Finite Group Databasebeta

← #29574712

Subgroups — #29574712

Order 15616.

600 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
1#21454845C211yesmaximal, characteristic
2C4×#14519761C211yesmaximal, characteristic
3#21445885C211yesmaximal, characteristic, complemented
4#21445885C211yesmaximal, characteristic, complemented
5#21444135C211yesmaximal, characteristic
6#21443667C211yesmaximal, characteristic, complemented
7C61×#1282C211yesmaximal, characteristic, complemented
8C2×#14519761C411yescharacteristic
9C2×C4×C22⋊D61C2211yescharacteristic
10C22×C4×Dic61C2211yescharacteristic
11#16839833—22no—
12#16839830—22no—
13#16839816—21no—
14#16839724—21no—
15C2×C22⋊#92418C2211yescharacteristic
16C2×C22⋊#92418C2211yescharacteristic
17#16839654—21no—
18#16839647—21no—
19#16839639C411yescharacteristic
20#16839637C411yescharacteristic
21#16839623C411yescharacteristic
22C4×C4⋊Dic61—21no—
23C4×#11758198—21no—
24#16839516—22no—
25#16839516—22no—
26#16839487—21no—
27#16839485—21no—
28C2×C22⋊C8×C61C2211yescharacteristic
29C2×C22⋊C8×C61C2211yescharacteristic
30C4×C22⋊C4×C61C2211yescharacteristic
31C22×C22⋊D61C2×C411yescharacteristic
32C23×Dic61C2×C411yescharacteristic
33C23×Dic61C2×C411yescharacteristic
34C22×C4×D61D411yescharacteristic
35C2×#11758189—22no—
36C22×#92418—21no—
37C22×#92418D412yes—
38#14519761—22no—
39#14519761—22no—
40C4×C22⋊D61—21no—
41C4×C22⋊D61—21no—
42C4×C22⋊D61—21no—
43C4×C22⋊D61—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61—21no—
46C2×#11758196D411yescharacteristic
47C2×C22⋊Dic61C2×C411yescharacteristic
48C2×#11758198—21no—
49C2×#11758198—21no—
50C2×#11758198D411yescharacteristic
51C2×C4×Dic61—21no—
52C2×C4×Dic61—21no—
53C2×C4×Dic61—21no—
54C2×C4×Dic61—21no—
55C2×C4×Dic61—21no—
56C2×C4×Dic61—21no—
57C2×C4×Dic61D411yescharacteristic
58#14519706—22no—
59C22⋊#92418—21no—
60C22⋊#92418—21no—
61C22⋊#92418—21no—
62C22⋊#92418—21no—
63#14519689—22no—
64#14519687—22no—
65#14519684—21no—
66#14519684—21no—
67#14519684—21no—
68#14519684—21no—
69C23×C4×C61C2311yescharacteristic
70C22×C8×C61—21no—
71C22×C8×C61D412yes—
72C2×C22⋊C4×C61C2×C411yescharacteristic
73C2×C22⋊C4×C61C2×C411yescharacteristic
74C2×C42×C61—21no—
75C22⋊C8×C61—21no—
76C22⋊C8×C61—21no—
77C22⋊C8×C61—21no—
78C22⋊C8×C61—21no—
79#95×C61—21no—
80C24×C61C22×C411yescharacteristic
81C23×D61C22⋊C411yescharacteristic
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—42no—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61#4111yescharacteristic
90C22×C4×C61#4111yescharacteristic
91C2×C22⋊D61—21no—
92C2×C22⋊D61—21no—
93C2×C22⋊D61—21no—
94C2×C22⋊D61—21no—
95C2×C22⋊D61—21no—
96C2×C22⋊D61—21no—
97C22×Dic61—21no—
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61—21no—
105C22×Dic61—21no—
106C22×Dic61—42no—
107C22×Dic61C22⋊C411yescharacteristic
108C22×Dic61C22⋊C411yescharacteristic
109C22×Dic61C22⋊C411yescharacteristic
110C22×Dic61M1612yescomplemented
111C2×C4×D61—21no—
112C2×C4×D61—21no—
113C2×C4×D61—42no—
114C2×C8×C61—22no—
115C2×C8×C61—42no—
116C2×C8×C61—42no—
117C2×C8×C61—41no—
118C22⋊C4×C61—21no—
119C22⋊C4×C61—21no—
120C22⋊C4×C61—21no—
121C22⋊C4×C61M1612yescomplemented
122C42×C61—42no—
123C22⋊Dic61—21no—
124C22⋊Dic61—21no—
125#11758198—24no—
126#11758198—21no—
127#11758198—21no—
128#11758198—42no—
129C4⋊Dic61—24no—
130C4⋊Dic61—42no—
131#11758196—21no—
132#11758196—21no—
133C4×Dic61—21no—
134C4×Dic61—21no—
135C4×Dic61—41no—
136C4×Dic61—41no—
137C4×Dic61—41no—
138C4×Dic61—41no—
139C2×#92418—22no—
140C2×#92418—42no—
141C2×#92418—42no—
142C2×#92418—41no—
143#11758189—82no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61C2×C22⋊C411yescharacteristic
147C23×C61C4×D411yescharacteristic
148C23×C61C4×D411yescharacteristic
149C23×C61C2×M1611yescharacteristic
150C23×C61C2×M1611yescharacteristic
151C23×C61#13111yescharacteristic
152C23×C61#13111yescharacteristic
153C22×D61—21no—
154C22×D61—21no—
155C22×D61—21no—
156C22×D61—41no—
157C22×D61—41no—
158C2×C4×C61—22no—
159C2×C4×C61—22no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61#12011yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#12311yescharacteristic
174C22⋊D61—41no—
175C22⋊D61—41no—
176C22⋊D61—41no—
177C22⋊D61—41no—
178C2×Dic61—22no—
179C2×Dic61—22no—
180C2×Dic61—22no—
181C2×Dic61—21no—
182C2×Dic61—21no—
183C2×Dic61—21no—
184C2×Dic61—21no—
185C2×Dic61—22no—
186C2×Dic61—21no—
187C2×Dic61—22no—
188C2×Dic61—22no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C4×D61—42no—
200C4×D61—81no—
201C8×C61—42no—
202C8×C61—81no—
203#92418—42no—
204#92418—81no—
205#8968—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—41no—
213C22×C61—41no—
214C22×C61—41no—
215C22×C61—41no—
216C22×C61#38611yescharacteristic
217C22×C61#40711yescharacteristic
218C22×C61#40811yescharacteristic
219C22×C61#43511yescharacteristic
220C22×C61#43511yescharacteristic
221C22×C61#43611yescharacteristic
222C22×C61#43611yescharacteristic
223C2×D61—42no—
224C2×D61—41no—
225C2×D61—41no—
226C2×D61—41no—
227C2×D61—81no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—42no—
232C4×C61—81no—
233C4×C61#45812yes—
234Dic61—42no—
235Dic61—41no—
236Dic61—41no—
237Dic61—42no—
238Dic61—41no—
239#2973—611no—
240C4×#123—611no—
241#1817—611no—
242#1817—611no—
243#1459—611no—
244#1282—611no—
245#1282D6111yescharacteristic, complemented
246C2×C61—21no—
247C2×C61—21no—
248C2×C61—21no—
249C2×C61—21no—
250C2×C61—21no—
251C2×C61—21no—
252C2×C61#187011yescharacteristic
253C2×C61#189011yescharacteristic
254C2×C61#189111yescharacteristic
255D61—81no—
256C2×#123—611no—
257C2×C4×D4—611no—
258C22×C42—611no—
259#435—1222no—
260#435—1222no—
261#432—1221no—
262#407—1221no—
263C2×C22⋊C8—611no—
264C2×C22⋊C8—611no—
265C2×C22⋊C8C2×D6111yescharacteristic
266C2×C22⋊C8C2×D6111yescharacteristic
267#388—1221no—
268#387—1221no—
269#386—611no—
270#385—611no—
271#381—611no—
272C4×C4⋊C4—1221no—
273C4×C22⋊C4—1221no—
274C4×C22⋊C4C2×D6111yescharacteristic
275#336—1222no—
276#336—1222no—
277#324—1221no—
278#323—1221no—
279C61#896811yescharacteristic, complemented, Sylow
280C22×D4—611no—
281C23×C4—611no—
282C23×C4—611no—
283C23×C4—611no—
284C23×C4C22×D6111yescharacteristic
285C2×M16—1222no—
286C22×C8—21no—
287C22×C8—612no—
288C22×C8—1221no—
289C22×C8D24412yes—
290#123—1222no—
291#123—1222no—
292C4×D4—1221no—
293C4×D4—1221no—
294C4×D4—1221no—
295C4×D4—1221no—
296C2×C4⋊C4—611no—
297C2×C4⋊C4—1221no—
298C2×C4⋊C4—1221no—
299C2×C22⋊C4—611no—
300C2×C22⋊C4—611no—
301C2×C22⋊C4—1221no—
302C2×C22⋊C4—1221no—
303C2×C22⋊C4C4×D6111yescharacteristic
304C2×C22⋊C4C4×D6111yescharacteristic
305C2×C42—21no—
306C2×C42—611no—
307C2×C42—1221no—
308C2×C42—1221no—
309C2×C42—1221no—
310C2×C42—1221no—
311C2×C42—1221no—
312C2×C42—1221no—
313C4⋊C8—1222no—
314C22⋊C8—21no—
315C22⋊C8—21no—
316C22⋊C8—21no—
317C22⋊C8—21no—
318C22⋊C8—1222no—
319C22⋊C8—1221no—
320C22⋊C8—1221no—
321C22⋊C8—1221no—
322C22⋊C8—1221no—
323#97—1222no—
324#95—21no—
325#95—1221no—
326#95—1221no—
327#95—1221no—
328#95—1221no—
329C24—611no—
330C24C2×C4×D6111yescharacteristic
331C2×D4—1221no—
332C2×D4—1221no—
333C2×D4—1221no—
334C2×D4—1221no—
335C2×D4—1221no—
336C2×D4—1221no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—42no—
340C22×C4—611no—
341C22×C4—612no—
342C22×C4—611no—
343C22×C4—611no—
344C22×C4—1221no—
345C22×C4—1221no—
346C22×C4—1221no—
347C22×C4—1221no—
348C22×C4—1221no—
349C22×C4—1221no—
350C22×C4—1221no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—1221no—
354C22×C4—1221no—
355C22×C4—2442no—
356C22×C4—2442no—
357C22×C4C2×D24411yescharacteristic
358C22×C4#1175822211yescharacteristic
359C22×C4D4×D6111yescharacteristic
360C22×C4D4×D6111yescharacteristic
361C22×C4D4×D6111yescharacteristic
362C22×C4#1175822611yescharacteristic
363M16—4882no—
364C2×C8—22no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—41no—
368C2×C8—1222no—
369C2×C8—2442no—
370C2×C8—2441no—
371C2×C8—2442no—
372C4⋊C4—1224no—
373C4⋊C4—1221no—
374C4⋊C4—1221no—
375C4⋊C4—2442no—
376C22⋊C4—21no—
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—1224no—
380C22⋊C4—1221no—
381C22⋊C4—1221no—
382C22⋊C4—1221no—
383C22⋊C4—1221no—
384C22⋊C4—2442no—
385C22⋊C4#1175818912yescomplemented
386C42—42no—
387C42—1221no—
388C42—1221no—
389C42—2441no—
390C42—2441no—
391C42—2441no—
392C42—2441no—
393C23—41no—
394C23—41no—
395C23—1221no—
396C23—1221no—
397C23—1221no—
398C23—2441no—
399C23—2441no—
400C23C22⋊C4×D6111yescharacteristic
401C23#1451973911yescharacteristic
402C23C4×D24411yescharacteristic
403C23M16×D6111yescharacteristic
404C23C2×#1175818911yescharacteristic
405C23#1451978311yescharacteristic
406C23#1451978511yescharacteristic
407D4—2441no—
408D4—2441no—
409D4—2441no—
410D4—2441no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—1222no—
424C2×C4—1222no—
425C2×C4—1222no—
426C2×C4—1222no—
427C2×C4—1221no—
428C2×C4—1221no—
429C2×C4—1222no—
430C2×C4—1222no—
431C2×C4—1221no—
432C2×C4—1221no—
433C2×C4—1221no—
434C2×C4—4881no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4—2441no—
440C2×C4—2441no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4—2442no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4#1451974711yescharacteristic
447C2×C4#1451975011yescharacteristic
448C2×C4#1451975211yescharacteristic
449C2×C4#1451976211yescharacteristic
450C8—42no—
451C8—81no—
452C8—4881no—
453C8—2442no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—4881no—
465C22—2442no—
466C22—2441no—
467C22—2441no—
468C22—2441no—
469C22#1683964011yescharacteristic
470C22#1683972411yescharacteristic
471C22#1683972711yescharacteristic
472C22#1683983211yescharacteristic
473C22#1683983511yescharacteristic
474C22#1683983911yescharacteristic
475C22#1683984111yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—42no—
480C4—81no—
481C4—2441no—
482C4—2442no—
483C4—2441no—
484C4—2442no—
485C4—2441no—
486C4#1683995312yes—
487C2—21no—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—21no—
492C2—21no—
493C2—4881no—
494C2#2144611511yescharacteristic
495C2#2144623111yescharacteristic
496C2#2144623611yescharacteristic
subgroupG/H
centerC2×C4#14519747
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1282C2
socleC22×C61#386
prime radicalC22×C22⋊D61C2×C4
cosocleC23×C4×C61C23
hypercenter#1282D61
hypocenterC61#8968
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
3. #21445885C2semidirect
4. #21445885C2semidirect
6. #21443667C2semidirect
7. C61×#1282C2semidirect
110. C22×Dic61M16semidirect
121. C22⋊C4×C61M16semidirect
245. #1282D61semidirect
279. C61#8968semidirect
385. C22⋊C4#11758189semidirect

Maximal subgroups

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

subgroupindex
#214548452
C4×#145197612
#214458852
#214441352
#214436672
C61×#12822
#896861