Finite Group Databasebeta

← #29663122

Subgroups — #29663122

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456746C211yesmaximal, characteristic
2#21456744C211yesmaximal, characteristic
3C2×#16840165C211yesmaximal, characteristic, complemented
4#21454530C211yesmaximal, characteristic
5#21445331C214yesmaximal, complemented
6#21445207C214yesmaximal, complemented
7#21444565C211yesmaximal, characteristic, complemented
8#21444505C211yesmaximal, characteristic, complemented
9C61×#1549C211yesmaximal, characteristic, complemented
10C2×#14519785—21no—
11#16840312C2211yescharacteristic
12C2×#14519777C2211yescharacteristic
13#16840165—21no—
14#16840165—21no—
15#16840165C2212yes—
16#16840165C2212yes—
17#16840136—21no—
18#16840135—21no—
19#16840113—21no—
20#16840112—21no—
21#16839968C2214yes—
22#16839964C2214yes—
23#16839903—22no—
24#16839901—22no—
25#16839839—21no—
26#16839839C2212yes—
27#16839827—21no—
28#16839827C2212yes—
29#16839815C2211yescharacteristic
30C2×#14519710—21no—
31C2×#14519710—21no—
32#16839730—21no—
33C2×C4×#92418C2211yescharacteristic
34#16839514C2212yes—
35#16839512C2212yes—
36#16839507C2214yes—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#423C2211yescharacteristic
40C61×#332C2214yes—
41C2×#11758222—21no—
42C22×Dic122—21no—
43C2×#11758190—24no—
44C2×#11758201—24no—
45#14519785—44no—
46C2×#11758189—22no—
47C22×#92418D411yescharacteristic
48C22×#92418D411yescharacteristic
49C2×C8×D61—22no—
50#14519777—21no—
51#14519777—21no—
52#14519777—21no—
53#14519777D411yescharacteristic
54#14519777C2312yes—
55#14519774D411yescharacteristic
56#14519765—21no—
57#14519751—21no—
58#14519751D412yes—
59C4×D244—21no—
60C4×D244C2312yes—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63#14519710—22no—
64#14519710—22no—
65#14519710—22no—
66#14519710—22no—
67C4⋊#92418C2312yes—
68#14519702—24no—
69#14519701—24no—
70C22⋊#92418—21no—
71#14519689—22no—
72#14519689—22no—
73C4×#92418—21no—
74C4×#92418D411yescharacteristic
75C4×#92418D411yescharacteristic
76C4×#92418C2311yescharacteristic
77C4×#92418C2311yescharacteristic
78C22×Q8×C61—21no—
79C2×M16×C61—21no—
80C4⋊Q8×C61—21no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61C2312yes—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C42×C61C2311yescharacteristic
87C4⋊C8×C61—21no—
88C4⋊C8×C61C2312yes—
89C4⋊C8×C61C2312yes—
90C22⋊C8×C61—21no—
91C4×C8×C61—21no—
92C2×Q8×C61—22no—
93C2×Q8×C61—41no—
94C2×Q8×C61—82no—
95C22×C4×C61—41no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61#4111yescharacteristic
99C2×C22⋊D61—22no—
100C22×Dic61—41no—
101#11758222—42no—
102#11758222—42no—
103C2×D244—21no—
104C2×D244—21no—
105C2×D244C2×D412yes—
106C2×C4×D61—22no—
107C2×Dic122—21no—
108C2×Dic122—22no—
109C2×Dic122—41no—
110C2×Dic122—82no—
111M16×C61—42no—
112C2×C8×C61—22no—
113C2×C8×C61—22no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—41no—
117C4⋊C4×C61—41no—
118C42×C61C2×D411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61#4111yescharacteristic
121C42×C61C2411yescharacteristic
122#11758201—84no—
123#11758198—22no—
124#11758198—22no—
125C4⋊Dic61—21no—
126C4⋊Dic61—22no—
127C4⋊Dic61—22no—
128C4⋊Dic61—41no—
129C4⋊Dic61—41no—
130C4⋊Dic61C2×D412yes—
131C2×#92418—22no—
132C2×#92418—22no—
133C2×#92418SD1614yescomplemented
134C2×#92418C2×D411yescharacteristic
135C2×#92418C2×D411yescharacteristic
136C2×#92418C2×D411yescharacteristic
137C2×#92418C2×D411yescharacteristic
138#11758190—84no—
139#11758189—44no—
140C8×D61—44no—
141C23×C61#12111yescharacteristic
142C22×D61—22no—
143Q8×C61—44no—
144Q8×C61—82no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—42no—
148C2×C4×C61—82no—
149C2×C4×C61#12111yescharacteristic
150C2×C4×C61#12111yescharacteristic
151C2×C4×C61#12111yescharacteristic
152C2×C4×C61C2×SD1612yes—
153C2×C4×C61#13712yes—
154C2×C4×C61C22×D411yescharacteristic
155C2×C4×C61C22×D411yescharacteristic
156C2×C4×C61C2×#4111yescharacteristic
157C22⋊D61—44no—
158C2×Dic61—22no—
159C2×Dic61—42no—
160C2×Dic61—82no—
161D244—24no—
162D244—42no—
163D244—42no—
164C4×D61—42no—
165C4×D61—42no—
166Dic122—42no—
167Dic122—44no—
168Dic122—82no—
169C8×C61—42no—
170C8×C61—42no—
171#92418—21no—
172#92418—21no—
173#92418—42no—
174#92418C2×SD1612yes—
175#92418C2×SD1612yes—
176#19192—611nomaximal, Sylow
177C22×C61—21no—
178C22×C61—21no—
179C22×C61#46112yes—
180C22×C61C2×#12111yescharacteristic
181C2×D61—42no—
182C2×D61—42no—
183C2×D61—42no—
184C4×C61—21no—
185C4×C61—21no—
186C4×C61—82no—
187C4×C61#46112yes—
188C4×C61#47312yes—
189C4×C61C22×SD1611yescharacteristic
190C4×C61C2×#13711yescharacteristic
191C4×C61#57612yes—
192Dic61—42no—
193Dic61—82no—
194#3206—611no—
195#3206—611no—
196C2×#500—611no—
197#2931—611no—
198#1693—614no—
199#1673—614no—
200#1564—611no—
201#1549—611no—
202#1549D6111yescharacteristic, complemented
203C2×C61—22no—
204C2×C61C2×#46111yescharacteristic
205C2×C61#305811yescharacteristic
206C2×C61C2×#47311yescharacteristic
207D61—82no—
208C2×#131—1221no—
209#532—611no—
210C2×C4⋊Q8—611no—
211C2×C4⋊Q8C2×D6111yescharacteristic
212#500—612no—
213#500—612no—
214#500—1221no—
215#500—1221no—
216#492—1221no—
217#492—1221no—
218#487—1221no—
219#487—1221no—
220#461—614no—
221#461—614no—
222#448—1222no—
223#448—1222no—
224#436—612no—
225#436—1221no—
226C8×D4—612no—
227C8×D4—1221no—
228#431—611no—
229#431C2×D6111yescharacteristic
230C2×#107—1221no—
231C2×#107—1221no—
232#423C2×D6111yescharacteristic
233#408—1221no—
234C2×C4×C8—611no—
235#335—612no—
236#334—612no—
237#332—614no—
238#332C2×D6114yes—
239C61#1919211yescharacteristic, complemented, Sylow
240C2×#41—1221no—
241C22×Q8—21no—
242C22×Q8—1221no—
243C2×SD16—1224no—
244C2×SD16—1224no—
245#131—2444no—
246C2×M16—21no—
247C2×M16—1222no—
248C22×C8—611no—
249C22×C8—611no—
250C22×C8—1222no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—611no—
255C4⋊Q8—612no—
256C4⋊Q8—1221no—
257C4⋊Q8—1221no—
258C4⋊Q8—1221no—
259C4⋊Q8C22×D6112yes—
260#127—611no—
261#124—1221no—
262#121—612no—
263#121—1221no—
264C4×D4—612no—
265C4×D4—1221no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—1221no—
269C2×C4⋊C4—1221no—
270C2×C42C22×D6111yescharacteristic
271#107—1222no—
272#107—1222no—
273#107—1222no—
274#107—1222no—
275C4⋊C8—21no—
276C4⋊C8—612no—
277C4⋊C8C22×D6112yes—
278C4⋊C8C22×D6112yes—
279#103—1224no—
280#103—1224no—
281C22⋊C8—21no—
282C22⋊C8—1222no—
283C22⋊C8—1222no—
284C22⋊C8—1221no—
285C4×C8—21no—
286C4×C8—611no—
287C4×C8—611no—
288C4×C8—611no—
289C4×C8—611no—
290C4×C8—1221no—
291#41—2442no—
292#41—2442no—
293C2×Q8—22no—
294C2×Q8—41no—
295C2×Q8—82no—
296C2×Q8—1221no—
297C2×Q8—1222no—
298C2×Q8—4882no—
299C2×Q8—2441no—
300C2×D4—612no—
301C2×D4—1222no—
302C2×D4—1221no—
303C2×D4—1221no—
304C22×C4—41no—
305C22×C4—1222no—
306C22×C4—2441no—
307C22×C4#1175822211yescharacteristic
308C22×C4D4×D6111yescharacteristic
309C22×C4D4×D6111yescharacteristic
310SD16—4884no—
311SD16—4884no—
312M16—42no—
313M16—2444no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—611no—
317C2×C8—611no—
318C2×C8—614no—
319C2×C8—611no—
320C2×C8—611no—
321C2×C8—1222no—
322C2×C8—1222no—
323C2×C8—2444no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—612no—
329C4⋊C4—1221no—
330C4⋊C4—1222no—
331C4⋊C4—1222no—
332C4⋊C4—2441no—
333C4⋊C4—2441no—
334C22⋊C4—1222no—
335C22⋊C4—1222no—
336C42#1175822211yescharacteristic
337C42D4×D6111yescharacteristic
338C42D4×D6111yescharacteristic
339C42C23×D6111yescharacteristic
340C23—1222no—
341C23#1451975011yescharacteristic
342Q8—44no—
343Q8—82no—
344Q8—4882no—
345Q8—2444no—
346Q8—2442no—
347D4—1224no—
348D4—2444no—
349D4—2442no—
350D4—2442no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—42no—
354C2×C4—82no—
355C2×C4—1222no—
356C2×C4—4882no—
357C2×C4—2442no—
358C2×C4—2442no—
359C2×C4—2442no—
360C2×C4#1451975011yescharacteristic
361C2×C4#1451975011yescharacteristic
362C2×C4#1451975011yescharacteristic
363C2×C4SD16×D6112yes—
364C2×C4#1451980912yes—
365C2×C4C2×D4×D6111yescharacteristic
366C2×C4C2×D4×D6111yescharacteristic
367C2×C4C2×#1175822211yescharacteristic
368C8—42no—
369C8—42no—
370C8—612no—
371C8—612no—
372C8—1221no—
373C8—1221no—
374C8—2442no—
375C22—21no—
376C22—21no—
377C22—2442no—
378C22—2442no—
379C22—2442no—
380C22#1683996612yes—
381C22C2×#1451975011yescharacteristic
382C4—21no—
383C4—21no—
384C4—82no—
385C4—4882no—
386C4—2442no—
387C4#1683996612yes—
388C4#1684003612yes—
389C4C2×SD16×D6111yescharacteristic
390C4C2×#1451980911yescharacteristic
391C4#1684056612yes—
392C2—22no—
393C2—4882no—
394C2C2×#1683996611yescharacteristic
395C2#2145546611yescharacteristic
396C2C2×#1684003611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1549C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#1549D61
hypocenterC61#19192
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
3. C2×#16840165C2semidirect
5. #21445331C2semidirect
6. #21445207C2semidirect
7. #21444565C2semidirect
8. #21444505C2semidirect
9. C61×#1549C2semidirect
133. C2×#92418SD16semidirect
202. #1549D61semidirect
239. C61#19192semidirect

Maximal subgroups

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

subgroupindex
#214567462
#214567442
C2×#168401652
#214545302
#214453312
#214452072
#214445652
#214445052
C61×#15492
#1919261