Finite Group Databasebeta

← #29572914

Subgroups — #29572914

Order 15616.

524 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839830C211yesmaximal, characteristic
2#21445846C211yesmaximal, characteristic, complemented
3#21445842C211yesmaximal, characteristic, complemented
4#21445804C211yesmaximal, characteristic
5#21445804C211yesmaximal, characteristic
6#21443687C211yesmaximal, characteristic, complemented
7C61×#1289C211yesmaximal, characteristic, complemented
8C22×#11758189C411yescharacteristic, complemented
9C23×#92418C2211yescharacteristic
10C2×C4×C22⋊D61C2211yescharacteristic
11#16839830—21no—
12#16839830—21no—
13#16839830—21no—
14#16839830—21no—
15C2×#14519710—21no—
16C2×#14519708—21no—
17C2×#14519706C411yescharacteristic, complemented
18C2×#14519691—21no—
19C2×#14519691—21no—
20C2×#14519689C411yescharacteristic, complemented
21C2×#14519687C411yescharacteristic, complemented
22#16839622C2211yescharacteristic
23#16839622C2211yescharacteristic
24#16839528—21no—
25#16839528—21no—
26#16839524—21no—
27#16839524—21no—
28C2×C22⋊C8×C61C2211yescharacteristic
29C61×#380C2211yescharacteristic
30C61×#380C2211yescharacteristic
31C22×C22⋊D61D411yescharacteristic
32C22×C4×D61C2×C411yescharacteristic
33C2×#11758189—21no—
34C2×#11758189—21no—
35C2×#11758189—42no—
36C22×#92418—42no—
37C22×#92418C2×C411yescharacteristic
38C22×#92418C2×C411yescharacteristic
39C22×#92418D411yescharacteristic
40C22×#92418D411yescharacteristic
41C22×#92418D411yescharacteristic
42C22×#92418D411yescharacteristic
43C4×C22⋊D61—21no—
44C4×C22⋊D61—21no—
45C4×C22⋊D61—21no—
46C4×C22⋊D61—21no—
47C2×C4⋊Dic61—21no—
48C2×C4⋊Dic61—21no—
49C2×#11758196Q811yescharacteristic
50C2×C22⋊Dic61—21no—
51C2×C22⋊Dic61—21no—
52C2×C22⋊Dic61D411yescharacteristic
53C2×#11758198Q811yescharacteristic
54C2×C4×Dic61C2×C411yescharacteristic
55#14519710—42no—
56#14519708—42no—
57#14519706—21no—
58#14519706—21no—
59#14519691—42no—
60#14519691—42no—
61#14519689—21no—
62#14519689—21no—
63#14519687—21no—
64#14519687—21no—
65#14519684—21no—
66#14519684—21no—
67C23×C4×C61C2311yescharacteristic
68C22×C8×C61C2×C411yescharacteristic
69C22×C8×C61C2×C411yescharacteristic
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C22⋊C4×C61—21no—
73C2×C22⋊C4×C61—21no—
74C22⋊C8×C61—21no—
75C22⋊C8×C61—21no—
76#95×C61—21no—
77#95×C61—21no—
78C24×C61C2×D411yescharacteristic
79C23×D61C4⋊C411yescharacteristic
80C22×C4×C61—41no—
81C22×C4×C61—41no—
82C22×C4×C61—41no—
83C22×C4×C61—41no—
84C22×C4×C61C22×C411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×Q811yescharacteristic
88C22×C4×C61#4111yescharacteristic
89C22×C4×C61#4111yescharacteristic
90C2×C22⋊D61—21no—
91C2×C22⋊D61—21no—
92C2×C22⋊D61—21no—
93C2×C22⋊D61—21no—
94C2×C22⋊D61—21no—
95C2×C22⋊D61—21no—
96C22×Dic61—41no—
97C22×Dic61—41no—
98C22×Dic61C4⋊C411yescharacteristic
99C22×Dic61C4⋊C411yescharacteristic
100C22×Dic61C4⋊C411yescharacteristic
101C2×C4×D61—21no—
102C2×C4×D61—21no—
103C2×C4×D61—41no—
104C2×C4×D61—41no—
105C2×C8×C61—21no—
106C2×C8×C61—21no—
107C2×C8×C61—41no—
108C2×C8×C61—41no—
109C4⋊C4×C61—41no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—41no—
113C22⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117C22⋊Dic61—21no—
118C22⋊Dic61—21no—
119C22⋊Dic61—41no—
120C22⋊Dic61—41no—
121C22⋊Dic61—41no—
122C22⋊Dic61—41no—
123#11758198—21no—
124#11758198—21no—
125C4⋊Dic61—41no—
126C4⋊Dic61—41no—
127C4⋊Dic61—41no—
128C4⋊Dic61—41no—
129#11758196—21no—
130#11758196—21no—
131C4×Dic61—21no—
132C4×Dic61—21no—
133C2×#92418—22no—
134C2×#92418—22no—
135C2×#92418—22no—
136C2×#92418—41no—
137C2×#92418—41no—
138C2×#92418—41no—
139C2×#92418C22⋊C411yescharacteristic
140C2×#92418C22⋊C411yescharacteristic
141C2×#92418C22⋊C411yescharacteristic
142C2×#92418C22⋊C411yescharacteristic
143#11758189—42no—
144#11758189—42no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61C2×C4⋊C411yescharacteristic
148C23×C61#12111yescharacteristic
149C23×C61#12111yescharacteristic
150C23×C61AGL(1,ℤ/8)11yescharacteristic
151C23×C61AGL(1,ℤ/8)11yescharacteristic
152C23×C61#13711yescharacteristic
153C23×C61#13711yescharacteristic
154C22×D61—21no—
155C22×D61—21no—
156C22×D61—21no—
157C22×D61—41no—
158C22×D61—41no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—81no—
169C2×C4×C61—81no—
170C2×C4×C61—81no—
171C2×C4×C61—81no—
172C2×C4×C61—81no—
173C2×C4×C61—81no—
174C2×C4×C61C2×C22⋊C411yescharacteristic
175C2×C4×C61#11711yescharacteristic
176C2×C4×C61C22⋊Q811yescharacteristic
177C2×C4×C61C22⋊Q811yescharacteristic
178C22⋊D61—41no—
179C22⋊D61—41no—
180C22⋊D61—41no—
181C22⋊D61—41no—
182C2×Dic61—21no—
183C2×Dic61—21no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—81no—
190C2×Dic61—81no—
191C2×Dic61—81no—
192C2×Dic61—81no—
193C2×Dic61—81no—
194C2×Dic61—81no—
195C4×D61—42no—
196C4×D61—41no—
197C4×D61—41no—
198C8×C61—41no—
199C8×C61—41no—
200#92418—21no—
201#92418—21no—
202#92418—21no—
203#92418#9912yescomplemented
204#8610—611nomaximal, Sylow
205C22×C61—21no—
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—41no—
212C22×C61—41no—
213C22×C61—41no—
214C22×C61—41no—
215C22×C61#38011yescharacteristic
216C22×C61#42811yescharacteristic
217C22×C61#42811yescharacteristic
218C22×C61#46811yescharacteristic
219C22×C61#46811yescharacteristic
220C22×C61#46911yescharacteristic
221C22×C61#47011yescharacteristic
222C2×D61—41no—
223C2×D61—41no—
224C2×D61—41no—
225C2×D61—81no—
226C2×D61—81no—
227C4×C61—21no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—161no—
231C4×C61—161no—
232C4×C61C2×#9911yescharacteristic
233C4×C61#41011yescharacteristic
234Dic61—41no—
235Dic61—41no—
236Dic61—41no—
237Dic61—161no—
238Dic61—161no—
239C2×#435—611no—
240#1808—611no—
241#1807—611no—
242#1799—611no—
243#1799—611no—
244#1289—611no—
245#1289D6111yescharacteristic, complemented
246C2×C61—21no—
247C2×C61—21no—
248C2×C61—21no—
249C2×C61—21no—
250C2×C61—21no—
251C2×C61—21no—
252C2×C61#179911yescharacteristic
253C2×C61#180911yescharacteristic
254C2×C61#181211yescharacteristic
255D61—81no—
256C22×M16—611no—
257C23×C8—611no—
258C2×C4×D4—611no—
259#435—1221no—
260#435—1221no—
261#435—1221no—
262#435—1221no—
263C2×#107—1221no—
264C2×#106—1221no—
265C2×C4⋊C8—611no—
266C2×#99—1221no—
267C2×#99—1221no—
268C2×C22⋊C8—611no—
269C2×C22⋊C8C2×D6111yescharacteristic
270C2×#97—611no—
271#380—611no—
272#380—611no—
273#380C2×D6111yescharacteristic
274#380C2×D6111yescharacteristic
275#342—1221no—
276#342—1221no—
277#340—1221no—
278#340—1221no—
279C61#861011yescharacteristic, complemented, Sylow
280C22×D4—611no—
281C23×C4—611no—
282C23×C4C22×D6111yescharacteristic
283C2×M16—1221no—
284C2×M16—1221no—
285C2×M16—2442no—
286C22×C8—611no—
287C22×C8—611no—
288C22×C8—611no—
289C22×C8—611no—
290C22×C8—611no—
291C22×C8—611no—
292C22×C8—2442no—
293C22×C8C4×D6111yescharacteristic
294C22×C8C4×D6111yescharacteristic
295C4×D4—1221no—
296C4×D4—1221no—
297C4×D4—1221no—
298C4×D4—1221no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—611no—
302C2×C4⋊C4—1221no—
303C2×C4⋊C4—1221no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—611no—
307C2×C22⋊C4—611no—
308C2×C22⋊C4—1221no—
309C2×C22⋊C4—1221no—
310C2×C42—611no—
311#107—2442no—
312#106—2442no—
313C4⋊C8—1221no—
314C4⋊C8—1221no—
315#99—2442no—
316#99—2442no—
317C22⋊C8—21no—
318C22⋊C8—21no—
319C22⋊C8—1221no—
320C22⋊C8—1221no—
321#97—1221no—
322#97—1221no—
323#95—21no—
324#95—21no—
325#95—1221no—
326#95—1221no—
327C24—611no—
328C24D4×D6111yescharacteristic
329C2×D4—1221no—
330C2×D4—1221no—
331C2×D4—1221no—
332C2×D4—1221no—
333C2×D4—1221no—
334C2×D4—1221no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—611no—
340C22×C4—611no—
341C22×C4—611no—
342C22×C4—1221no—
343C22×C4—1221no—
344C22×C4—2441no—
345C22×C4—2441no—
346C22×C4—2441no—
347C22×C4—2441no—
348C22×C4C2×C4×D6111yescharacteristic
349C22×C4#1175822211yescharacteristic
350C22×C4#1175822211yescharacteristic
351C22×C4D4×D6111yescharacteristic
352C22×C4D4×D6111yescharacteristic
353C22×C4Q8×D6111yescharacteristic
354M16—2442no—
355M16—2442no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—611no—
361C2×C8—611no—
362C2×C8—611no—
363C2×C8—611no—
364C2×C8—1222no—
365C2×C8—1222no—
366C2×C8—1222no—
367C2×C8—2441no—
368C2×C8—2441no—
369C2×C8—2441no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—1221no—
375C4⋊C4—1221no—
376C4⋊C4—2441no—
377C4⋊C4—2441no—
378C4⋊C4—2441no—
379C4⋊C4—2441no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—1221no—
385C22⋊C4—1221no—
386C22⋊C4—1221no—
387C22⋊C4—1221no—
388C22⋊C4—2441no—
389C22⋊C4—2441no—
390C22⋊C4—2441no—
391C22⋊C4—2441no—
392C42—1221no—
393C42—1221no—
394C23—41no—
395C23—41no—
396C23—1221no—
397C23—1221no—
398C23—1221no—
399C23—2441no—
400C23—2441no—
401C23C4⋊C4×D6111yescharacteristic
402C23#1451975011yescharacteristic
403C23#1451975011yescharacteristic
404C23#1451980111yescharacteristic
405C23#1451980411yescharacteristic
406C23#1451980611yescharacteristic
407C23#1451980911yescharacteristic
408D4—2441no—
409D4—2441no—
410D4—2441no—
411D4—2441no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—1221no—
428C2×C4—1221no—
429C2×C4—1221no—
430C2×C4—1221no—
431C2×C4—1221no—
432C2×C4—4881no—
433C2×C4—4881no—
434C2×C4—4881no—
435C2×C4—4881no—
436C2×C4—4881no—
437C2×C4—4881no—
438C2×C4—2442no—
439C2×C4—2441no—
440C2×C4—2441no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4C22⋊C4×D6111yescharacteristic
444C2×C4#1451973411yescharacteristic
445C2×C4#1451975511yescharacteristic
446C2×C4#1451975511yescharacteristic
447C8—41no—
448C8—41no—
449C8—612no—
450C8—1221no—
451C8—1221no—
452C8—1221no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—4881no—
464C22—4881no—
465C22—2441no—
466C22—2441no—
467C22—2441no—
468C22#1683962111yescharacteristic
469C22#1683980111yescharacteristic
470C22#1683980311yescharacteristic
471C22#1684000711yescharacteristic
472C22#1684001111yescharacteristic
473C22#1684001411yescharacteristic
474C22#1684001911yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—161no—
479C4—161no—
480C4—9761no—
481C4—9761no—
482C4—2441no—
483C4—2441no—
484C4—2441no—
485C4#99×D6111yescharacteristic
486C4#1683973411yescharacteristic
487C2—21no—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—21no—
492C2—21no—
493C2—4881no—
494C2#2144580611yescharacteristic
495C2#2144585111yescharacteristic
496C2#2144586411yescharacteristic
subgroupG/H
centerC22#16839621
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1289C2
socleC22×C61#380
prime radicalC22×C22⋊D61D4
cosocleC23×C4×C61C23
hypercenter#1289D61
hypocenterC61#8610
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
2. #21445846C2semidirect
3. #21445842C2semidirect
6. #21443687C2semidirect
7. C61×#1289C2semidirect
8. C22×#11758189C4semidirect
17. C2×#14519706C4semidirect
20. C2×#14519689C4semidirect
21. C2×#14519687C4semidirect
203. #92418#99semidirect
245. #1289D61semidirect
279. C61#8610semidirect

Maximal subgroups

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

subgroupindex
C2×#168398302
#214458462
#214458422
#214458042
#214436872
C61×#12892
#861061