Finite Group Databasebeta

← #30670233

Subgroups — #30670233

Order 16768.

672 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862203C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D131C211yesmaximal, characteristic
3#22553248C211yesmaximal, characteristic, complemented
4#22553247C211yesmaximal, characteristic, complemented
5#22553241C211yesmaximal, characteristic, complemented
6#22553240C211yesmaximal, characteristic, complemented
7#22553204C211yesmaximal, characteristic, complemented
8#22553203C211yesmaximal, characteristic, complemented
9#22553200C211yesmaximal, characteristic, complemented
10#22553186C211yesmaximal, characteristic, complemented
11#22553183C211yesmaximal, characteristic, complemented
12#22553174C211yesmaximal, characteristic, complemented
13#22553167C211yesmaximal, characteristic
14#22553167C211yesmaximal, characteristic
15#388×C131C211yesmaximal, characteristic, complemented
16C24×D131C2211yescharacteristic
17C22×C4×D131C2211yescharacteristic
18C22×C4×D131C2211yescharacteristic
19#16862203—24no—
20C2×C4⋊Dic131C2211yescharacteristic, complemented
21C2×Dic131⋊C4C2211yescharacteristic, complemented
22C2×Dic131⋊C4C2211yescharacteristic, complemented
23C2×Dic131⋊C4C2211yescharacteristic, complemented
24C2×Dic131⋊C4C2211yescharacteristic, complemented
25C2×C22⋊Dic131C2211yescharacteristic
26C2×C22⋊Dic131C2211yescharacteristic
27C2×C22⋊Dic131C2211yescharacteristic
28C2×(C2×D131)⋊C4C2211yescharacteristic
29C2×(C2×D131)⋊C4C2211yescharacteristic
30C2×(C2×D131)⋊C4C2211yescharacteristic
31C2×(C2×D131)⋊C4C2211yescharacteristic
32C2×(C2×D131)⋊C4C2211yescharacteristic
33C2×(C2×D131)⋊C4C2211yescharacteristic
34C2×(C2×D131)⋊C4C2211yescharacteristic
35C2×(C2×D131)⋊C4C2211yescharacteristic
36C2×(C2×D131)⋊C4C2211yescharacteristic
37C2×(C2×D131)⋊C4C2211yescharacteristic
38C22⋊C4×D131—24no—
39C2×C4×Dic131C2211yescharacteristic, complemented
40C131⋊#95C2211yescharacteristic, complemented
41C131⋊#95C2211yescharacteristic, complemented
42C131⋊#95C2211yescharacteristic, complemented
43C131⋊#95C2211yescharacteristic, complemented
44C131⋊#95C2211yescharacteristic, complemented
45C131⋊#95C2211yescharacteristic, complemented
46C2×C4⋊C4×C131C2211yescharacteristic, complemented
47C2×C22⋊C4×C131C2211yescharacteristic
48C2×C22⋊C4×C131C2211yescharacteristic
49C2×C22⋊C4×C131C2211yescharacteristic
50C2×C42×C131C2211yescharacteristic, complemented
51#95×C131C2211yescharacteristic, complemented
52#95×C131C2211yescharacteristic, complemented
53C23×D131—22no—
54C23×D131—22no—
55C23×D131—22no—
56C23×D131—22no—
57C23×D131—22no—
58C23×D131—41no—
59C23×D131—41no—
60C23×D131C2311yescharacteristic
61C23×D131C2311yescharacteristic
62C22×Dic131C2311yescharacteristic
63C22×Dic131C2311yescharacteristic
64C22×Dic131C2311yescharacteristic
65C22×Dic131C2311yescharacteristic
66C22×Dic131C2311yescharacteristic
67C22×Dic131C2311yescharacteristic
68C2×C4×D131—22no—
69C2×C4×D131—22no—
70C2×C4×D131—22no—
71C2×C4×D131—41no—
72C2×C4×D131—41no—
73C2×C4×D131D414yes—
74C4⋊Dic131—41no—
75Dic131⋊C4—22no—
76Dic131⋊C4—22no—
77Dic131⋊C4—41no—
78Dic131⋊C4—41no—
79C22⋊Dic131—22no—
80C22⋊Dic131—22no—
81C22⋊Dic131—41no—
82(C2×D131)⋊C4—22no—
83(C2×D131)⋊C4—22no—
84(C2×D131)⋊C4—22no—
85(C2×D131)⋊C4—22no—
86(C2×D131)⋊C4—22no—
87(C2×D131)⋊C4—22no—
88(C2×D131)⋊C4—22no—
89(C2×D131)⋊C4—22no—
90(C2×D131)⋊C4—41no—
91(C2×D131)⋊C4—41no—
92C4×Dic131—22no—
93C24×C131C2311yescharacteristic
94C22×C4×C131C2311yescharacteristic
95C22×C4×C131C2311yescharacteristic
96C22×C4×C131C2311yescharacteristic
97C22×C4×C131C2311yescharacteristic
98C22×C4×C131C2311yescharacteristic
99C22×C4×C131C2311yescharacteristic
100C4⋊C4×C131—41no—
101C22⋊C4×C131—22no—
102C22⋊C4×C131—22no—
103C22⋊C4×C131—41no—
104C42×C131—22no—
105C22×D131—21no—
106C22×D131—21no—
107C22×D131—21no—
108C22×D131—21no—
109C22×D131—28no—
110C22×D131—21no—
111C22×D131—21no—
112C22×D131—21no—
113C22×D131—21no—
114C22×D131—21no—
115C22×D131—21no—
116C22×D131—44no—
117C22×D131—44no—
118C22×D131—44no—
119C22×D131—44no—
120C22×D131—44no—
121C22×D131—44no—
122C22×D131C2×D412yes—
123C22×D131#4112yes—
124C22×D131#4112yes—
125C22×D131#4112yes—
126C2×Dic131—21no—
127C2×Dic131—21no—
128C2×Dic131—21no—
129C2×Dic131—21no—
130C2×Dic131—21no—
131C2×Dic131—21no—
132C2×Dic131—21no—
133C2×Dic131—21no—
134C2×Dic131—21no—
135C2×Dic131—21no—
136C2×Dic131—21no—
137C2×Dic131—21no—
138C2×Dic131—21no—
139C2×Dic131—21no—
140C2×Dic131—21no—
141C2×Dic131—21no—
142C2×Dic131—21no—
143C2×Dic131C2×D412yes—
144C4×D131—28no—
145C4×D131—44no—
146C23×C131—21no—
147C23×C131—21no—
148C23×C131—21no—
149C23×C131—21no—
150C23×C131—21no—
151C23×C131—21no—
152C23×C131#4112yes—
153C23×C131C2411yescharacteristic
154C2×C4×C131—21no—
155C2×C4×C131—21no—
156C2×C4×C131—21no—
157C2×C4×C131—21no—
158C2×C4×C131—21no—
159C2×C4×C131—21no—
160C2×C4×C131—21no—
161C2×C4×C131—21no—
162C2×C4×C131—21no—
163C2×C4×C131—21no—
164C2×C4×C131—21no—
165C2×C4×C131—21no—
166C2×C4×C131—21no—
167C2×C4×C131—21no—
168C2×C4×C131—21no—
169C2×C4×C131—21no—
170C2×C4×C131—21no—
171C2×C4×C131C2×D412yes—
172C22×C131—22no—
173C22×C131—22no—
174C22×C131—22no—
175C22×C131—41no—
176C22×C131—41no—
177C22×C131—41no—
178C22×C131—41no—
179C22×C131C22×D411yescharacteristic
180C22×C131C2×#4111yescharacteristic
181C22×C131C2×#4111yescharacteristic
182C22×C131C2×#4111yescharacteristic
183C22×C131C2×#4111yescharacteristic
184C22×C131H(5,𝔽2)11yescharacteristic
185C22×C131H(5,𝔽2)11yescharacteristic
186C2×D131—22no—
187C2×D131—22no—
188C2×D131—22no—
189C2×D131—22no—
190C2×D131—22no—
191C2×D131—22no—
192C2×D131—22no—
193C2×D131—22no—
194C2×D131—22no—
195C2×D131—416no—
196C2×D131—41no—
197C2×D131—41no—
198C2×D131—41no—
199C2×D131—41no—
200C2×D131#12314yes—
201C4×C131—22no—
202C4×C131—41no—
203C4×C131—41no—
204C4×C131—41no—
205C4×C131—41no—
206C4×C131—41no—
207Dic131—22no—
208Dic131—41no—
209Dic131—41no—
210Dic131—41no—
211Dic131—41no—
212Dic131—41no—
213C2×C131—42no—
214C2×C131C2×#12311yescharacteristic
215C2×C131#52511yescharacteristic
216C2×C131#53511yescharacteristic
217C2×C131#55111yescharacteristic
218C2×C131#55111yescharacteristic
219C2×C131#55111yescharacteristic
220C2×C131#55111yescharacteristic
221D131—24no—
222D131—42no—
223C131#248211yescharacteristic, complemented, Sylow
224#2482—1311nomaximal, Sylow
225C2×#117—1311no—
226C22×C22⋊C4—1311no—
227#399—1311no—
228#399—1311no—
229#397—1311no—
230#397—1311no—
231#388—1311no—
232#388—1311no—
233#388—1311no—
234#388D13111yescharacteristic, complemented
235#385—1311no—
236#385—1311no—
237#381—1311no—
238#379—1311no—
239#379—1311no—
240C25—1311no—
241C23×C4—1311no—
242C23×C4—1311no—
243#117—2624no—
244C2×C4⋊C4—1311no—
245C2×C4⋊C4—1311no—
246C2×C4⋊C4—1311no—
247C2×C4⋊C4—1311no—
248C2×C4⋊C4—1311no—
249C2×C4⋊C4C2×D13111yescharacteristic, complemented
250C2×C22⋊C4—2624no—
251C2×C22⋊C4—1311no—
252C2×C22⋊C4—1311no—
253C2×C22⋊C4—1311no—
254C2×C22⋊C4—1311no—
255C2×C22⋊C4—1311no—
256C2×C22⋊C4—1311no—
257C2×C22⋊C4—1311no—
258C2×C22⋊C4—1311no—
259C2×C22⋊C4—1311no—
260C2×C22⋊C4—1311no—
261C2×C22⋊C4—1311no—
262C2×C22⋊C4—1311no—
263C2×C22⋊C4—1311no—
264C2×C22⋊C4C2×D13111yescharacteristic
265C2×C22⋊C4C2×D13111yescharacteristic
266C2×C22⋊C4C2×D13111yescharacteristic
267C2×C42—1311no—
268C2×C42C2×D13111yescharacteristic, complemented
269#95—1311no—
270#95—1311no—
271#95—1311no—
272#95—1311no—
273#95—1311no—
274#95—1311no—
275#95C2×D13111yescharacteristic, complemented
276#95C2×D13111yescharacteristic, complemented
277C24—2622no—
278C24—2622no—
279C24—2622no—
280C24—2622no—
281C24—2622no—
282C24—5241no—
283C24—5241no—
284C24—1311no—
285C24—1311no—
286C24C22×D13111yescharacteristic
287C22×C4—2622no—
288C22×C4—2622no—
289C22×C4—2622no—
290C22×C4—5241no—
291C22×C4—5241no—
292C22×C4—1311no—
293C22×C4—1311no—
294C22×C4—1311no—
295C22×C4—1311no—
296C22×C4—1311no—
297C22×C4—1311no—
298C22×C4—1314no—
299C22×C4C22×D13111yescharacteristic
300C22×C4C22×D13111yescharacteristic
301C22×C4C22×D13111yescharacteristic
302C22×C4C22×D13111yescharacteristic
303C22×C4C22×D13111yescharacteristic
304C22×C4C22×D13111yescharacteristic
305C4⋊C4—41no—
306C4⋊C4—2622no—
307C4⋊C4—2622no—
308C4⋊C4—5241no—
309C4⋊C4—5241no—
310C4⋊C4—5241no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—41no—
314C22⋊C4—2622no—
315C22⋊C4—2622no—
316C22⋊C4—2622no—
317C22⋊C4—2622no—
318C22⋊C4—2622no—
319C22⋊C4—2622no—
320C22⋊C4—2622no—
321C22⋊C4—2622no—
322C22⋊C4—2622no—
323C22⋊C4—2622no—
324C22⋊C4—5241no—
325C22⋊C4—5241no—
326C22⋊C4—5241no—
327C42—22no—
328C42—2622no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—2628no—
336C23—2621no—
337C23—2621no—
338C23—2621no—
339C23—2621no—
340C23—2621no—
341C23—2621no—
342C23—2621no—
343C23—2621no—
344C23—2621no—
345C23—2621no—
346C23—5244no—
347C23—5244no—
348C23—5244no—
349C23—5244no—
350C23—5244no—
351C23—5244no—
352C23—1312no—
353C23—1312no—
354C23—1312no—
355C23—1312no—
356C23#1452351312yes—
357C23C23×D13111yescharacteristic
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—2628no—
376C2×C4—2621no—
377C2×C4—2621no—
378C2×C4—2621no—
379C2×C4—2621no—
380C2×C4—2621no—
381C2×C4—2621no—
382C2×C4—2621no—
383C2×C4—2621no—
384C2×C4—2621no—
385C2×C4—2621no—
386C2×C4—2621no—
387C2×C4—2621no—
388C2×C4—2621no—
389C2×C4—2621no—
390C2×C4—2621no—
391C2×C4—2621no—
392C2×C4—2621no—
393C2×C4—5244no—
394C2×C4—1312no—
395C2×C4D4×D13112yes—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—2622no—
404C22—2622no—
405C22—2622no—
406C22—2622no—
407C22—2622no—
408C22—2622no—
409C22—2622no—
410C22—2622no—
411C22—2622no—
412C22—52416no—
413C22—5241no—
414C22—5241no—
415C22—5241no—
416C22—5241no—
417C22—1314no—
418C22C2×D4×D13111yescharacteristic
419C22#41×D13111yescharacteristic
420C22#41×D13111yescharacteristic
421C22#41×D13111yescharacteristic
422C22C2×#1452351311yescharacteristic
423C22#1686229111yescharacteristic
424C22#1686229211yescharacteristic
425C4—22no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—2622no—
432C4—5241no—
433C4—5241no—
434C4—5241no—
435C4—5241no—
436C4—5241no—
437C2—42no—
438C2—2624no—
439C2—5242no—
440C2#123×D13111yescharacteristic
441C2#2255382111yescharacteristic
442C2#2255387811yescharacteristic
443C2#2255398611yescharacteristic
444C2#2255398711yescharacteristic
445C2#2255399111yescharacteristic
446C2#2255399211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#388×C131C2
prime radicalC24×D131C22
hypercenter#388D131
hypocenterC131#2482
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#16862203C2semidirect
3. #22553248C2semidirect
4. #22553247C2semidirect
5. #22553241C2semidirect
6. #22553240C2semidirect
7. #22553204C2semidirect
8. #22553203C2semidirect
9. #22553200C2semidirect
10. #22553186C2semidirect
11. #22553183C2semidirect
12. #22553174C2semidirect
15. #388×C131C2semidirect
20. C2×C4⋊Dic131C22semidirect
21. C2×Dic131⋊C4C22semidirect
22. C2×Dic131⋊C4C22semidirect
23. C2×Dic131⋊C4C22semidirect
24. C2×Dic131⋊C4C22semidirect
39. C2×C4×Dic131C22semidirect
40. C131⋊#95C22semidirect
41. C131⋊#95C22semidirect
42. C131⋊#95C22semidirect
43. C131⋊#95C22semidirect
44. C131⋊#95C22semidirect
45. C131⋊#95C22semidirect
46. C2×C4⋊C4×C131C22semidirect
50. C2×C42×C131C22semidirect
51. #95×C131C22semidirect
52. #95×C131C22semidirect
223. C131#2482semidirect
234. #388D131semidirect
249. C2×C4⋊C4C2×D131semidirect
268. C2×C42C2×D131semidirect
275. #95C2×D131semidirect
276. #95C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622032
C2×C22⋊C4×D1312
#225532482
#225532472
#225532412
#225532402
#225532042
#225532032
#225532002
#225531862
#225531832
#225531742
#225531672
#388×C1312
#2482131