Finite Group Databasebeta

← #29834829

Subgroups — #29834829

Order 15616.

584 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455396C211yesmaximal, characteristic
2C2×#16839799C211yesmaximal, characteristic, complemented
3#21448026C212yesmaximal, complemented
4#21448025C212yesmaximal, complemented
5#21448009C212yesmaximal, complemented
6#21448008C212yesmaximal, complemented
7#21447607C211yesmaximal, characteristic, complemented
8#21447605C211yesmaximal, characteristic, complemented
9#21447531C211yesmaximal, characteristic, complemented
10#21447429C211yesmaximal, characteristic, complemented
11C61×#2139C211yesmaximal, characteristic, complemented
12C22×C8×D61C2211yescharacteristic
13#16840269—21no—
14C2×#14519735—21no—
15C2×#14519717C2211yescharacteristic
16#16840000C2211yescharacteristic
17#16840000C2211yescharacteristic
18#16839995C2211yescharacteristic
19#16839995C2211yescharacteristic
20#16839988—21no—
21#16839988—21no—
22#16839800C2211yescharacteristic
23#16839799—21no—
24#16839799—21no—
25#16839799C2212yes—
26#16839799C2212yes—
27C2×#14519711C2211yescharacteristic
28C2×#14519708—21no—
29#16839758—21no—
30#16839590C2212yescomplemented
31#16839589C2212yescomplemented
32#16839588C2212yescomplemented
33#16839587C2212yescomplemented
34#16839566C2212yescomplemented
35#16839563C2212yescomplemented
36#16839541C2212yes—
37#16839541C2212yes—
38C2×#135×C61C2211yescharacteristic
39C2×M32×C61C2211yescharacteristic
40C2×#107×C61C2211yescharacteristic
41C61×#358C2212yescomplemented
42C61×#357C2212yescomplemented
43C22×C4×D61D411yescharacteristic
44C22×#92418D411yescharacteristic
45C2×C8×D61—22no—
46C2×C8×D61—22no—
47C2×C8×D61D411yescharacteristic
48C2×C8×D61D411yescharacteristic
49C2×C8×D61C2311yescharacteristic
50C2×C8×D61C2311yescharacteristic
51#14519758—41no—
52#14519753—21no—
53#14519753—21no—
54#14519748—21no—
55#14519748—21no—
56#14519745—41no—
57C4×C22⋊D61—41no—
58C4×C22⋊D61—41no—
59#14519736—21no—
60#14519735—22no—
61#14519735—22no—
62#14519735—41no—
63#14519735—41no—
64C2×C4⋊Dic61—21no—
65C2×#11758198—41no—
66C2×C4×Dic61—41no—
67#14519717—22no—
68C2×#11758185C2312yes—
69#14519711—21no—
70#14519711C2311yescharacteristic
71#14519711C2311yescharacteristic
72#14519711C2312yes—
73#14519709—21no—
74#14519708—22no—
75#14519708—22no—
76#14519701—21no—
77#14519701—21no—
78#14519698—21no—
79#14519698—21no—
80C2×#41×C61—21no—
81#135×C61—22no—
82#135×C61—22no—
83C2×Q16×C61C2311yescharacteristic
84C2×SD16×C61—21no—
85C2×D8×C61C2311yescharacteristic
86C22×C8×C61C2311yescharacteristic
87C2×C4⋊C4×C61—21no—
88M32×C61—22no—
89C2×C16×C61C2312yes—
90#107×C61—21no—
91#107×C61C2312yes—
92C23×D61D811yescharacteristic
93#41×C61—42no—
94#41×C61—42no—
95C2×Q8×C61—21no—
96C2×D4×C61—21no—
97C2×D4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61C2×D411yescharacteristic
101C2×C22⋊D61—41no—
102C2×C22⋊D61—41no—
103C22×Dic61—41no—
104C22×Dic61D811yescharacteristic
105C2×C4×D61—22no—
106C2×C4×D61—22no—
107C2×C4×D61D811yescharacteristic
108C2×C4×D61D811yescharacteristic
109C2×C4×D61C2×D411yescharacteristic
110C2×C4×D61C2×D411yescharacteristic
111Q16×C61—21no—
112Q16×C61—21no—
113SD16×C61—42no—
114D8×C61—21no—
115D8×C61—21no—
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61#4112yes—
118C2×C8×C61#4112yes—
119C2×C8×C61C2411yescharacteristic
120C4⋊C4×C61—22no—
121C4⋊C4×C61—41no—
122C22⋊Dic61—41no—
123C22⋊Dic61—41no—
124C22⋊Dic61—41no—
125#11758198—42no—
126#11758198—42no—
127#11758198—41no—
128#11758198—41no—
129C4⋊Dic61—21no—
130C4⋊Dic61—21no—
131C4⋊Dic61—22no—
132C4⋊Dic61—41no—
133#11758196—41no—
134#11758196—41no—
135C4×Dic61—41no—
136C4×Dic61—42no—
137C4×Dic61—42no—
138C2×#92418—22no—
139C2×#92418C2×D411yescharacteristic
140C2×#92418C2×D411yescharacteristic
141C8×D61—22no—
142C8×D61—22no—
143C8×D61—21no—
144C8×D61—21no—
145C8×D61—21no—
146C8×D61—21no—
147C16×C61—22no—
148#11758185—22no—
149C23×C61—41no—
150C23×C61C2×D811yescharacteristic
151C22×D61—22no—
152C22×D61—41no—
153C22×D61—41no—
154C22×D61C2×D811yescharacteristic
155C22×D61C2×D811yescharacteristic
156Q8×C61—41no—
157Q8×C61—41no—
158D4×C61—41no—
159D4×C61—41no—
160D4×C61—82no—
161C2×C4×C61—42no—
162C2×C4×C61—41no—
163C2×C4×C61—82no—
164C2×C4×C61—82no—
165C2×C4×C61#12312yes—
166C2×C4×C61C2×D811yescharacteristic
167C2×C4×C61#13712yes—
168C2×C4×C61C22×D411yescharacteristic
169C22⋊D61—82no—
170C22⋊D61—82no—
171C2×Dic61—22no—
172C2×Dic61—42no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—82no—
176C2×Dic61C2×D811yescharacteristic
177C2×Dic61C2×D811yescharacteristic
178C4×D61—21no—
179C4×D61—21no—
180C4×D61—21no—
181C4×D61—21no—
182C4×D61—22no—
183C4×D61—22no—
184C8×C61#12312yes—
185C8×C61C2×#4111yescharacteristic
186C8×C61C2×#4111yescharacteristic
187#92418—21no—
188#92418—21no—
189#33647—611nomaximal, Sylow
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—81no—
193C22×C61—81no—
194C22×C61—81no—
195C22×C61#48012yes—
196C22×C61C22×D811yescharacteristic
197C2×D61—21no—
198C2×D61—21no—
199C2×D61—21no—
200C2×D61—21no—
201C2×D61—21no—
202C2×D61—21no—
203C2×D61—44no—
204C4×C61—81no—
205C4×C61—82no—
206C4×C61#48012yes—
207C4×C61C2×#12311yescharacteristic
208C4×C61C2×#13711yescharacteristic
209Dic61—21no—
210Dic61—21no—
211Dic61—82no—
212Dic61—81no—
213Dic61—81no—
214#3050—611no—
215C2×#427—611no—
216#2237—612no—
217#2237—612no—
218#2234—612no—
219#2234—612no—
220#2159—611no—
221#2159—611no—
222#2139—611no—
223#2139D6111yescharacteristic, complemented
224#2112—611no—
225C2×C61—22no—
226C2×C61—161no—
227C2×C61C2×#48011yescharacteristic
228C2×C61#341411yescharacteristic
229C2×C61#341411yescharacteristic
230D61—41no—
231D61—41no—
232C2×#135C2×D6111yescharacteristic
233C23×C8—611no—
234#525—1221no—
235C2×#117—1221no—
236C2×M32—611no—
237C2×M32C2×D6111yescharacteristic
238#467—611no—
239#467—611no—
240#466—611no—
241#466—611no—
242#465—1221no—
243#465—1221no—
244#427—611no—
245#427—612no—
246#427—612no—
247#427—1221no—
248#427—1221no—
249C2×#107—611no—
250C2×#107C2×D6111yescharacteristic
251C2×#106—1221no—
252#416—1221no—
253#367—612no—
254#367—612no—
255#366—612no—
256#366—612no—
257#358—612no—
258#358C2×D6112yescomplemented
259#357—612no—
260#357C2×D6112yescomplemented
261C22⋊C16—612no—
262C22⋊C16—612no—
263C61#3364711yescharacteristic, complemented, Sylow
264C2×#41—21no—
265C23×C4—611no—
266#135—22no—
267#135—22no—
268C2×Q16C22×D6111yescharacteristic
269C2×SD16—21no—
270C2×D8C22×D6111yescharacteristic
271C22×C8—611no—
272C22×C8—611no—
273C22×C8—611no—
274C22×C8—611no—
275C22×C8—611no—
276C22×C8—1222no—
277C22×C8—1222no—
278C22×C8C22×D6111yescharacteristic
279#123—2441no—
280C22⋊Q8—1221no—
281C22⋊Q8—1221no—
282#121—1221no—
283#121—1221no—
284#120—2441no—
285C4×D4—2441no—
286C4×D4—2441no—
287#117—1222no—
288#117—1222no—
289#117—1221no—
290#117—2441no—
291#117—2441no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—1221no—
294C2×C22⋊C4—2441no—
295C2×C42—2441no—
296M32—22no—
297M32—1222no—
298C2×C16—612no—
299C2×C16C22×D6112yes—
300#107—21no—
301#107—612no—
302#107—611no—
303#107—611no—
304#107—1221no—
305#107C22×D6112yes—
306#106—1222no—
307#106—1222no—
308#106—1221no—
309#103—1221no—
310#103—1221no—
311#102—1221no—
312#102—1221no—
313C24—611no—
314#41—42no—
315#41—42no—
316C2×Q8—21no—
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—2441no—
320C2×D4—2441no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—611no—
324C22×C4—611no—
325C22×C4—611no—
326C22×C4—611no—
327C22×C4—611no—
328C22×C4—1222no—
329C22×C4—1222no—
330C22×C4—2441no—
331C22×C4D4×D6111yescharacteristic
332Q16—21no—
333Q16—21no—
334SD16—42no—
335D8—21no—
336D8—21no—
337C2×C8—611no—
338C2×C8—611no—
339C2×C8—1222no—
340C2×C8—1221no—
341C2×C8—1221no—
342C2×C8—1221no—
343C2×C8—1221no—
344C2×C8—1222no—
345C2×C8—1222no—
346C2×C8#1175822212yes—
347C2×C8D4×D6111yescharacteristic
348C2×C8#1175822412yes—
349C2×C8C23×D6111yescharacteristic
350C4⋊C4—22no—
351C4⋊C4—41no—
352C4⋊C4—1221no—
353C4⋊C4—1221no—
354C4⋊C4—1222no—
355C4⋊C4—2441no—
356C4⋊C4—2441no—
357C4⋊C4—2441no—
358C22⋊C4—2442no—
359C22⋊C4—2442no—
360C22⋊C4—2441no—
361C22⋊C4—2441no—
362C22⋊C4—2441no—
363C22⋊C4—2441no—
364C22⋊C4—2441no—
365C42—2442no—
366C42—2442no—
367C42—2441no—
368C16—22no—
369C16—1222no—
370C23—41no—
371C23—611no—
372C23—611no—
373C23—1222no—
374C23—2441no—
375C23—2441no—
376C23D8×D6111yescharacteristic
377Q8—41no—
378Q8—41no—
379D4—41no—
380D4—41no—
381D4—82no—
382D4—4882no—
383D4—4882no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—611no—
389C2×C4—611no—
390C2×C4—1222no—
391C2×C4—1221no—
392C2×C4—1221no—
393C2×C4—1221no—
394C2×C4—1221no—
395C2×C4—1222no—
396C2×C4—1222no—
397C2×C4—4882no—
398C2×C4—2442no—
399C2×C4—2441no—
400C2×C4—2441no—
401C2×C4#1451975912yes—
402C2×C4D8×D6111yescharacteristic
403C2×C4#1451980612yes—
404C2×C4C2×D4×D6111yescharacteristic
405C8—1221no—
406C8—1221no—
407C8#1451975912yes—
408C8C2×#1175822411yescharacteristic
409C8C2×#1175822211yescharacteristic
410C22—21no—
411C22—21no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—1221no—
416C22—1221no—
417C22—1221no—
418C22—1221no—
419C22—1221no—
420C22—1221no—
421C22—2444no—
422C22#1684007412yes—
423C22C2×D8×D6111yescharacteristic
424C4—82no—
425C4—81no—
426C4—1221no—
427C4—1221no—
428C4—4881no—
429C4—4882no—
430C4—4881no—
431C4#1684007412yes—
432C4C2×#1451975911yescharacteristic
433C4C2×#1451980611yescharacteristic
434C2—22no—
435C2—161no—
436C2—2441no—
437C2—2441no—
438C2C2×#1684007411yescharacteristic
439C2#2145895111yescharacteristic
440C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2139C2
socleC22×C61C22×D8
prime radical#21455396C2
hypercenter#2139D61
hypocenterC61#33647
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2139

Complements

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

NH ≅ G/Ntype
2. C2×#16839799C2semidirect
3. #21448026C2semidirect
4. #21448025C2semidirect
5. #21448009C2semidirect
6. #21448008C2semidirect
7. #21447607C2semidirect
8. #21447605C2semidirect
9. #21447531C2semidirect
10. #21447429C2semidirect
11. C61×#2139C2semidirect
30. #16839590C22semidirect
31. #16839589C22semidirect
32. #16839588C22semidirect
33. #16839587C22semidirect
34. #16839566C22semidirect
35. #16839563C22semidirect
41. C61×#358C22semidirect
42. C61×#357C22semidirect
223. #2139D61semidirect
258. #358C2×D61semidirect
260. #357C2×D61semidirect
263. C61#33647semidirect

Maximal subgroups

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

subgroupindex
#214553962
C2×#168397992
#214480262
#214480252
#214480092
#214480082
#214476072
#214476052
#214475312
#214474292
C61×#21392
#3364761