Finite Group Databasebeta

← #29663531

Subgroups — #29663531

Order 15616.

640 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457136C211yesmaximal, characteristic
2#21456949C211yesmaximal, characteristic
3#21456365C211yesmaximal, characteristic, complemented
4#21454565C211yesmaximal, characteristic
5#21445343C212yesmaximal, complemented
6#21445319C212yesmaximal, complemented
7#21445219C212yesmaximal, complemented
8#21445195C212yesmaximal, complemented
9#21444474C211yesmaximal, characteristic, complemented
10C2×#16839510C211yesmaximal, characteristic, complemented
11C61×#1565C211yesmaximal, characteristic, complemented
12C2×#14519785—21no—
13#16840319C2211yescharacteristic
14#16840312C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16#16840171—22no—
17#16840167C2212yes—
18#16840160—21no—
19#16840160C2212yes—
20#16840109—21no—
21#16840081—22no—
22#16840053C2212yes—
23#16840053C2212yes—
24#16840053C2212yes—
25#16840048—21no—
26#16839991—22no—
27#16839968C2212yes—
28#16839963—21no—
29#16839841C2212yes—
30#16839835—22no—
31C8×D244C2212yes—
32#16839804—21no—
33#16839804—21no—
34C2×C4⋊#92418C2211yescharacteristic
35#16839730—21no—
36#16839510C2212yes—
37#16839510C2211yescharacteristic
38#16839510C2212yes—
39#16839510C2211yescharacteristic
40#16839509C2211yescharacteristic
41#16839509C2211yescharacteristic
42C61×#431C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C4×M16×C61C2211yescharacteristic
45C61×#335C2212yes—
46C61×#334C2212yes—
47C2×#11758222—21no—
48C22×Dic122—21no—
49C2×#11758190—22no—
50#14519785—44no—
51C2×#11758189—22no—
52C22×#92418—21no—
53C2×C8×D61—22no—
54#14519777—21no—
55#14519777—21no—
56#14519777—21no—
57#14519777D411yescharacteristic
58#14519777C2311yescharacteristic
59#14519777C2312yes—
60#14519777C2311yescharacteristic
61#14519774D411yescharacteristic
62#14519769—21no—
63#14519765—21no—
64#14519756—21no—
65#14519756—21no—
66#14519751—21no—
67#14519751D412yes—
68C4×Dic122—21no—
69C4×Dic122—21no—
70C4×D244—21no—
71C4×D244C2312yes—
72#14519736—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75#14519711—22no—
76#14519711—22no—
77#14519709—22no—
78#14519709—22no—
79#14519709—22no—
80#14519709—22no—
81#14519709—22no—
82C4⋊#92418—21no—
83C4⋊#92418C2311yescharacteristic
84C4⋊#92418C2311yescharacteristic
85#14519702—22no—
86#14519699—22no—
87#14519699—22no—
88#14519699—22no—
89#14519689—22no—
90#14519689—22no—
91C2×M16×C61D411yescharacteristic
92C2×M16×C61D411yescharacteristic
93C22×C8×C61—21no—
94C2×C42×C61C2311yescharacteristic
95C4⋊C8×C61—21no—
96C4⋊C8×C61C2312yes—
97C4⋊C8×C61C2311yescharacteristic
98C4⋊C8×C61C2311yescharacteristic
99C22⋊C8×C61—21no—
100#97×C61D412yes—
101C4×C8×C61—21no—
102C4×C8×C61C2312yes—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61#4111yescharacteristic
106C2×C22⋊D61—22no—
107C22×Dic61—41no—
108#11758222—42no—
109#11758222—42no—
110C2×D244—21no—
111C2×D244—21no—
112C2×D244C2×D412yes—
113C2×C4×D61—22no—
114C2×Dic122—21no—
115C2×Dic122—21no—
116C2×Dic122—22no—
117C2×Dic122—21no—
118C2×Dic122—41no—
119C2×Dic122—82no—
120M16×C61—22no—
121M16×C61SD1614yescomplemented
122C2×C8×C61—21no—
123C2×C8×C61—21no—
124C2×C8×C61—22no—
125C2×C8×C61—81no—
126C2×C8×C61C2×D412yes—
127C2×C8×C61C2×D412yes—
128C42×C61C2×D411yescharacteristic
129C42×C61C2×D411yescharacteristic
130C42×C61#4111yescharacteristic
131C42×C61C2411yescharacteristic
132C22⋊Dic61—41no—
133#11758198—22no—
134#11758198—22no—
135C4⋊Dic61—21no—
136C4⋊Dic61—22no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139C4⋊Dic61—22no—
140C4⋊Dic61—21no—
141C4⋊Dic61—21no—
142C4⋊Dic61—41no—
143C4⋊Dic61—41no—
144C4⋊Dic61C2×D412yes—
145#11758196—41no—
146#11758196—41no—
147C4×Dic61—41no—
148C2×#92418—21no—
149C2×#92418—21no—
150C2×#92418—42no—
151#11758190—82no—
152#11758189—44no—
153C8×D61—44no—
154C23×C61#12111yescharacteristic
155C22×D61—22no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—41no—
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61C2×SD1612yes—
163C2×C4×C61#13712yes—
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C22×D411yescharacteristic
166C2×C4×C61C2×#4111yescharacteristic
167C22⋊D61—44no—
168C2×Dic61—22no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—42no—
172C2×Dic61—82no—
173D244—22no—
174D244—22no—
175D244—42no—
176D244—81no—
177C4×D61—42no—
178C4×D61—42no—
179Dic122—42no—
180Dic122—41no—
181Dic122—41no—
182Dic122—82no—
183Dic122—81no—
184Dic122—82no—
185C8×C61—22no—
186C8×C61—42no—
187C8×C61—41no—
188C8×C61—41no—
189C8×C61C2×SD1614yes—
190#92418—41no—
191#92418—41no—
192#19233—611nomaximal, Sylow
193C22×C61—21no—
194C22×C61—21no—
195C22×C61#46112yes—
196C22×C61C2×#12111yescharacteristic
197C2×D61—42no—
198C2×D61—42no—
199C2×D61—42no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61#46112yes—
204C4×C61C22×SD1611yescharacteristic
205C4×C61C2×#13711yescharacteristic
206C4×C61#57711yescharacteristic
207C4×C61#57811yescharacteristic
208Dic61—42no—
209Dic61—81no—
210Dic61—82no—
211Dic61—81no—
212#3246—611no—
213#3227—611no—
214#3165—611no—
215#2935—611no—
216#1695—612no—
217#1691—612no—
218#1675—612no—
219#1671—612no—
220#1565D6111yescharacteristic, complemented
221#1542—611no—
222C2×#333—611no—
223C2×C61—22no—
224C2×C61C2×#46111yescharacteristic
225C2×C61#306311yescharacteristic
226C2×C61#306811yescharacteristic
227D61—82no—
228C2×#131—1221no—
229#533—611no—
230#532—611no—
231C2×C4⋊Q8—611no—
232#501—1222no—
233#500—612no—
234#498—612no—
235#498—1221no—
236#486—1221no—
237#481—1222no—
238#476—612no—
239#476—612no—
240#476—612no—
241#475—1221no—
242#465—1222no—
243#461—612no—
244#460—1221no—
245#436—612no—
246#435—1222no—
247C8×D4—612no—
248#431C2×D6111yescharacteristic
249#428—1221no—
250#428—1221no—
251C2×C4⋊C8—611no—
252C2×C4⋊C8C2×D6111yescharacteristic
253#408—1221no—
254C4×M16C2×D6111yescharacteristic
255#335C2×D6112yes—
256#334C2×D6112yes—
257#333—612no—
258#333—612no—
259#333—611no—
260#333—611no—
261#332—611no—
262#332—611no—
263C61#1923311yescharacteristic, complemented, Sylow
264C2×#41—1221no—
265C22×Q8—1221no—
266C2×SD16—1222no—
267#131—2444no—
268C2×M16—1222no—
269C2×M16D24411yescharacteristic
270C2×M16D24411yescharacteristic
271C22×C8—21no—
272C22×C8—1221no—
273C22×C8—1222no—
274C4⋊Q8—611no—
275C4⋊Q8—612no—
276C4⋊Q8—611no—
277C4⋊Q8—611no—
278C4⋊Q8—1221no—
279C4⋊Q8—1221no—
280C4⋊Q8—1221no—
281#127—611no—
282#125—1221no—
283#124—1221no—
284C22⋊Q8—1221no—
285C22⋊Q8—1221no—
286#121—612no—
287#121—1221no—
288C4×Q8—1221no—
289C4×Q8—1221no—
290C4×D4—612no—
291C4×D4—1221no—
292#117—1221no—
293C2×C4⋊C4—1221no—
294C2×C4⋊C4—1221no—
295C2×C42C22×D6111yescharacteristic
296#107—1222no—
297#107—1222no—
298#106—1222no—
299#106—1222no—
300#106—1222no—
301#106—1222no—
302#106—1222no—
303C4⋊C8—21no—
304C4⋊C8—611no—
305C4⋊C8—611no—
306C4⋊C8—1221no—
307C4⋊C8C22×D6111yescharacteristic
308C4⋊C8C22×D6111yescharacteristic
309C4⋊C8C22×D6112yes—
310#103—1222no—
311#102—1222no—
312#102—1222no—
313#102—1222no—
314C22⋊C8—21no—
315C22⋊C8—1222no—
316C22⋊C8—1222no—
317#97D24412yes—
318C4×C8—21no—
319C4×C8C22×D6112yes—
320#41—2442no—
321#41—2442no—
322C2×Q8—1221no—
323C2×Q8—1221no—
324C2×Q8—1221no—
325C2×Q8—1222no—
326C2×Q8—4882no—
327C2×Q8—2441no—
328C2×D4—612no—
329C2×D4—1222no—
330C2×D4—1221no—
331C2×D4—1221no—
332C22×C4—1222no—
333C22×C4—2441no—
334C22×C4C2×D24411yescharacteristic
335C22×C4D4×D6111yescharacteristic
336C22×C4#1175822611yescharacteristic
337SD16—4882no—
338M16—22no—
339M16—2444no—
340M16#1175819014yescomplemented
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—22no—
344C2×C8—81no—
345C2×C8—1221no—
346C2×C8—1221no—
347C2×C8—2442no—
348C2×C8—2444no—
349C2×C8C2×D24412yes—
350C2×C8C2×D24412yes—
351C4⋊C4—612no—
352C4⋊C4—1221no—
353C4⋊C4—1222no—
354C4⋊C4—1221no—
355C4⋊C4—1221no—
356C4⋊C4—1222no—
357C4⋊C4—1221no—
358C4⋊C4—1221no—
359C4⋊C4—2441no—
360C4⋊C4—2441no—
361C4⋊C4—2441no—
362C4⋊C4—2441no—
363C22⋊C4—1222no—
364C22⋊C4—1222no—
365C22⋊C4—2441no—
366C42—2441no—
367C42C2×D24411yescharacteristic
368C42D4×D6111yescharacteristic
369C42#1175822611yescharacteristic
370C42C23×D6111yescharacteristic
371C23—1222no—
372C23#1451975211yescharacteristic
373Q8—4881no—
374Q8—4882no—
375Q8—4882no—
376Q8—2441no—
377Q8—2441no—
378Q8—2442no—
379D4—1222no—
380D4—1222no—
381D4—4881no—
382D4—2444no—
383D4—2442no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—1222no—
388C2×C4—4882no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4—2442no—
392C2×C4—2442no—
393C2×C4—2442no—
394C2×C4#1451975211yescharacteristic
395C2×C4#1451975211yescharacteristic
396C2×C4#1451975211yescharacteristic
397C2×C4C2×#1175819012yes—
398C2×C4#1451981012yes—
399C2×C4C2×D4×D6111yescharacteristic
400C2×C4C22×D24411yescharacteristic
401C2×C4C2×#1175822611yescharacteristic
402C8—22no—
403C8—41no—
404C8—42no—
405C8—41no—
406C8—2441no—
407C8—2441no—
408C8C2×#1175819014yes—
409C22—21no—
410C22—21no—
411C22—2442no—
412C22—2442no—
413C22—2442no—
414C22#1683996812yes—
415C22C2×#1451975211yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—4881no—
420C4—4882no—
421C4—4881no—
422C4—2442no—
423C4#1683996812yes—
424C4C22×#1175819011yescharacteristic
425C4C2×#1451981011yescharacteristic
426C4#1684057211yescharacteristic
427C4#1684057811yescharacteristic
428C2—22no—
429C2—4882no—
430C2C2×#1683996811yescharacteristic
431C2#2145551111yescharacteristic
432C2#2145554811yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1565C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#1565D61
hypocenterC61#19233
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
3. #21456365C2semidirect
5. #21445343C2semidirect
6. #21445319C2semidirect
7. #21445219C2semidirect
8. #21445195C2semidirect
9. #21444474C2semidirect
10. C2×#16839510C2semidirect
11. C61×#1565C2semidirect
121. M16×C61SD16semidirect
220. #1565D61semidirect
263. C61#19233semidirect
340. M16#11758190semidirect

Maximal subgroups

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

subgroupindex
#214571362
#214569492
#214563652
#214545652
#214453432
#214453192
#214452192
#214451952
#214444742
C2×#168395102
C61×#15652
#1923361