Finite Group Databasebeta

← #29690558

Subgroups — #29690558

Order 15616.

806 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839814C211yesmaximal, characteristic
2#21448224C211yesmaximal, characteristic
3#21448215C211yesmaximal, characteristic
4#21446583C211yesmaximal, characteristic, complemented
5#21446577C211yesmaximal, characteristic, complemented
6#21446575C211yesmaximal, characteristic, complemented
7#21446019C212yesmaximal
8#21446017C212yesmaximal
9#21445980C212yesmaximal, complemented
10#21445948C212yesmaximal, complemented
11C61×#1948C211yesmaximal, characteristic, complemented
12C2×#14519735C2212yes—
13C2×C42×D61C2211yescharacteristic
14#16839814C418yescomplemented
15C2×#14519710C2212yes—
16C2×#14519710C2212yes—
17C2×#14519709C2212yes—
18C2×C4⋊#92418C2211yescharacteristic
19C2×#14519689C2212yes—
20C2×C8×Dic61C2212yes—
21#16839632C2211yescharacteristic
22#16839629—21no—
23#16839627C2211yescharacteristic
24#16839627C2211yescharacteristic
25#16839627C2212yes—
26#16839624—21no—
27#16839619—21no—
28#16839619—21no—
29#16839619C2211yescharacteristic
30#16839619C2211yescharacteristic
31C4⋊C4×Dic61—21no—
32C4⋊C4×Dic61—21no—
33C4×C4⋊Dic61C2211yescharacteristic
34C4×#11758198—21no—
35#16839608—21no—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39C2×#106×C61C2212yes—
40C2×C4⋊C8×C61C2211yescharacteristic
41C61×#384C2211yescharacteristic
42C4×C4⋊C4×C61C2211yescharacteristic
43C61×#340C2212yes—
44C22×C4×D61D411yescharacteristic
45C22×C4×D61D411yescharacteristic
46C22×C4×D61C2311yescharacteristic
47C22×#92418C2312yes—
48#14519735—42no—
49#14519735—42no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61C2312yes—
53C2×C4⋊Dic61C2311yescharacteristic
54C2×C4⋊Dic61C2311yescharacteristic
55C2×#11758198—22no—
56C2×#11758198—21no—
57C2×#11758198—21no—
58C2×C4×Dic61—21no—
59C2×C4×Dic61—22no—
60C2×C4×Dic61D411yescharacteristic
61C2×C4×Dic61D411yescharacteristic
62C2×C4×Dic61C2311yescharacteristic
63C42×D61C2×C414yes—
64C42×D61Q814yes—
65#14519710—42no—
66#14519710—42no—
67#14519709—42no—
68C4⋊#92418C2×C414yes—
69#14519689C2×C418yes—
70C8×Dic61C2×C418yes—
71#14519684—21no—
72#14519684—21no—
73#14519684—21no—
74#14519684—21no—
75#14519684—21no—
76#14519684—21no—
77#14519684—21no—
78#14519684—22no—
79#14519684—22no—
80C22×C8×C61C2312yes—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—21no—
83C2×C4⋊C4×C61C2312yes—
84C2×C4⋊C4×C61C2311yescharacteristic
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C42×C61—21no—
87C2×C42×C61C2311yescharacteristic
88#106×C61—42no—
89C4⋊C8×C61C2×C414yes—
90#95×C61—21no—
91C23×D61C2×D411yescharacteristic
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—22no—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2411yescharacteristic
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61—22no—
101C22×Dic61C2×D411yescharacteristic
102C22×Dic61C2×D411yescharacteristic
103C22×Dic61C2×D411yescharacteristic
104C2×C4×D61—22no—
105C2×C4×D61—22no—
106C2×C4×D61—21no—
107C2×C4×D61—21no—
108C2×C4×D61—21no—
109C2×C4×D61—21no—
110C2×C4×D61—41no—
111C2×C4×D61—41no—
112C2×C4×D61C4⋊C414yes—
113C2×C4×D61C4⋊C414yes—
114C2×C4×D61C22×C412yes—
115C2×C4×D61C2×Q812yes—
116C2×C8×C61—22no—
117C2×C8×C61—22no—
118C2×C8×C61C22×C414yes—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—21no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—42no—
126C4⋊C4×C61—42no—
127C42×C61—42no—
128C42×C61C22×C412yes—
129C42×C61C2×Q812yes—
130#11758198—42no—
131#11758198—42no—
132#11758198—44no—
133C4⋊Dic61—21no—
134C4⋊Dic61—22no—
135C4⋊Dic61—21no—
136C4⋊Dic61—22no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139C4⋊Dic61—42no—
140C4⋊Dic61—42no—
141C4×Dic61—44no—
142C4×Dic61—42no—
143C4×Dic61C4⋊C414yes—
144C4×Dic61C4⋊C414yes—
145C4×Dic61C22×C412yes—
146C4×Dic61C2×Q812yes—
147C2×#92418—22no—
148C2×#92418—22no—
149C2×#92418C22×C414yes—
150C23×C61C22×D411yescharacteristic
151C22×D61—21no—
152C22×D61—21no—
153C22×D61—21no—
154C22×D61—21no—
155C22×D61—21no—
156C22×D61—21no—
157C22×D61C2×C4⋊C412yes—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—42no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—42no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—42no—
169C2×C4×C61C2×C4⋊C412yes—
170C2×C4×C61C2×C4⋊C412yes—
171C2×C4×C61#13512yes—
172C2×C4×C61#13512yes—
173C2×C4×C61C23×C411yescharacteristic
174C2×C4×C61C22×Q811yescharacteristic
175C2×C4×C61H(5,𝔽2)11yescharacteristic
176C2×C4×C61H(5,𝔽2)11yescharacteristic
177C2×C4×C612−1+411yescharacteristic
178C2×C4×C612−1+411yescharacteristic
179C2×Dic61—21no—
180C2×Dic61—21no—
181C2×Dic61—21no—
182C2×Dic61—21no—
183C2×Dic61—21no—
184C2×Dic61—21no—
185C2×Dic61—42no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—42no—
191C2×Dic61—42no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61C2×C4⋊C412yes—
195C2×Dic61C2×C4⋊C412yes—
196C2×Dic61C2×C4⋊C412yes—
197C4×D61—22no—
198C4×D61—22no—
199C4×D61—22no—
200C4×D61—22no—
201C4×D61—28no—
202C4×D61—44no—
203C8×C61—24no—
204#92418—24no—
205#21985—611nomaximal, Sylow
206C22×C61C22×C4⋊C411yescharacteristic
207C22×C61#53511yescharacteristic
208C22×C61#53611yescharacteristic
209C22×C61C2×#13511yescharacteristic
210C22×C61C2×#13511yescharacteristic
211C22×C61#57511yescharacteristic
212C22×C61#57511yescharacteristic
213C2×D61—22no—
214C2×D61—22no—
215C2×D61—22no—
216C2×D61—41no—
217C2×D61—41no—
218C2×D61—41no—
219C2×D61—41no—
220C4×C61—22no—
221C4×C61—81no—
222C4×C61—82no—
223C4×C61—81no—
224C4×C61#42714yes—
225C4×C61#52011yescharacteristic
226C4×C61#52011yescharacteristic
227C4×C61#54411yescharacteristic
228C4×C61#54411yescharacteristic
229Dic61—22no—
230Dic61—22no—
231Dic61—22no—
232Dic61—82no—
233Dic61—81no—
234Dic61—81no—
235C2×#431—611no—
236#2285—611no—
237#2283—611no—
238#1948—611no—
239#1948D6111yescharacteristic, complemented
240#1947—611no—
241#1947—611no—
242#1854—612no—
243#1854—612no—
244#1846—612no—
245#1835—612no—
246C2×C61#231711yescharacteristic
247C2×C61C2×#42711yescharacteristic
248C2×C61#291111yescharacteristic
249C2×C61#321211yescharacteristic
250C2×C61#321211yescharacteristic
251C2×C61#323311yescharacteristic
252C2×C61#323311yescharacteristic
253D61—42no—
254C2×#117—612no—
255C22×C42—611no—
256#431—618no—
257C2×#107—612no—
258C2×#107—612no—
259C2×#106—612no—
260C2×#106C2×D6112yes—
261C2×C4⋊C8—611no—
262C2×C4⋊C8C2×D6111yescharacteristic
263C2×C22⋊C8—612no—
264C2×C4×C8—612no—
265#384—611no—
266#384C2×D6111yescharacteristic
267#383—1221no—
268#382—611no—
269#382—612no—
270#382—611no—
271#381—1221no—
272#380—611no—
273#380—611no—
274#380—1221no—
275#380—1221no—
276C4×C4⋊C4—611no—
277C4×C4⋊C4—1221no—
278C4×C4⋊C4—1221no—
279C4×C4⋊C4C2×D6111yescharacteristic
280C4×C22⋊C4—1221no—
281#376—1221no—
282#340—612no—
283#340—612no—
284#340—612no—
285#340C2×D6112yes—
286C61#2198511yescharacteristic, complemented, Sylow
287C23×C4—611no—
288C23×C4—611no—
289C23×C4—611no—
290C22×C8—612no—
291C22×C8C22×D6112yes—
292#117—2442no—
293#117—2442no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—612no—
297C2×C4⋊C4—611no—
298C2×C4⋊C4—611no—
299C2×C4⋊C4—1221no—
300C2×C4⋊C4—1221no—
301C2×C4⋊C4C22×D6112yes—
302C2×C4⋊C4C22×D6111yescharacteristic
303C2×C4⋊C4C22×D6111yescharacteristic
304C2×C22⋊C4—1222no—
305C2×C22⋊C4—1221no—
306C2×C22⋊C4—1221no—
307C2×C42—21no—
308C2×C42—611no—
309C2×C42—611no—
310C2×C42—611no—
311C2×C42—614no—
312C2×C42—614no—
313C2×C42—1222no—
314C2×C42—1221no—
315C2×C42C22×D6111yescharacteristic
316#107—2442no—
317#107—2442no—
318#106—42no—
319#106—2442no—
320C4⋊C8—614no—
321C4⋊C8C4×D6114yes—
322C22⋊C8—618no—
323C4×C8—618no—
324#95—21no—
325#95—1221no—
326#95—1221no—
327#95—1221no—
328#95—1221no—
329#95—1221no—
330#95—1221no—
331#95—1222no—
332#95—1222no—
333#95—1221no—
334C24—611no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—611no—
339C22×C4—611no—
340C22×C4—611no—
341C22×C4—612no—
342C22×C4—612no—
343C22×C4—614no—
344C22×C4—614no—
345C22×C4—1221no—
346C22×C4—1221no—
347C22×C4—1221no—
348C22×C4—1221no—
349C22×C4—1222no—
350C22×C4—1222no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—1222no—
354C22×C4—2441no—
355C22×C4—2441no—
356C22×C4D4×D6111yescharacteristic
357C22×C4D4×D6111yescharacteristic
358C22×C4C23×D6111yescharacteristic
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—614no—
362C2×C8—1222no—
363C2×C8—1222no—
364C2×C8C2×C4×D6114yes—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—1221no—
374C4⋊C4—1222no—
375C4⋊C4—1221no—
376C4⋊C4—1222no—
377C4⋊C4—1221no—
378C4⋊C4—1221no—
379C4⋊C4—2442no—
380C4⋊C4—2442no—
381C22⋊C4—2444no—
382C22⋊C4—2442no—
383C22⋊C4—2442no—
384C42—42no—
385C42—612no—
386C42—612no—
387C42—614no—
388C42—614no—
389C42—2444no—
390C42—2442no—
391C42C2×C4×D6112yes—
392C42Q8×D6112yes—
393C23—612no—
394C23—1221no—
395C23—1221no—
396C23—1221no—
397C23—1221no—
398C23—1221no—
399C23—1221no—
400C23C2×D4×D6111yescharacteristic
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—612no—
413C2×C4—612no—
414C2×C4—612no—
415C2×C4—1221no—
416C2×C4—1221no—
417C2×C4—1221no—
418C2×C4—1221no—
419C2×C4—1221no—
420C2×C4—1221no—
421C2×C4—1222no—
422C2×C4—1222no—
423C2×C4—1228no—
424C2×C4—1222no—
425C2×C4—1222no—
426C2×C4—2444no—
427C2×C4—2441no—
428C2×C4—2441no—
429C2×C4—2442no—
430C2×C4—2441no—
431C2×C4—2441no—
432C2×C4—2442no—
433C2×C4—2442no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4C4⋊C4×D6112yes—
437C2×C4C4⋊C4×D6112yes—
438C2×C4#1451979812yes—
439C2×C4#1451979812yes—
440C2×C4C22×C4×D6111yescharacteristic
441C2×C4C2×Q8×D6111yescharacteristic
442C2×C4#1451982211yescharacteristic
443C2×C4#1451982311yescharacteristic
444C2×C4#1451982411yescharacteristic
445C2×C4#1451982511yescharacteristic
446C8—24no—
447C8—1224no—
448C22—1222no—
449C22—1222no—
450C22—1222no—
451C22—2441no—
452C22—2441no—
453C22—2441no—
454C22—2441no—
455C22C2×C4⋊C4×D6111yescharacteristic
456C22#1684032811yescharacteristic
457C22#1684033511yescharacteristic
458C22C2×#1451979811yescharacteristic
459C22C2×#1451979811yescharacteristic
460C22#1684055811yescharacteristic
461C22#1684056111yescharacteristic
462C4—22no—
463C4—81no—
464C4—81no—
465C4—82no—
466C4—1222no—
467C4—1222no—
468C4—1222no—
469C4—4881no—
470C4—4882no—
471C4—4881no—
472C4#1683979714yes—
473C4#1684024111yescharacteristic
474C4#1684024411yescharacteristic
475C4#1684038911yescharacteristic
476C4#1684039211yescharacteristic
477C2—2442no—
478C2#2144844111yescharacteristic
479C2C2×#1683979711yescharacteristic
480C2#2145440611yescharacteristic
481C2#2145679411yescharacteristic
482C2#2145679911yescharacteristic
483C2#2145699711yescharacteristic
484C2#2145700211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1948C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1948D61
hypocenterC61#21985
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
4. #21446583C2semidirect
5. #21446577C2semidirect
6. #21446575C2semidirect
9. #21445980C2semidirect
10. #21445948C2semidirect
11. C61×#1948C2semidirect
14. #16839814C4semidirect
239. #1948D61semidirect
286. C61#21985semidirect

Maximal subgroups

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

subgroupindex
C2×#168398142
#214482242
#214482152
#214465832
#214465772
#214465752
#214460192
#214460172
#214459802
#214459482
C61×#19482
#2198561