Finite Group Databasebeta

← #29667911

Subgroups — #29667911

Order 15616.

720 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456942C211yesmaximal, characteristic
2#21456769C211yesmaximal, characteristic, complemented
3#21456378C211yesmaximal, characteristic
4C2×#16840160C211yesmaximal, characteristic, complemented
5#21456213C211yesmaximal, characteristic
6#21456200C211yesmaximal, characteristic, complemented
7#21445479C212yesmaximal, complemented
8#21445465C212yesmaximal, complemented
9#21445420C212yesmaximal, complemented
10#21445416C212yesmaximal, complemented
11C8×M16×C61C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13#16840312C2211yescharacteristic
14C2×#14519777C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16#16840167C2212yes—
17#16840160—22no—
18#16840160C2211yescharacteristic
19#16840160C2212yes—
20#16840160C2212yes—
21#16840142C2212yes—
22#16840135C2211yescharacteristic
23#16840131—22no—
24#16840122—22no—
25#16840113C2212yes—
26#16840113C2212yes—
27#16840113C2212yes—
28#16840113C2212yes—
29#16840113C2211yescharacteristic
30#16840109C2212yes—
31#16840109C2212yes—
32#16840109C2211yescharacteristic
33#16840076—22no—
34#16840043—22no—
35#16839991—22no—
36#16839973—22no—
37#16839920—22no—
38C4×#11758190—22no—
39C2×#14519709—22no—
40#16839772—22no—
41#16839764—22no—
42C61×#431C2211yescharacteristic
43C4×M16×C61C2211yescharacteristic
44C2×C4×C8×C61C2211yescharacteristic
45C61×#322C2212yes—
46C82×C61C2212yes—
47C2×#11758222—21no—
48C22×Dic122—21no—
49C22×Dic122—21no—
50C2×Dic244—24no—
51C2×#11758190—22no—
52#14519777—21no—
53#14519777—21no—
54#14519777—21no—
55#14519777—21no—
56#14519777—21no—
57#14519777—21no—
58#14519777C2312yes—
59#14519777C2312yes—
60#14519777C2311yescharacteristic
61#14519777C2311yescharacteristic
62#14519777C2311yescharacteristic
63#14519774C2311yescharacteristic
64#14519769—21no—
65#14519765—21no—
66#14519756—21no—
67#14519756—21no—
68#14519751—21no—
69#14519751—21no—
70C4×Dic122—21no—
71C4×Dic122—21no—
72C4×D244—21no—
73C4×D244—21no—
74#14519736—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79#14519711—24no—
80#14519709—22no—
81#14519709—24no—
82#14519709—24no—
83#14519702—24no—
84#14519702—22no—
85#14519702—24no—
86#14519702—24no—
87#14519702—24no—
88#14519699—22no—
89#14519699—24no—
90#14519699—24no—
91C2×M16×C61D412yes—
92C22×C8×C61D412yes—
93C2×C42×C61C2311yescharacteristic
94C4⋊C8×C61D412yes—
95C22⋊C8×C61D412yes—
96#97×C61D412yes—
97C4×C8×C61D412yes—
98C4×C8×C61C2311yescharacteristic
99C4×C8×C61C2312yes—
100C4×C8×C61C2312yes—
101C4×C8×C61C2311yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C2×C22⋊D61—41no—
106C22×Dic61—41no—
107C22×Dic61—41no—
108#11758222—44no—
109C2×D244—21no—
110C2×D244—21no—
111C2×D244—21no—
112C2×C4×D61—41no—
113C2×Dic122—22no—
114C2×Dic122—22no—
115C2×Dic122—21no—
116C2×Dic122—21no—
117C2×Dic122—21no—
118C2×Dic122—44no—
119C2×Dic122—41no—
120C2×Dic122—41no—
121C2×Dic122—82no—
122M16×C61—24no—
123C2×C8×C61—24no—
124C2×C8×C61C2×D412yes—
125C2×C8×C61C2×D414yes—
126C2×C8×C61C2×D414yes—
127C2×C8×C61C2×D412yes—
128C42×C61C2×D411yescharacteristic
129C42×C61C2×D411yescharacteristic
130C42×C61C2×D411yescharacteristic
131C42×C61C2411yescharacteristic
132C22⋊Dic61—41no—
133#11758198—41no—
134#11758198—41no—
135C4⋊Dic61—22no—
136C4⋊Dic61—21no—
137C4⋊Dic61—22no—
138C4⋊Dic61—21no—
139C4⋊Dic61—21no—
140C4⋊Dic61—21no—
141C4⋊Dic61—21no—
142C4⋊Dic61—22no—
143C4⋊Dic61—22no—
144C4⋊Dic61—21no—
145C4⋊Dic61—41no—
146C4⋊Dic61—41no—
147C4⋊Dic61—41no—
148C4⋊Dic61—41no—
149#11758196—41no—
150#11758196—41no—
151C4×Dic61—41no—
152Dic244—84no—
153#11758190—44no—
154C23×C61#12711yescharacteristic
155C22×D61—41no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61#12711yescharacteristic
159C2×C4×C61#12711yescharacteristic
160C2×C4×C61#12711yescharacteristic
161C2×C4×C61#13514yes—
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C22×D411yescharacteristic
165C22⋊D61—82no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170C2×Dic61—41no—
171C2×Dic61—82no—
172C2×Dic61—82no—
173D244—42no—
174D244—42no—
175D244—42no—
176C4×D61—82no—
177Dic122—44no—
178Dic122—42no—
179Dic122—42no—
180Dic122—42no—
181Dic122—44no—
182Dic122—82no—
183Dic122—82no—
184C8×C61—24no—
185C8×C61—22no—
186C8×C61—24no—
187C8×C61#13714yes—
188#19626—611nomaximal, Sylow
189C22×C61—21no—
190C22×C61—21no—
191C22×C61#49512yes—
192C22×C61C2×#12711yescharacteristic
193C2×D61—81no—
194C2×D61—81no—
195C2×D61—81no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61#49512yes—
199C4×C61C2×#13512yes—
200C4×C61C2×#13712yes—
201C4×C61#57712yes—
202Dic61—82no—
203Dic61—81no—
204Dic61—81no—
205Dic61—82no—
206Dic61—81no—
207#3226—611no—
208#3209—611no—
209#3167—611no—
210C2×#498—611no—
211#3141—611no—
212#3139—611no—
213#1720—612no—
214#1718—612no—
215#1708—612no—
216#1707—612no—
217C8×M16D6111yescharacteristic, complemented
218C2×C61—22no—
219C2×C61C2×#49511yescharacteristic
220C2×C61#315211yescharacteristic
221C2×C61#315411yescharacteristic
222D61—161no—
223#533—611no—
224#532—611no—
225C2×C4⋊Q8—611no—
226C2×C4⋊Q8—611no—
227#500—612no—
228#498—612no—
229#498—612no—
230#498—611no—
231#498—1222no—
232#494—612no—
233#492—611no—
234#491—1222no—
235#489—1222no—
236#487—612no—
237#487—612no—
238#487—612no—
239#487—611no—
240#487—612no—
241#486—612no—
242#486—611no—
243#486—612no—
244#480—1222no—
245#474—1222no—
246#465—1222no—
247#462—1222no—
248#451—1222no—
249C4×SD16—1222no—
250#431C2×D6111yescharacteristic
251C2×#106—1222no—
252#419—1222no—
253#417—1222no—
254C4×M16C2×D6111yescharacteristic
255C2×C4×C8C2×D6111yescharacteristic
256#322C2×D6112yes—
257C82C2×D6112yes—
258C61#1962611yescharacteristic, complemented, Sylow
259C2×#41—1221no—
260C22×Q8—1221no—
261C22×Q8—1221no—
262C2×Q16—1224no—
263C2×SD16—1222no—
264C2×M16D24412yes—
265C22×C8D24412yes—
266C4⋊Q8—612no—
267C4⋊Q8—612no—
268C4⋊Q8—611no—
269C4⋊Q8—611no—
270C4⋊Q8—611no—
271C4⋊Q8—1221no—
272C4⋊Q8—1221no—
273C4⋊Q8—1221no—
274C4⋊Q8—1221no—
275C4⋊Q8—1221no—
276C4⋊Q8—1221no—
277#127—611no—
278#125—1221no—
279#124—1221no—
280C22⋊Q8—1221no—
281C22⋊Q8—1221no—
282#121—1221no—
283#121—1221no—
284C4×Q8—1221no—
285C4×Q8—1221no—
286C4×D4—1221no—
287C4×D4—1221no—
288#117—1221no—
289C2×C4⋊C4—1221no—
290C2×C4⋊C4—1221no—
291C2×C4⋊C4—1221no—
292C2×C4⋊C4—1221no—
293C2×C42C22×D6111yescharacteristic
294#107—1224no—
295#106—1224no—
296#106—1224no—
297#106—1222no—
298C4⋊C8D24412yes—
299#103—1224no—
300#103—1224no—
301#103—1224no—
302#103—1222no—
303#103—1224no—
304#102—1224no—
305#102—1222no—
306#102—1224no—
307C22⋊C8D24412yes—
308#97D24412yes—
309C4×C8D24412yes—
310C4×C8C22×D6111yescharacteristic
311C4×C8C22×D6112yes—
312C4×C8C22×D6112yes—
313C4×C8C22×D6111yescharacteristic
314#41—2444no—
315C2×Q8—1221no—
316C2×Q8—1222no—
317C2×Q8—1221no—
318C2×Q8—1222no—
319C2×Q8—1221no—
320C2×Q8—4882no—
321C2×Q8—2444no—
322C2×Q8—2441no—
323C2×Q8—2441no—
324C2×D4—1221no—
325C2×D4—1221no—
326C2×D4—1221no—
327C2×D4—2441no—
328C22×C4—2441no—
329C22×C4—2441no—
330C22×C4—2441no—
331C22×C4C2×D24411yescharacteristic
332C22×C4C2×D24411yescharacteristic
333C22×C4C2×D24411yescharacteristic
334Q16—4884no—
335SD16—2444no—
336M16—24no—
337C2×C8—24no—
338C2×C8C2×D24412yes—
339C2×C8C2×D24414yes—
340C2×C8C2×D24414yes—
341C2×C8C2×D24412yes—
342C4⋊C4—1222no—
343C4⋊C4—1221no—
344C4⋊C4—1222no—
345C4⋊C4—1221no—
346C4⋊C4—1221no—
347C4⋊C4—1221no—
348C4⋊C4—1221no—
349C4⋊C4—1222no—
350C4⋊C4—1222no—
351C4⋊C4—1221no—
352C4⋊C4—2441no—
353C4⋊C4—2441no—
354C4⋊C4—2441no—
355C4⋊C4—2441no—
356C4⋊C4—2441no—
357C4⋊C4—2441no—
358C22⋊C4—2441no—
359C22⋊C4—2441no—
360C22⋊C4—2441no—
361C42—2441no—
362C42C2×D24411yescharacteristic
363C42C2×D24411yescharacteristic
364C42C2×D24411yescharacteristic
365C42C23×D6111yescharacteristic
366C23—2441no—
367C23#1451977411yescharacteristic
368Q8—4882no—
369Q8—4882no—
370Q8—2444no—
371Q8—2442no—
372Q8—2442no—
373Q8—2442no—
374Q8—2444no—
375D4—4882no—
376D4—2442no—
377D4—2442no—
378D4—2442no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—4882no—
382C2×C4—4882no—
383C2×C4—4882no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2442no—
388C2×C4—2442no—
389C2×C4#1451977411yescharacteristic
390C2×C4#1451977411yescharacteristic
391C2×C4#1451977411yescharacteristic
392C2×C4#1451979914yes—
393C2×C4C22×D24411yescharacteristic
394C2×C4C22×D24411yescharacteristic
395C2×C4C22×D24411yescharacteristic
396C8—24no—
397C8—22no—
398C8—24no—
399C8#1451981014yes—
400C22—21no—
401C22—21no—
402C22—4881no—
403C22—4881no—
404C22—4881no—
405C22#1684014612yes—
406C22C2×#1451977411yescharacteristic
407C4—21no—
408C4—21no—
409C4—4881no—
410C4—4882no—
411C4—4881no—
412C4—4882no—
413C4—4881no—
414C4#1684014612yes—
415C4C2×#1451979912yes—
416C4C2×#1451981012yes—
417C4#1684057412yes—
418C2—22no—
419C2—9761no—
420C2C2×#1684014611yescharacteristic
421C2#2145627811yescharacteristic
422C2#2145629511yescharacteristic
subgroupG/H
centerC22C2×#14519774
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC8×M16×C61C2
socleC22×C61C2×#127
prime radical#16840312C22
hypercenterC8×M16D61
hypocenterC61#19626
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×M16

Complements

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

NH ≅ G/Ntype
2. #21456769C2semidirect
4. C2×#16840160C2semidirect
6. #21456200C2semidirect
7. #21445479C2semidirect
8. #21445465C2semidirect
9. #21445420C2semidirect
10. #21445416C2semidirect
11. C8×M16×C61C2semidirect
217. C8×M16D61semidirect
258. C61#19626semidirect

Maximal subgroups

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

subgroupindex
#214569422
#214567692
#214563782
C2×#168401602
#214562132
#214562002
#214454792
#214454652
#214454202
#214454162
C8×M16×C612
#1962661