22×Q8C22×D6112yes—
285C23×C4—21no—
286C23×C4—21no—
287C23×C4C22×D6111yescharacteristic
288C22×C8—22no—
289C22×C8—42no—
290C22×C8—611no—
291C22×C8—611no—
292C22×C8C22×D6112yes—
293C4⋊Q8—1224no—
294C4⋊Q8—2442no—
295C22⋊Q8—1222no—
296C22⋊Q8—1222no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—611no—
300C2×C4⋊C4—612no—
301C2×C4⋊C4—1222no—
302C2×C4⋊C4—1222no—
303C2×C4⋊C4—1222no—
304C2×C4⋊C4—1221no—
305C2×C4⋊C4C22⋊D6114yes—
306C2×C4⋊C4C22×D6111yescharacteristic
307C2×C4⋊C4C22×D6111yescharacteristic
308C2×C22⋊C4—611no—
309C2×C22⋊C4—611no—
310C2×C22⋊C4—1221no—
311C2×C22⋊C4—1221no—
312C2×C42—612no—
313C2×C42—1222no—
314#107—1222no—
315#107—1222no—
316C4⋊C8—2442no—
317#103—24no—
318#103—24no—
319#103—1224no—
320#103—1222no—
321#103—1222no—
322#103—2442no—
323#103—2442no—
324C22⋊C8—1222no—
325#95—612no—
326#95—1222no—
327#95—1222no—
328#95—1222no—
329#95—1222no—
330#95—1221no—
331C24D4×D6111yescharacteristic
332C2×Q8—22no—
333C2×Q8—28no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—44no—
339C2×Q8—48no—
340C2×Q8—44no—
341C2×Q8—42no—
342C2×Q8—1221no—
343C2×Q8—1221no—
344C2×Q8—1222no—
345C2×Q8—1222no—
346C2×Q8#1175822214yes—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—82no—
356C22×C4—612no—
357C22×C4—612no—
358C22×C4—1221no—
359C22×C4—1222no—
360C22×C4—1221no—
361C22×C4D4×D6111yescharacteristic
362C22×C4#1175822412yes—
363C22×C4C2×C22⋊D6112yes—
364C22×C4C23×D6111yescharacteristic
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—44no—
369C2×C8—1221no—
370C2×C8—1221no—
371C2×C8—1222no—
372C2×C8—1222no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—24no—
376C4⋊C4—1221no—
377C4⋊C4—1221no—
378C4⋊C4—1222no—
379C4⋊C4—1222no—
380C4⋊C4—4882no—
381C4⋊C4—2444no—
382C4⋊C4—2444no—
383C4⋊C4—2442no—
384C4⋊C4C2×C22⋊D6112yes—
385C4⋊C4C2×C22⋊D6112yes—
386C22⋊C4—614no—
387C22⋊C4—1222no—
388C22⋊C4—2442no—
389C22⋊C4—2442no—
390C42—1222no—
391C42—1222no—
392C42—4882no—
393C23—22no—
394C23—22no—
395C23#1451974912yes—
396C23Q16×D6112yes—
397C23#1451980912yes—
398C23C2×D4×D6111yescharacteristic
399Q8—24no—
400Q8—24no—
401Q8—48no—
402Q8—42no—
403Q8—42no—
404Q8—2441no—
405Q8—2441no—
406Q8—2442no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—48no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—84no—
420C2×C4—614no—
421C2×C4—1222no—
422C2×C4—1222no—
423C2×C4—1222no—
424C2×C4—1222no—
425C2×C4—1222no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4—2442no—
429C2×C4—2442no—
430C2×C4—2442no—
431C2×C4—2442no—
432C2×C4—2442no—
433C2×C4#1451974912yes—
434C2×C4C22×C22⋊D6111yescharacteristic
435C2×C4C2×#1175822411yescharacteristic
436C2×C4C2×#1175822212yes—
437C2×C4#1451982212yes—
438C8—42no—
439C8—2441no—
440C8—2441no—
441C22—24no—
442C22—24no—
443C22—22no—
444C22—22no—
445C22#1683997014yes—
446C22C2×#1451974911yescharacteristic
447C22#1684040212yes—
448C22C2×Q16×D6111yescharacteristic
449C22C2×#1451980911yescharacteristic
450C22#1684056812yes—
451C4—22no—
452C4—42no—
453C4—42no—
454C4—44no—
455C4—1224no—
456C4—4882no—
457C4—4881no—
458C4—4881no—
459C4—2442no—
460C4#1684027311yescharacteristic
461C4#1684032911yescharacteristic
462C4#1684043812yes—
463C2—24no—
464C2#2144920411yescharacteristic
465C2C2×#1683997011yescharacteristic
466C2#2145532811yescharacteristic
467C2#2145769112yes—
468C2#2145798612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#103×C61C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenterC22×#103D61
hypocenterC61#24482
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
4. #21447210C2semidirect
5. #21447064C2semidirect
6. #21446740C2semidirect
7. #21446463C2semidirect
277. C61#24482semidirect

Maximal subgroups

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

subgroupindex
C2×#168400972
#214494592
#214486962
#214472102
#214470642
#214467402
#214464632
#214462422
#214459972
#214457542
C22×#103×C612
#2448261