Finite Group Databasebeta

← #29646299

Subgroups — #29646299

Order 15616.

922 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840452C211yesmaximal, characteristic
2#21454909C212yesmaximal, complemented
3#21444185C214yesmaximal, complemented
4#21444176C212yesmaximal, complemented
5#21444150C212yesmaximal, complemented
6#21444140C211yesmaximal, characteristic
7#21444130C212yesmaximal, complemented
8C61×#1466C211yesmaximal, characteristic
9#16840452—22no—
10#16840452—22no—
11#16840452—24no—
12C2×#14519770C2212yes—
13C2×#14519769C2211yescharacteristic
14C2×#14519767C412yes—
15C2×#14519758C412yes—
16C2×#14519756C411yescharacteristic
17C2×#14519756C411yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×#14519736C411yescharacteristic
20C2×#14519733C2212yes—
21C22×C4⋊Dic61C411yescharacteristic
22#16839890—24no—
23#16839889—22no—
24#16839886—22no—
25#16839816C2214yes—
26#16839815C2212yes—
27#16839812C2214yes—
28#16839487C2214yes—
29#16839487C2214yes—
30#16839486C2211yescharacteristic
31#16839486C2211yescharacteristic
32#16839486C2212yes—
33C2×C4×D4×C61C2211yescharacteristic
34C61×#432C2212yes—
35C61×#323C2212yes—
36C61×#323C2212yes—
37C22×Dic122C2×C411yescharacteristic
38C23×Dic61C2×C412yes—
39#14519770—24no—
40#14519770—24no—
41#14519769—22no—
42#14519769—22no—
43#14519767—24no—
44#14519767—24no—
45#14519758—24no—
46#14519758—24no—
47#14519756—22no—
48#14519756—22no—
49#14519756—22no—
50#14519756—22no—
51C4×Dic122D412yes—
52C4×Dic122D412yes—
53C4×Dic122D414yes—
54#14519736—22no—
55#14519736—21no—
56#14519736—21no—
57#14519733—24no—
58#14519733—24no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61C2×C411yescharacteristic
63C2×C4⋊Dic61C2×C412yes—
64C2×C4⋊Dic61C2×C411yescharacteristic
65C2×C4⋊Dic61C2×C411yescharacteristic
66C2×#11758196C2×C412yes—
67C2×#11758196C2×C412yes—
68C2×#11758196C2×C412yes—
69C2×#11758196C2×C412yes—
70C2×#11758196C2312yes—
71C2×C22⋊Dic61C2×C412yes—
72C2×C22⋊Dic61C2×C412yes—
73C2×C22⋊Dic61C2×C412yes—
74C2×C4×Dic61C2×C412yes—
75C2×C4×Dic61C2312yes—
76C4⋊#92418—22no—
77#14519706—24no—
78#14519706—24no—
79C22⋊#92418C2312yes—
80C22⋊#92418C2312yes—
81#14519687—24no—
82C4×#92418—22no—
83C8×Dic61—24no—
84C22×D4×C61C2×C411yescharacteristic
85C23×C4×C61C2×C411yescharacteristic
86C23×C4×C61C2×C411yescharacteristic
87C4×D4×C61—22no—
88C4×D4×C61—22no—
89C2×C4⋊C4×C61C2×C411yescharacteristic
90C2×C22⋊C4×C61C2311yescharacteristic
91C2×C22⋊C4×C61C2311yescharacteristic
92C2×C42×C61C2311yescharacteristic
93C4⋊C8×C61—22no—
94C22⋊C8×C61C2314yes—
95#97×C61—22no—
96C24×C61C22×C411yescharacteristic
97C24×C61C22×C411yescharacteristic
98C2×D4×C61—22no—
99C2×D4×C61—22no—
100C2×D4×C61—41no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—42no—
106C22×C4×C61—42no—
107C22×C4×C61C22×C411yescharacteristic
108C22×C4×C61C22×C411yescharacteristic
109C22×C4×C61C22×C411yescharacteristic
110C22×C4×C61C22×C411yescharacteristic
111C22×C4×C61C2411yescharacteristic
112C22×Dic61—24no—
113C22×Dic61—22no—
114C22×Dic61—22no—
115C22×Dic61—42no—
116C22×Dic61C22×C412yes—
117C22×Dic61C22×C412yes—
118C22×Dic61C22×C412yes—
119C22×Dic61C22×C412yes—
120C2×Dic122—42no—
121C2×Dic122C22⋊C412yes—
122C2×Dic122C22⋊C412yes—
123C2×C8×C61—24no—
124C4⋊C4×C61—22no—
125C22⋊C4×C61—22no—
126C22⋊C4×C61—22no—
127C42×C61C2×D412yes—
128C42×C61C2×D412yes—
129C22⋊Dic61—24no—
130C22⋊Dic61—22no—
131C22⋊Dic61—22no—
132C22⋊Dic61—24no—
133C4⋊Dic61—21no—
134C4⋊Dic61—21no—
135C4⋊Dic61—21no—
136C4⋊Dic61—21no—
137C4⋊Dic61—22no—
138C4⋊Dic61—22no—
139C4⋊Dic61C22⋊C412yes—
140C4⋊Dic61C22⋊C411yescharacteristic
141C4⋊Dic61C22⋊C411yescharacteristic
142#11758196—24no—
143#11758196—24no—
144#11758196—24no—
145#11758196—24no—
146#11758196C22⋊C414yes—
147#11758196C22⋊C414yes—
148C4×Dic61—22no—
149C4×Dic61—22no—
150C4×Dic61C2×D414yes—
151C4×Dic61C2×D414yes—
152C2×#92418—22no—
153C2×#92418—22no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—21no—
157C23×C61—21no—
158C23×C61—21no—
159C23×C61—41no—
160C23×C61—41no—
161C23×C61—41no—
162C23×C61—41no—
163C23×C61#13112yes—
164C23×C61C23×C411yescharacteristic
165D4×C61—44no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—22no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—41no—
172C2×C4×C61—42no—
173C2×C4×C61—42no—
174C2×C4×C61—42no—
175C2×C4×C61—42no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61C2×C22⋊C412yes—
179C2×C4×C61C2×C22⋊C411yescharacteristic
180C2×C4×C61C2×C22⋊C411yescharacteristic
181C2×C4×C61C22×D412yes—
182C2×Dic61—22no—
183C2×Dic61—22no—
184C2×Dic61—22no—
185C2×Dic61—22no—
186C2×Dic61—22no—
187C2×Dic61—22no—
188C2×Dic61—42no—
189C2×Dic61—44no—
190C2×Dic61—44no—
191C2×Dic61—42no—
192C2×Dic61C2×C22⋊C412yes—
193C2×Dic61C2×C22⋊C412yes—
194C2×Dic61C2×C22⋊C414yes—
195Dic122—24no—
196Dic122—42no—
197C8×C61—44no—
198#92418—42no—
199#92418—42no—
200#17423—611nomaximal, Sylow
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—22no—
204C22×C61—42no—
205C22×C61—42no—
206C22×C61—41no—
207C22×C61—41no—
208C22×C61—41no—
209C22×C61—41no—
210C22×C61—41no—
211C22×C61—81no—
212C22×C61—81no—
213C22×C61C22×C22⋊C411yescharacteristic
214C22×C61C2×#13111yescharacteristic
215C22×C61#56811yescharacteristic
216C4×C61—22no—
217C4×C61—22no—
218C4×C61—21no—
219C4×C61—41no—
220C4×C61—41no—
221C4×C61#41312yes—
222Dic61—22no—
223Dic61—22no—
224Dic61—24no—
225Dic61—42no—
226Dic61—42no—
227C2×#552—611no—
228#2984—612no—
229#1473—614no—
230#1470—612no—
231#1466D6111yescharacteristic
232#1464—612no—
233#1460—611no—
234#1458—612no—
235C2×C61—21no—
236C2×C61—21no—
237C2×C61—42no—
238C2×C61—81no—
239C2×C61#288011yescharacteristic
240C2×C61#288411yescharacteristic
241C2×C61C2×#41311yescharacteristic
242#552—1222no—
243#552—1224no—
244#552—1222no—
245C2×#126—612no—
246C2×#125—612no—
247C2×#125—611no—
248C2×#123—612no—
249C2×C22⋊Q8—611no—
250C2×C22⋊Q8—611no—
251C2×C4×Q8—611no—
252C2×C4×D4C2×D6111yescharacteristic
253C2×#117—611no—
254C2×#117—612no—
255C22×C4⋊C4—611no—
256#446—1224no—
257#446—1222no—
258C8×Q8—1222no—
259#432—614no—
260#432C2×D6112yes—
261#431—612no—
262#431—614no—
263#324—614no—
264#324—614no—
265#323—611no—
266#323—611no—
267#323—612no—
268#323C2×D6112yes—
269#323C2×D6112yes—
270C61#1742311yescharacteristic, complemented, Sylow
271C22×Q8—611no—
272C22×D4C4×D6111yescharacteristic
273C23×C4—612no—
274C23×C4C4×D6111yescharacteristic
275C23×C4C4×D6111yescharacteristic
276#126—1224no—
277#126—1224no—
278#125—1224no—
279#125—1224no—
280#125—1222no—
281#125—1222no—
282#123—1224no—
283#123—1224no—
284C22⋊Q8—1222no—
285C22⋊Q8—1222no—
286C22⋊Q8—1222no—
287C22⋊Q8—1222no—
288C4×Q8—612no—
289C4×Q8—614no—
290C4×Q8—612no—
291C4×D4—22no—
292C4×D4—22no—
293#117—1224no—
294#117—1224no—
295#117—1222no—
296#117—1221no—
297#117—1221no—
298C2×C4⋊C4—611no—
299C2×C4⋊C4—612no—
300C2×C4⋊C4—611no—
301C2×C4⋊C4—612no—
302C2×C4⋊C4—612no—
303C2×C4⋊C4—611no—
304C2×C4⋊C4—612no—
305C2×C4⋊C4—612no—
306C2×C4⋊C4—612no—
307C2×C4⋊C4—1222no—
308C2×C4⋊C4—1221no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4C4×D6111yescharacteristic
311C2×C22⋊C4—612no—
312C2×C22⋊C4—612no—
313C2×C22⋊C4—612no—
314C2×C22⋊C4C22×D6111yescharacteristic
315C2×C22⋊C4C22×D6111yescharacteristic
316C2×C42—612no—
317C2×C42—612no—
318C2×C42C22×D6111yescharacteristic
319C4⋊C8—22no—
320C4⋊C8—1224no—
321C4⋊C8—1224no—
322C4⋊C8—1222no—
323C22⋊C8—612no—
324C22⋊C8—612no—
325C22⋊C8C22×D6114yes—
326#97—22no—
327#97—1224no—
328C4×C8—1224no—
329C4×C8—1222no—
330C24C2×C4×D6111yescharacteristic
331C24C2×C4×D6111yescharacteristic
332C2×Q8—612no—
333C2×Q8—612no—
334C2×Q8—2442no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—41no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—612no—
345C22×C4—612no—
346C22×C4—612no—
347C22×C4—612no—
348C22×C4—1222no—
349C22×C4—1222no—
350C22×C4—1224no—
351C22×C4—2442no—
352C22×C4C2×C4×D6111yescharacteristic
353C22×C4C2×C4×D6111yescharacteristic
354C22×C4C2×C4×D6111yescharacteristic
355C22×C4C2×C4×D6111yescharacteristic
356C22×C4C23×D6111yescharacteristic
357C2×C8—24no—
358C2×C8—1222no—
359C2×C8—1222no—
360C4⋊C4—22no—
361C4⋊C4—612no—
362C4⋊C4—611no—
363C4⋊C4—611no—
364C4⋊C4—614no—
365C4⋊C4—614no—
366C4⋊C4—1222no—
367C4⋊C4—1222no—
368C4⋊C4—1224no—
369C4⋊C4—1224no—
370C4⋊C4—1224no—
371C4⋊C4—1221no—
372C4⋊C4—1221no—
373C4⋊C4—1224no—
374C4⋊C4—1221no—
375C4⋊C4—1221no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—1224no—
379C22⋊C4—1222no—
380C22⋊C4—1222no—
381C22⋊C4—1224no—
382C42—614no—
383C42—614no—
384C42—1222no—
385C42—1222no—
386C42D4×D6112yes—
387C42D4×D6112yes—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23#1451978512yes—
398C23C22×C4×D6111yescharacteristic
399Q8—1224no—
400Q8—2442no—
401D4—44no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—612no—
415C2×C4—612no—
416C2×C4—614no—
417C2×C4—1222no—
418C2×C4—1222no—
419C2×C4—1222no—
420C2×C4—1222no—
421C2×C4—1222no—
422C2×C4—1222no—
423C2×C4—2442no—
424C2×C4—2444no—
425C2×C4—2444no—
426C2×C4—2442no—
427C2×C4C22⋊C4×D6112yes—
428C2×C4C22⋊C4×D6111yescharacteristic
429C2×C4C22⋊C4×D6111yescharacteristic
430C2×C4C2×D4×D6112yes—
431C8—44no—
432C8—2442no—
433C8—2442no—
434C22—21no—
435C22—21no—
436C22—22no—
437C22—42no—
438C22—42no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—81no—
445C22—81no—
446C22C2×C22⋊C4×D6111yescharacteristic
447C22C2×#1451978511yescharacteristic
448C22#1684052811yescharacteristic
449C4—22no—
450C4—22no—
451C4—21no—
452C4—41no—
453C4—41no—
454C4—1222no—
455C4—1222no—
456C4—1224no—
457C4—2442no—
458C4—2442no—
459C4#1683974312yes—
460C2—21no—
461C2—21no—
462C2—42no—
463C2—81no—
464C2#2145426911yescharacteristic
465C2#2145428111yescharacteristic
466C2C2×#1683974311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1466C2
socleC22×C61C22×C22⋊C4
prime radicalC22×D4×C61C2×C4
cosocleC22×C4×C61C24
hypercenter#1466D61
hypocenterC61#17423
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
2. #21454909C2semidirect
3. #21444185C2semidirect
4. #21444176C2semidirect
5. #21444150C2semidirect
7. #21444130C2semidirect
270. C61#17423semidirect

Maximal subgroups

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

subgroupindex
C2×#168404522
#214549092
#214441852
#214441762
#214441502
#214441402
#214441302
C61×#14662
#1742361