Finite Group Databasebeta

← #29575797

Subgroups — #29575797

Order 15616.

648 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454151C211yesmaximal, characteristic
2C2×#16839629C211yesmaximal, characteristic, complemented
3#21445885C211yesmaximal, characteristic, complemented
4C2×#16839497C211yesmaximal, characteristic
5C2×#16839495C211yesmaximal, characteristic
6#21443697C211yesmaximal, characteristic, complemented
7C61×#1294C211yesmaximal, characteristic, complemented
8C2×#14519761—21no—
9C2×#14519751C2211yescharacteristic
10C22×C4×Dic61C2211yescharacteristic
11C2×C22⋊#92418C2211yescharacteristic
12#16839663C412yescomplemented
13#16839661C412yescomplemented
14#16839654—22no—
15#16839639—21no—
16#16839629—22no—
17#16839629—22no—
18C2×#14519684—21no—
19C2×#14519684—21no—
20C2×#14519684C2211yescharacteristic
21#16839516—22no—
22#16839497—22no—
23#16839495—22no—
24C2×C22⋊C8×C61C2211yescharacteristic
25C61×#380C2211yescharacteristic
26C2×#95×C61C2211yescharacteristic
27C22×D244C2×C411yescharacteristic
28C22×C22⋊D61—21no—
29C23×Dic61—21no—
30C23×Dic61D411yescharacteristic
31C23×Dic61D411yescharacteristic
32C22×#92418—21no—
33#14519761—42no—
34#14519761—42no—
35#14519751—21no—
36#14519751—21no—
37#14519751—21no—
38#14519751—21no—
39C2×C4⋊Dic61—22no—
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×#11758198—21no—
42C2×#11758198—22no—
43C2×#11758198—22no—
44C2×C4×Dic61—21no—
45C2×C4×Dic61—21no—
46C2×C4×Dic61—21no—
47C2×C4×Dic61—21no—
48C2×C4×Dic61C2×C412yes—
49C2×C4×Dic61D412yes—
50C22⋊#92418—21no—
51C22⋊#92418—21no—
52#14519684—24no—
53#14519684—24no—
54#14519684—21no—
55#14519684—22no—
56#14519684D412yes—
57C23×C4×C61—21no—
58C23×C4×C61C2311yescharacteristic
59C22×C8×C61—21no—
60C2×C4⋊C4×C61C2×C412yes—
61C2×C22⋊C4×C61—21no—
62C22⋊C8×C61—21no—
63C22⋊C8×C61—21no—
64#95×C61—21no—
65#95×C61—21no—
66#95×C61D412yes—
67C24×C61C2×D411yescharacteristic
68C23×D61—21no—
69C22×C4×C61—22no—
70C22×C4×C61—22no—
71C22×C4×C61—42no—
72C22×C4×C61—42no—
73C22×C4×C61—42no—
74C22×C4×C61—41no—
75C22×C4×C61—41no—
76C22×C4×C61C22×C411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61#4111yescharacteristic
81C22×C4×C61#4111yescharacteristic
82C2×C22⋊D61—41no—
83C2×C22⋊D61—41no—
84C2×C22⋊D61—41no—
85C2×C22⋊D61—41no—
86C2×C22⋊D61—41no—
87C2×C22⋊D61—41no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C22×Dic61—22no—
98C22×Dic61—42no—
99C22×Dic61—42no—
100C22×Dic61—42no—
101C22×Dic61—41no—
102C22×Dic61—41no—
103C22×Dic61C22⋊C412yes—
104C22×Dic61C22⋊C412yes—
105C2×D244—21no—
106C2×D244—21no—
107C2×D244—42no—
108C2×C8×C61—42no—
109C2×C8×C61—41no—
110C4⋊C4×C61—42no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113#11758198—44no—
114#11758198—44no—
115#11758198—41no—
116#11758198—41no—
117C4⋊Dic61—21no—
118C4⋊Dic61—21no—
119C4⋊Dic61—44no—
120C4×Dic61—22no—
121C4×Dic61—22no—
122C4×Dic61—22no—
123C4×Dic61—22no—
124C2×#92418—42no—
125C2×#92418—41no—
126C23×C61—22no—
127C23×C61—22no—
128C23×C61C2×C22⋊C411yescharacteristic
129C23×C61#12011yescharacteristic
130C23×C61#12311yescharacteristic
131C23×C61#13511yescharacteristic
132C23×C61#13511yescharacteristic
133C23×C61AGL(1,ℤ/8)11yescharacteristic
134C23×C61AGL(1,ℤ/8)11yescharacteristic
135C22×D61—41no—
136C22×D61—42no—
137C22×D61—41no—
138C22×D61—41no—
139C22×D61—42no—
140C2×C4×C61—22no—
141C2×C4×C61—22no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—42no—
147C2×C4×C61—42no—
148C2×C4×C61—42no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61—82no—
153C2×C4×C61—82no—
154C2×C4×C61C4×D411yescharacteristic
155C2×C4×C61C4×D411yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#12111yescharacteristic
158C22⋊D61—81no—
159C22⋊D61—81no—
160C22⋊D61—81no—
161C22⋊D61—81no—
162C2×Dic61—22no—
163C2×Dic61—22no—
164C2×Dic61—22no—
165C2×Dic61—22no—
166C2×Dic61—22no—
167C2×Dic61—22no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—42no—
175C2×Dic61—42no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C2×Dic61—41no—
179C2×Dic61—42no—
180C2×Dic61—82no—
181C2×Dic61—82no—
182D244—81no—
183D244—81no—
184C8×C61—81no—
185#92418—81no—
186#9157—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—22no—
192C22×C61—22no—
193C22×C61—22no—
194C22×C61—22no—
195C22×C61#38611yescharacteristic
196C22×C61#41012yes—
197C22×C61#41611yescharacteristic
198C22×C61#41811yescharacteristic
199C22×C61#42112yes—
200C22×C61#44911yescharacteristic
201C22×C61#44911yescharacteristic
202C22×C61#48211yescharacteristic
203C22×C61#48211yescharacteristic
204C2×D61—81no—
205C2×D61—81no—
206C2×D61—81no—
207C2×D61—82no—
208C2×D61—82no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—82no—
214C4×C61—82no—
215Dic61—42no—
216Dic61—42no—
217Dic61—82no—
218Dic61—81no—
219#2854—611no—
220C2×#383—611no—
221#1817—611no—
222C2×#327—611no—
223C2×#327—611no—
224#1294—611no—
225#1294D6111yescharacteristic, complemented
226C2×C61—21no—
227C2×C61—21no—
228C2×C61—21no—
229C2×C61—21no—
230C2×C61#151412yes—
231C2×C61#152112yes—
232C2×C61#187511yescharacteristic
233C2×C61#189711yescharacteristic
234C2×C61#189911yescharacteristic
235D61—161no—
236C2×#123—1221no—
237C2×#121—611no—
238C22×C42—611no—
239C2×C22⋊C8—611no—
240C2×C22⋊C8C2×D6111yescharacteristic
241#390—612no—
242#389—612no—
243#388—1222no—
244#386—1221no—
245#383—1222no—
246#383—1222no—
247#380C2×D6111yescharacteristic
248C2×#95—611no—
249C2×#95—1221no—
250C2×#95—1221no—
251C2×#95C2×D6111yescharacteristic
252#336—1222no—
253#327—1222no—
254#327—1222no—
255C61#915711yescharacteristic, complemented, Sylow
256C22×D4—611no—
257C22×D4—1221no—
258C23×C4—21no—
259C23×C4—611no—
260C23×C4—611no—
261C23×C4—1221no—
262C23×C4C22×D6111yescharacteristic
263C22×C8—21no—
264C22×C8—1221no—
265#123—2442no—
266#123—2442no—
267#121—1221no—
268#121—1221no—
269#121—1221no—
270#121—1221no—
271C2×C4⋊C4—611no—
272C2×C4⋊C4—1222no—
273C2×C4⋊C4C4×D6112yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—1222no—
276C2×C22⋊C4—1222no—
277C2×C22⋊C4—1221no—
278C2×C42—612no—
279C2×C42—612no—
280C2×C42—1221no—
281C2×C42—1221no—
282C2×C42—1221no—
283C2×C42—1221no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—1221no—
287C22⋊C8—1221no—
288#95—21no—
289#95—21no—
290#95—612no—
291#95—1222no—
292#95—1224no—
293#95—1224no—
294#95—1221no—
295#95D24412yes—
296C24—1221no—
297C24D4×D6111yescharacteristic
298C2×D4—1221no—
299C2×D4—1221no—
300C2×D4—2442no—
301C2×D4—2441no—
302C2×D4—2441no—
303C2×D4—2441no—
304C2×D4—2441no—
305C2×D4—2441no—
306C2×D4—2441no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—612no—
315C22×C4—612no—
316C22×C4—1221no—
317C22×C4—1221no—
318C22×C4—1221no—
319C22×C4—1221no—
320C22×C4—1221no—
321C22×C4—1221no—
322C22×C4—1221no—
323C22×C4—1221no—
324C22×C4—1221no—
325C22×C4—1222no—
326C22×C4—2442no—
327C22×C4—2442no—
328C22×C4—2442no—
329C22×C4—2441no—
330C22×C4—2441no—
331C22×C4C2×C4×D6111yescharacteristic
332C22×C4C2×D24411yescharacteristic
333C22×C4#1175822211yescharacteristic
334C22×C4D4×D6111yescharacteristic
335C22×C4D4×D6111yescharacteristic
336C22×C4#1175822611yescharacteristic
337C2×C8—42no—
338C2×C8—41no—
339C2×C8—2442no—
340C2×C8—2441no—
341C4⋊C4—42no—
342C4⋊C4—1221no—
343C4⋊C4—1221no—
344C4⋊C4—2444no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—2444no—
348C22⋊C4—2444no—
349C22⋊C4—2441no—
350C22⋊C4—2441no—
351C42—1222no—
352C42—1222no—
353C42—1222no—
354C42—1222no—
355C23—22no—
356C23—22no—
357C23—2442no—
358C23—2441no—
359C23—2441no—
360C23—2442no—
361C23—2441no—
362C23C22⋊C4×D6111yescharacteristic
363C23#1451974711yescharacteristic
364C23#1451976211yescharacteristic
365C23#1451979811yescharacteristic
366C23#1451980011yescharacteristic
367C23#1451980111yescharacteristic
368C23#1451980411yescharacteristic
369D4—4881no—
370D4—4881no—
371D4—4881no—
372D4—4881no—
373D4—4881no—
374D4—4881no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—1222no—
390C2×C4—1222no—
391C2×C4—1222no—
392C2×C4—1222no—
393C2×C4—1222no—
394C2×C4—1222no—
395C2×C4—4882no—
396C2×C4—4882no—
397C2×C4—2442no—
398C2×C4—2442no—
399C2×C4—2441no—
400C2×C4—2441no—
401C2×C4—2441no—
402C2×C4—2442no—
403C2×C4—2442no—
404C2×C4—2442no—
405C2×C4—2441no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4—2441no—
409C2×C4#1451973911yescharacteristic
410C2×C4C4×D24411yescharacteristic
411C2×C4#1451975011yescharacteristic
412C2×C4#1451975211yescharacteristic
413C8—81no—
414C8—4881no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—4882no—
424C22—4882no—
425C22—4881no—
426C22—4881no—
427C22—4881no—
428C22#1683964011yescharacteristic
429C22#1683973412yes—
430C22#1683975711yescharacteristic
431C22#1683976511yescharacteristic
432C22#1683977712yes—
433C22#1683990811yescharacteristic
434C22#1683991011yescharacteristic
435C22#1684008811yescharacteristic
436C22#1684008911yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—82no—
442C4—82no—
443C4—4882no—
444C4—4881no—
445C4—2442no—
446C4—2442no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—9761no—
452C2#2144436412yes—
453C2#2144439512yes—
454C2#2144614011yescharacteristic
455C2#2144626611yescharacteristic
456C2#2144627611yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1294C2
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1294D61
hypocenterC61#9157
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
2. C2×#16839629C2semidirect
3. #21445885C2semidirect
6. #21443697C2semidirect
7. C61×#1294C2semidirect
12. #16839663C4semidirect
13. #16839661C4semidirect
225. #1294D61semidirect
255. C61#9157semidirect

Maximal subgroups

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

subgroupindex
#214541512
C2×#168396292
#214458852
C2×#168394972
C2×#168394952
#214436972
C61×#12942
#915761