Finite Group Databasebeta

← #29647742

Subgroups — #29647742

Order 15616.

918 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459684C211yesmaximal, characteristic
2#21454900C211yesmaximal, characteristic, complemented
3#21444394C214yesmaximal, complemented
4#21444382C211yesmaximal, characteristic, complemented
5#21444378C211yesmaximal, characteristic
6#21444317C211yesmaximal, characteristic, complemented
7#21444313C211yesmaximal, characteristic
8#21444303C214yesmaximal, complemented
9C61×#2356C211yesmaximal, characteristic, complemented
10#16840497C412yescomplemented
11#16840490C412yescomplemented
12#16840485C412yescomplemented
13#16840484—22no—
14#16840472C412yescomplemented
15#16840453—22no—
16#16840359—22no—
17#16840336—21no—
18#16840319C2211yescharacteristic
19#16840312C2211yescharacteristic
20#16840296C2214yes—
21#16840243—21no—
22Q8×#92418—22no—
23Q8×#92418—22no—
24#16839815C2211yescharacteristic
25#16839815C2211yescharacteristic
26#16839815C2214yes—
27#16839498C2214yes—
28#16839498C2212yes—
29#16839498C2212yes—
30#16839495C2214yes—
31#16839495C2212yes—
32#16839495C2212yes—
33C2×C4×Q8×C61C2211yescharacteristic
34C61×#391—21no—
35C61×#387—22no—
36C61×#383C2212yes—
37C61×#382C2211yescharacteristic
38C4×C4⋊C4×C61—21no—
39C4×C4⋊C4×C61C2211yescharacteristic
40C61×#376C2212yes—
41C2×#11758222—21no—
42#14519777C2×C411yescharacteristic
43#14519777C2×C411yescharacteristic
44#14519777C2×C411yescharacteristic
45#14519775—22no—
46#14519774C2×C411yescharacteristic
47#14519772—24no—
48#14519771—24no—
49#14519769—21no—
50#14519769C2×C414yes—
51#14519767—24no—
52#14519766—22no—
53#14519765—21no—
54#14519765C2×C414yes—
55#14519762—24no—
56#14519760—24no—
57#14519756—21no—
58#14519756C2312yes—
59#14519755—24no—
60#14519753—22no—
61#14519753—22no—
62#14519751—21no—
63#14519751C2312yes—
64C4×Dic122—21no—
65C4×Dic122C2×C414yes—
66C4×Dic122C2×C412yes—
67#14519742—22no—
68C4×D244—21no—
69C4×D244C2×C412yes—
70C4×D244C2×C414yes—
71#14519739—22no—
72C4×C22⋊D61—24no—
73#14519736—21no—
74#14519735—22no—
75#14519734—24no—
76C4⋊C4×D61—22no—
77C4⋊#92418—24no—
78C4⋊#92418—21no—
79C4⋊#92418—21no—
80C22⋊#92418C2312yes—
81C22⋊#92418C2312yes—
82C4×#92418—24no—
83C4×#92418—21no—
84C4×#92418—21no—
85C22×Q8×C61D411yescharacteristic
86C4×Q8×C61C2×C412yes—
87C4×Q8×C61C2×C412yes—
88C4×Q8×C61D412yes—
89C4×Q8×C61D412yes—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61D411yescharacteristic
93C2×C4⋊C4×C61D412yes—
94C2×C42×C61—22no—
95C2×C42×C61—21no—
96C2×C42×C61C2312yes—
97C2×C42×C61C2311yescharacteristic
98#95×C61—22no—
99#95×C61—22no—
100#95×C61—21no—
101#95×C61—22no—
102#95×C61C2311yescharacteristic
103#95×C61C2312yes—
104#95×C61C2311yescharacteristic
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61C22⋊C412yes—
108C2×Q8×C61C22⋊C412yes—
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61C2×D412yes—
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C2×C22⋊D61—22no—
118#11758222—42no—
119#11758222—42no—
120C2×D244—21no—
121C2×D244—21no—
122C2×D244C22×C412yes—
123C2×C4×D61—22no—
124C2×C4×D61—24no—
125C2×Dic122—21no—
126C2×Dic122—21no—
127C2×Dic122C22×C412yes—
128C4⋊C4×C61—42no—
129C4⋊C4×C61—42no—
130C4⋊C4×C61C22⋊C412yes—
131C4⋊C4×C61C22⋊C412yes—
132C4⋊C4×C61C22⋊C414yes—
133C4⋊C4×C61C22⋊C414yes—
134C42×C61—44no—
135C42×C61—42no—
136C42×C61C22×C411yescharacteristic
137C42×C61C22×C414yes—
138C42×C61C22×C411yescharacteristic
139C42×C61C2×D414yes—
140C42×C61C2×D411yescharacteristic
141C42×C61C2×D411yescharacteristic
142C22⋊Dic61—22no—
143#11758198—24no—
144#11758198—24no—
145#11758198—22no—
146#11758198—24no—
147#11758198—22no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61C22×C412yes—
153C4⋊Dic61C22×C412yes—
154#11758196—24no—
155#11758196—24no—
156#11758196—24no—
157#11758196—22no—
158#11758196—22no—
159C4×Dic61—24no—
160C4×Dic61—22no—
161C2×#92418—22no—
162C2×#92418—22no—
163C23×C61C22×D411yescharacteristic
164C22×D61—22no—
165Q8×C61—24no—
166Q8×C61—24no—
167C2×C4×C61—22no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—42no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—42no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—42no—
179C2×C4×C61—41no—
180C2×C4×C61—42no—
181C2×C4×C61C2×C22⋊C414yes—
182C2×C4×C61C2×C22⋊C411yescharacteristic
183C2×C4×C61C2×C22⋊C414yes—
184C2×C4×C61C2×C22⋊C411yescharacteristic
185C2×C4×C61C2×C22⋊C411yescharacteristic
186C2×C4×C61C2×C22⋊C411yescharacteristic
187C2×C4×C61#13512yes—
188C2×C4×C61#13512yes—
189C2×C4×C61C23×C411yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C22⋊D61—42no—
192C22⋊D61—42no—
193C2×Dic61—22no—
194C2×Dic61—22no—
195C2×Dic61—22no—
196D244—42no—
197D244—42no—
198D244—81no—
199C4×D61—44no—
200C4×D61—84no—
201Dic122—42no—
202Dic122—42no—
203Dic122—81no—
204#92418—42no—
205#92418—42no—
206#17616—611nomaximal, Sylow
207C22×C61—21no—
208C22×C61—21no—
209C22×C61C22×C22⋊C411yescharacteristic
210C22×C61C2×#13511yescharacteristic
211C22×C61C2×#13511yescharacteristic
212C2×D61—42no—
213C2×D61—42no—
214C2×D61—42no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—24no—
218C4×C61—21no—
219C4×C61—24no—
220C4×C61—21no—
221C4×C61—42no—
222C4×C61—81no—
223C4×C61—82no—
224C4×C61#41612yes—
225C4×C61#41612yes—
226Dic61—42no—
227Dic61—42no—
228Dic61—42no—
229#3507—611no—
230#2982—611no—
231#2356D6111yescharacteristic, complemented
232#1521—614no—
233#1518—611no—
234#1517—611no—
235#1503—611no—
236#1502—611no—
237#1500—614no—
238C2×C61—21no—
239C2×C61—21no—
240C2×C61#288411yescharacteristic
241C2×C61C2×#41611yescharacteristic
242C2×C61#290111yescharacteristic
243D61—82no—
244#559—612no—
245#557—612no—
246#556—612no—
247#556—1222no—
248#554—612no—
249#552—1222no—
250#539—1222no—
251#536—1221no—
252#533—611no—
253#532—611no—
254#529—614no—
255#520—1221no—
256C2×C4×Q8C2×D6111yescharacteristic
257C8×Q8—1222no—
258C8×Q8—1222no—
259#431—614no—
260#431—611no—
261#431—611no—
262#391—21no—
263#387—22no—
264#383C2×D6112yes—
265#382C2×D6111yescharacteristic
266C4×C4⋊C4—21no—
267C4×C4⋊C4C2×D6111yescharacteristic
268#376C2×D6112yes—
269#328—612no—
270#328—614no—
271#328—612no—
272#327—612no—
273#327—614no—
274#327—612no—
275C61#1761611yescharacteristic, complemented, Sylow
276C2×#41—1221no—
277C22×Q8C22⋊D6111yescharacteristic
278C4⋊Q8—611no—
279C4⋊Q8—611no—
280C4⋊Q8—611no—
281C4⋊Q8—1222no—
282#127—611no—
283#126—1224no—
284#126—1224no—
285#125—614no—
286#125—1224no—
287#125—1221no—
288#124—614no—
289#124—1222no—
290#124—1221no—
291#123—1224no—
292#123—1224no—
293C22⋊Q8—612no—
294C22⋊Q8—1222no—
295C22⋊Q8—1222no—
296C22⋊Q8—1221no—
297C22⋊Q8—1224no—
298#121—612no—
299#121—1221no—
300C4×Q8—614no—
301C4×Q8—612no—
302C4×Q8—1222no—
303C4×Q8—1221no—
304C4×Q8C4×D6112yes—
305C4×Q8C4×D6112yes—
306C4×Q8D24412yes—
307C4×Q8D24412yes—
308C4×D4—614no—
309C4×D4—612no—
310C4×D4—1224no—
311C4×D4—1222no—
312C4×D4—1221no—
313#117—1221no—
314#117—1222no—
315#117—1224no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—1222no—
319C2×C4⋊C4C22⋊D6111yescharacteristic
320C2×C4⋊C4C22⋊D6112yes—
321C2×C42—22no—
322C2×C42—21no—
323C2×C42C22×D6112yes—
324C2×C42C22×D6111yescharacteristic
325C4⋊C8—1224no—
326C4⋊C8—1221no—
327C4⋊C8—1221no—
328C22⋊C8—612no—
329C22⋊C8—612no—
330C4×C8—1224no—
331C4×C8—1221no—
332C4×C8—1221no—
333#95—22no—
334#95—22no—
335#95—21no—
336#95—22no—
337#95C22×D6111yescharacteristic
338#95C22×D6112yes—
339#95C22×D6111yescharacteristic
340#41—2442no—
341#41—2442no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—612no—
345C2×Q8—1221no—
346C2×Q8—1221no—
347C2×Q8#1175819812yes—
348C2×Q8#1175819812yes—
349C2×D4—612no—
350C2×D4—1222no—
351C2×D4—1221no—
352C2×D4—1221no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—1224no—
357C22×C4—1222no—
358C22×C4C2×C22⋊D6112yes—
359C22×C4C2×C22⋊D6112yes—
360C22×C4C2×C22⋊D6111yescharacteristic
361C22×C4C2×C22⋊D6111yescharacteristic
362C22×C4C23×D6111yescharacteristic
363C2×C8—1222no—
364C2×C8—1222no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—612no—
368C4⋊C4—612no—
369C4⋊C4—1222no—
370C4⋊C4—1221no—
371C4⋊C4—1224no—
372C4⋊C4—1224no—
373C4⋊C4—1222no—
374C4⋊C4—1224no—
375C4⋊C4—1221no—
376C4⋊C4—1221no—
377C4⋊C4—1221no—
378C4⋊C4#1175819812yes—
379C4⋊C4#1175819812yes—
380C4⋊C4#1175819814yes—
381C4⋊C4#1175819814yes—
382C22⋊C4—1222no—
383C22⋊C4—1222no—
384C22⋊C4—1224no—
385C22⋊C4—1224no—
386C22⋊C4—1222no—
387C22⋊C4—1224no—
388C42—44no—
389C42—42no—
390C42—1224no—
391C42—1222no—
392C42C2×C4×D6111yescharacteristic
393C42C2×C4×D6114yes—
394C42C2×C4×D6111yescharacteristic
395C42C2×D24414yes—
396C42C2×D24411yescharacteristic
397C42C2×D24411yescharacteristic
398C23—1222no—
399C23C22×C22⋊D6111yescharacteristic
400Q8—24no—
401Q8—24no—
402Q8—4881no—
403Q8—2442no—
404Q8—2442no—
405D4—4881no—
406D4—2442no—
407D4—2442no—
408D4—2442no—
409D4—2442no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—1222no—
425C2×C4—1222no—
426C2×C4—1222no—
427C2×C4—4884no—
428C2×C4—2444no—
429C2×C4C2×#1175819814yes—
430C2×C4C2×#1175819814yes—
431C2×C4C2×#1175819811yescharacteristic
432C2×C4C2×#1175819811yescharacteristic
433C2×C4C2×#1175819811yescharacteristic
434C2×C4C2×#1175819811yescharacteristic
435C2×C4#1451979712yes—
436C2×C4#1451979712yes—
437C2×C4C22×C4×D6111yescharacteristic
438C2×C4C22×D24411yescharacteristic
439C8—2442no—
440C8—2442no—
441C22—21no—
442C22—21no—
443C22—2442no—
444C22—2442no—
445C22—2442no—
446C22C22×#1175819811yescharacteristic
447C22C2×#1451979711yescharacteristic
448C22C2×#1451979711yescharacteristic
449C4—24no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—24no—
455C4—42no—
456C4—81no—
457C4—82no—
458C4—2442no—
459C4—2442no—
460C4—2442no—
461C4#1683975612yes—
462C4#1683975612yes—
463C2—21no—
464C2—21no—
465C2—4882no—
466C2#2145428211yescharacteristic
467C2C2×#1683975611yescharacteristic
468C2#2145436711yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2356C2
socleC22×C61C22×C22⋊C4
prime radical#16840312C22
cosocleC22×C4×C61C24
hypercenter#2356D61
hypocenterC61#17616
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2356

Complements

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

NH ≅ G/Ntype
2. #21454900C2semidirect
3. #21444394C2semidirect
4. #21444382C2semidirect
6. #21444317C2semidirect
8. #21444303C2semidirect
9. C61×#2356C2semidirect
10. #16840497C4semidirect
11. #16840490C4semidirect
12. #16840485C4semidirect
14. #16840472C4semidirect
231. #2356D61semidirect
275. C61#17616semidirect

Maximal subgroups

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

subgroupindex
#214596842
#214549002
#214443942
#214443822
#214443782
#214443172
#214443132
#214443032
C61×#23562
#1761661