Finite Group Databasebeta

← #29698501

Subgroups — #29698501

Order 15616.

846 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839877C211yesmaximal, characteristic
2#21454700C212yesmaximal
3#21454416C211yesmaximal, characteristic, complemented
4#21446596C212yesmaximal, complemented
5#21446052C212yesmaximal, complemented
6#21446049C212yesmaximal, complemented
7#21445982C211yesmaximal, characteristic, complemented
8#21445884C211yesmaximal, characteristic, complemented
9#21445883C211yesmaximal, characteristic, complemented
10#21445700C211yesmaximal, characteristic, complemented
11C61×#2913C211yesmaximal, characteristic, complemented
12C2×#14519797C2211yescharacteristic
13C2×#14519783C2211yescharacteristic
14#16840242C2212yes—
15#16839877C418yescomplemented
16#16839872—22no—
17#16839867—22no—
18C4×#11758201—22no—
19C4×#11758200—22no—
20C2×#14519713C2211yescharacteristic
21C2×#14519712C2211yescharacteristic
22#16839805C2212yes—
23#16839801—22no—
24#16839799—21no—
25C2×#14519711C2211yescharacteristic
26#107×D61—21no—
27C2×#14519708C2212yes—
28C2×#14519704C2211yescharacteristic
29C2×#14519703C2211yescharacteristic
30#16839763C2212yes—
31#16839761C2212yes—
32#16839718C2212yes—
33C2×C8×Dic61C2211yescharacteristic
34#16839627C2212yes—
35#16839531C2212yes—
36#16839530C2212yes—
37#16839521C2212yes—
38C2×#131×C61C2211yescharacteristic
39C61×#520C2212yes—
40C61×#428—22no—
41C61×#428C2212yes—
42C61×#427—21no—
43C61×#427C2211yescharacteristic
44C2×#107×C61—21no—
45C2×#107×C61C2211yescharacteristic
46C2×#11758222C2311yescharacteristic
47#14519797C2×C414yes—
48#14519797Q814yes—
49C2×#11758202D411yescharacteristic
50C2×#11758200D411yescharacteristic
51C2×#11758201D411yescharacteristic
52C2×#11758199D411yescharacteristic
53#14519783—22no—
54#14519783C2×C414yescomplemented
55C2×#11758194C2312yes—
56C2×#11758194C2311yescharacteristic
57C2×#11758189—21no—
58C22×#92418C2311yescharacteristic
59C2×C8×D61—21no—
60C4×Dic122—22no—
61C4×D244—22no—
62#14519737—42no—
63#14519735—22no—
64C2×C4⋊Dic61C2312yes—
65C4⋊C4×D61—22no—
66C2×C4×Dic61C2311yescharacteristic
67#14519713—22no—
68#14519712C2×C414yescomplemented
69#14519711—21no—
70#14519711—21no—
71#14519710—22no—
72#14519708—22no—
73#14519708—22no—
74#14519708—22no—
75#14519704C2×C414yescomplemented
76#14519703C2×C414yescomplemented
77#14519701—22no—
78#14519700—22no—
79#14519698—22no—
80#14519697—22no—
81#14519688—22no—
82C4×#92418—22no—
83C8×Dic61C2×C414yescomplemented
84#14519684—42no—
85C2×#41×C61C2311yescharacteristic
86#131×C61—22no—
87#131×C61C2×C414yescomplemented
88C2×M16×C61—21no—
89C2×M16×C61C2311yescharacteristic
90C22×C8×C61—21no—
91C22×C8×C61C2311yescharacteristic
92C4×Q8×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61—42no—
95#117×C61—22no—
96#117×C61C2312yes—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61C2312yes—
99#107×C61—21no—
100#107×C61—21no—
101#107×C61—22no—
102#107×C61D411yescharacteristic
103#107×C61D411yescharacteristic
104#106×C61—22no—
105#106×C61D412yes—
106#41×C61—21no—
107#41×C61—21no—
108#41×C61C22×C412yes—
109#41×C61C2×Q812yes—
110C2×Q8×C61C2×D411yescharacteristic
111C2×D4×C61—21no—
112C2×D4×C61C2×D411yescharacteristic
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—42no—
116C22×C4×C61C2411yescharacteristic
117C2×C22⋊D61—21no—
118C22×Dic61—21no—
119C22×Dic61—22no—
120#11758222—21no—
121#11758222—21no—
122#11758222C22×C412yes—
123#11758222C2×Q812yes—
124C2×D244C2×D411yescharacteristic
125C2×C4×D61—21no—
126C2×C4×D61—42no—
127C2×Dic122C2×D411yescharacteristic
128M16×C61—21no—
129M16×C61—24no—
130M16×C61C22×C412yes—
131C2×C8×C61—21no—
132C2×C8×C61—24no—
133C2×C8×C61C22×C412yes—
134C2×C8×C61C2×D411yescharacteristic
135C2×C8×C61C2×D411yescharacteristic
136C2×C8×C61#4112yes—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—42no—
140C4⋊C4×C61C2×D412yes—
141C4⋊C4×C61C2×D412yes—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—42no—
144C42×C61—22no—
145C42×C61—42no—
146#11758202C4⋊C414yes—
147#11758201C4⋊C414yes—
148#11758200C4⋊C414yes—
149#11758199C4⋊C414yes—
150#11758198—42no—
151C4⋊Dic61—22no—
152C4⋊Dic61—22no—
153#11758196—42no—
154C4×Dic61—21no—
155C4×Dic61—42no—
156C4×Dic61C22×C412yes—
157#11758194—21no—
158#11758194—22no—
159#11758194—22no—
160#11758194C22×C412yes—
161C2×#92418—21no—
162C2×#92418—22no—
163C2×#92418C22×C412yes—
164C2×#92418C2×D411yescharacteristic
165C2×#92418C2×D411yescharacteristic
166C2×#92418C2×Q812yes—
167#11758189—24no—
168C8×D61—24no—
169C23×C61—21no—
170C23×C61H(5,𝔽2)11yescharacteristic
171C22×D61—21no—
172Q8×C61—21no—
173Q8×C61C2×C4⋊C412yes—
174D4×C61—22no—
175D4×C61—22no—
176D4×C61—21no—
177D4×C61C2×C4⋊C412yes—
178C2×C4×C61—21no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—42no—
184C2×C4×C61—82no—
185C2×C4×C61—162no—
186C2×C4×C61C23×C411yescharacteristic
187C2×C4×C61C22×D411yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C22×Q811yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C2×C4×C612−1+411yescharacteristic
192C22⋊D61—22no—
193C22⋊D61—22no—
194C2×Dic61—21no—
195C2×Dic61—22no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—42no—
199C2×Dic61—82no—
200D244—21no—
201D244C2×C4⋊C412yes—
202C4×D61—22no—
203C4×D61—22no—
204C4×D61—162no—
205Dic122—21no—
206Dic122C2×C4⋊C412yes—
207C8×C61—22no—
208C8×C61C4×D412yes—
209C8×C61C4×D412yes—
210#92418—22no—
211#92418—42no—
212#92418C2×C4⋊C412yes—
213#92418C2×C4⋊C412yes—
214#22691—611nomaximal, Sylow
215C22×C61—21no—
216C22×C61—21no—
217C22×C61—22no—
218C22×C61—41no—
219C22×C61—41no—
220C22×C61#52011yescharacteristic
221C22×C61D4211yescharacteristic
222C22×C61#55411yescharacteristic
223C2×D61—22no—
224C2×D61—41no—
225C2×D61—41no—
226C4×C61—22no—
227C4×C61—42no—
228C4×C61—42no—
229C4×C61—82no—
230C4×C61C22×C4⋊C411yescharacteristic
231C4×C61C2×C4×D411yescharacteristic
232C4×C61#54711yescharacteristic
233C4×C61D4×Q811yescharacteristic
234Dic61—22no—
235Dic61—42no—
236Dic61—82no—
237C2×#443—611no—
238#2952—612no—
239#2913—611no—
240#2913D6111yescharacteristic, complemented
241#1951—612no—
242#1860—612no—
243#1860—612no—
244#1846—611no—
245#1816—611no—
246#1816—611no—
247#1772—611no—
248C2×C61—21no—
249C2×C61—21no—
250C2×C61—42no—
251C2×C61D4×C4⋊C411yescharacteristic
252C2×C61#329011yescharacteristic
253C2×C61#329011yescharacteristic
254D61—42no—
255C2×#135—611no—
256C2×#131—611no—
257C2×#131C2×D6111yescharacteristic
258#520—612no—
259#520C2×D6112yes—
260#443—618no—
261#442—1222no—
262#441—1222no—
263C4×SD16—1222no—
264C4×SD16—1222no—
265C2×#108—611no—
266C2×#108—611no—
267#428—22no—
268#428—612no—
269#428—1222no—
270#428C2×D6112yes—
271#427—21no—
272#427—1221no—
273#427C2×D6111yescharacteristic
274C2×#107—21no—
275C2×#107—611no—
276C2×#107—1221no—
277C2×#107C2×D6111yescharacteristic
278C2×#106—612no—
279C2×#104—611no—
280C2×#104—611no—
281#417—612no—
282#417—612no—
283#405—612no—
284C2×C4×C8—611no—
285#382—612no—
286#343—612no—
287#343—612no—
288#339—612no—
289C61#2269111yescharacteristic, complemented, Sylow
290C2×#41—611no—
291C2×#41C22×D6111yescharacteristic
292#135—614no—
293#135—614no—
294C2×Q16—611no—
295C2×SD16—611no—
296C2×SD16—611no—
297C2×D8—611no—
298#131—22no—
299#131—614no—
300#131—1222no—
301#131C4×D6114yescomplemented
302C2×M16—21no—
303C2×M16—611no—
304C2×M16—612no—
305C2×M16—1221no—
306C2×M16C22×D6111yescharacteristic
307C22×C8—21no—
308C22×C8—611no—
309C22×C8—1221no—
310C22×C8C22×D6111yescharacteristic
311C4×Q8—22no—
312C4×Q8—1222no—
313C4×D4—22no—
314C4×D4—42no—
315C4×D4—1222no—
316C4×D4—2442no—
317#117—22no—
318#117—1222no—
319#117C22×D6112yes—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—612no—
322C2×C4⋊C4—1222no—
323C2×C4⋊C4C22×D6112yes—
324C2×C42—611no—
325#108—614no—
326#108—1222no—
327#107—21no—
328#107—21no—
329#107—22no—
330#107—1221no—
331#107—1221no—
332#107—1222no—
333#107C22⋊D6111yescharacteristic
334#107C22⋊D6111yescharacteristic
335#106—22no—
336#106—1222no—
337#106—1222no—
338#106—1222no—
339#106C22⋊D6112yes—
340#104—614no—
341#104—614no—
342#103—1222no—
343#103—1222no—
344#102—1222no—
345#102—1222no—
346#97—1222no—
347C4×C8—614no—
348C4×C8—1222no—
349#95—2442no—
350#41—21no—
351#41—21no—
352#41—612no—
353#41—612no—
354#41—1221no—
355#41—1221no—
356#41C2×C4×D6112yes—
357#41Q8×D6112yes—
358C2×Q8—611no—
359C2×Q8D4×D6111yescharacteristic
360C2×D4—21no—
361C2×D4—611no—
362C2×D4—1221no—
363C2×D4D4×D6111yescharacteristic
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—1221no—
368C22×C4—1221no—
369C22×C4—1222no—
370C22×C4—2442no—
371C22×C4C23×D6111yescharacteristic
372Q16—614no—
373SD16—614no—
374SD16—614no—
375D8—614no—
376M16—21no—
377M16—24no—
378M16—612no—
379M16—1221no—
380M16—1224no—
381M16—1222no—
382M16—1222no—
383M16C2×C4×D6112yes—
384C2×C8—21no—
385C2×C8—24no—
386C2×C8—612no—
387C2×C8—611no—
388C2×C8—611no—
389C2×C8—612no—
390C2×C8—1221no—
391C2×C8—1224no—
392C2×C8—1222no—
393C2×C8C2×C4×D6112yes—
394C2×C8#1175822212yes—
395C2×C8C2×C22⋊D6111yescharacteristic
396C2×C8C2×C22⋊D6111yescharacteristic
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—42no—
400C4⋊C4—1222no—
401C4⋊C4—1222no—
402C4⋊C4—2442no—
403C4⋊C4C2×C22⋊D6112yes—
404C4⋊C4C2×C22⋊D6112yes—
405C22⋊C4—22no—
406C22⋊C4—42no—
407C22⋊C4—2442no—
408C42—22no—
409C42—42no—
410C42—612no—
411C42—1221no—
412C42—2442no—
413C23—21no—
414C23—1221no—
415C23#1451982211yescharacteristic
416Q8—21no—
417Q8—612no—
418Q8—1221no—
419Q8C4⋊C4×D6112yes—
420D4—22no—
421D4—22no—
422D4—21no—
423D4—612no—
424D4—1222no—
425D4—1222no—
426D4—1221no—
427D4C4⋊C4×D6112yes—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—42no—
434C2×C4—82no—
435C2×C4—162no—
436C2×C4—1221no—
437C2×C4—1222no—
438C2×C4—1222no—
439C2×C4—1222no—
440C2×C4—9762no—
441C2×C4—4882no—
442C2×C4—2442no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4C22×C4×D6111yescharacteristic
446C2×C4C2×D4×D6111yescharacteristic
447C2×C4C22×C22⋊D6111yescharacteristic
448C2×C4C2×Q8×D6111yescharacteristic
449C2×C4C2×#1175822211yescharacteristic
450C2×C4#1451982511yescharacteristic
451C8—22no—
452C8—612no—
453C8—612no—
454C8—1222no—
455C8—2442no—
456C8C4×C22⋊D6112yes—
457C8C4×C22⋊D6112yes—
458C22—21no—
459C22—21no—
460C22—22no—
461C22—41no—
462C22—41no—
463C22—1222no—
464C22—2441no—
465C22—2441no—
466C22#1684024311yescharacteristic
467C22D4×C22⋊D6111yescharacteristic
468C22#1684046711yescharacteristic
469C4—22no—
470C4—42no—
471C4—42no—
472C4—82no—
473C4—1222no—
474C4—4882no—
475C4—2442no—
476C4C2×C4⋊C4×D6111yescharacteristic
477C4C2×C4×C22⋊D6111yescharacteristic
478C4#1684041311yescharacteristic
479C4Q8×C22⋊D6111yescharacteristic
480C2—21no—
481C2—21no—
482C2—42no—
483C2—2442no—
484C2C4⋊C4×C22⋊D6111yescharacteristic
485C2#2145760111yescharacteristic
486C2#2145760511yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2913C2
socleC22×C61D42
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenter#2913D61
hypocenterC61#22691
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2913

Complements

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

NH ≅ G/Ntype
3. #21454416C2semidirect
4. #21446596C2semidirect
5. #21446052C2semidirect
6. #21446049C2semidirect
7. #21445982C2semidirect
8. #21445884C2semidirect
9. #21445883C2semidirect
10. #21445700C2semidirect
11. C61×#2913C2semidirect
15. #16839877C4semidirect
54. #14519783C2×C4semidirect
68. #14519712C2×C4semidirect
75. #14519704C2×C4semidirect
76. #14519703C2×C4semidirect
83. C8×Dic61C2×C4semidirect
87. #131×C61C2×C4semidirect
240. #2913D61semidirect
289. C61#22691semidirect
301. #131C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398772
#214547002
#214544162
#214465962
#214460522
#214460492
#214459822
#214458842
#214458832
#214457002
C61×#29132
#2269161