Finite Group Databasebeta

← #16854038

Subgroups — #16854038

Order 3968.

688 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758471C211yesmaximal, characteristic, complemented
2C2×#11758460C211yesmaximal, characteristic, complemented
3C2×#11758461C211yesmaximal, characteristic
4C2×C42×D31C211yesmaximal, characteristic, complemented
5#14520949C211yesmaximal, characteristic, complemented
6#14520949C211yesmaximal, characteristic, complemented
7#14520946C211yesmaximal, characteristic, complemented
8#14520944C211yesmaximal, characteristic, complemented
9#14520941C211yesmaximal, characteristic, complemented
10#14520940C211yesmaximal, characteristic, complemented
11#14520937C211yesmaximal, characteristic, complemented
12#14520927C211yesmaximal, characteristic, complemented
13#14520925C211yesmaximal, characteristic, complemented
14#14520908C211yesmaximal, characteristic, complemented
15C31×#388C211yesmaximal, characteristic, complemented
16C22×C22⋊D31C2211yescharacteristic
17C22×C22⋊D31C2211yescharacteristic
18C22×D124C2211yescharacteristic
19C22×C4×D31C2211yescharacteristic
20C22×C4×D31C2211yescharacteristic
21C22×C4×D31C2211yescharacteristic
22C2×C4⋊C4×C31C2211yescharacteristic
23C2×C22⋊C4×C31C2211yescharacteristic
24C2×C22⋊C4×C31C2211yescharacteristic
25C2×C22⋊C4×C31C2211yescharacteristic
26C2×C42×C31C2211yescharacteristic
27C2×C22⋊Dic31C2211yescharacteristic
28C2×C22⋊Dic31C2211yescharacteristic
29C2×(C2×D31)⋊C4C2211yescharacteristic
30C2×(C2×D31)⋊C4C2211yescharacteristic
31C2×(C2×D31)⋊C4C2211yescharacteristic
32C2×(C2×D31)⋊C4C2211yescharacteristic
33C2×(C2×D31)⋊C4C2211yescharacteristic
34C2×(C2×D31)⋊C4C2211yescharacteristic
35C2×(C2×D31)⋊C4C2211yescharacteristic
36C2×C4⋊Dic31C2211yescharacteristic
37C2×Dic31⋊C4C2211yescharacteristic
38C2×Dic31⋊C4C2211yescharacteristic
39C2×Dic31⋊C4C2211yescharacteristic
40C2×Dic31⋊C4C2211yescharacteristic
41C2×Dic31⋊C4C2211yescharacteristic
42C2×C4×Dic31C2211yescharacteristic
43C2×C4×Dic31C2211yescharacteristic
44C2×C4×Dic31C2211yescharacteristic
45#11758471—24no—
46#11758461—24no—
47#11758460—24no—
48C42×D31—24no—
49C31×#95C2211yescharacteristic
50C31×#95C2211yescharacteristic
51C31⋊#95C2211yescharacteristic
52C31⋊#95C2211yescharacteristic
53C31⋊#95C2211yescharacteristic
54C31⋊#95C2211yescharacteristic
55C24×C31C2311yescharacteristic
56C23×D31C2311yescharacteristic
57C23×D31C2311yescharacteristic
58C22×C4×C31C2311yescharacteristic
59C22×C4×C31C2311yescharacteristic
60C22×C4×C31C2311yescharacteristic
61C22×C4×C31C2311yescharacteristic
62C22×C4×C31C2311yescharacteristic
63C22×C4×C31C2311yescharacteristic
64C2×C22⋊D31—22no—
65C2×C22⋊D31—22no—
66C2×C22⋊D31—22no—
67C2×C22⋊D31—22no—
68C2×C22⋊D31—22no—
69C2×C22⋊D31—22no—
70C22×Dic31C2311yescharacteristic
71C22×Dic31C2311yescharacteristic
72C22×Dic31C2311yescharacteristic
73C22×Dic31C2311yescharacteristic
74C22×Dic31C2311yescharacteristic
75C22×Dic31C2311yescharacteristic
76C2×D124—22no—
77C2×D124—22no—
78C2×D124—22no—
79C2×C4×D31—22no—
80C2×C4×D31—22no—
81C2×C4×D31—22no—
82C2×C4×D31—22no—
83C2×C4×D31—22no—
84C2×C4×D31—22no—
85C2×C4×D31—22no—
86C2×C4×D31D414yes—
87C2×C4×D31D414yescomplemented
88C4⋊C4×C31—22no—
89C22⋊C4×C31—22no—
90C22⋊C4×C31—22no—
91C22⋊C4×C31—22no—
92C42×C31—22no—
93C22⋊Dic31—22no—
94C22⋊Dic31—22no—
95(C2×D31)⋊C4—22no—
96(C2×D31)⋊C4—22no—
97(C2×D31)⋊C4—22no—
98(C2×D31)⋊C4—22no—
99(C2×D31)⋊C4—22no—
100(C2×D31)⋊C4—22no—
101(C2×D31)⋊C4—22no—
102C4⋊Dic31—22no—
103Dic31⋊C4—22no—
104Dic31⋊C4—22no—
105Dic31⋊C4—22no—
106Dic31⋊C4—22no—
107Dic31⋊C4—22no—
108C4×Dic31—22no—
109C4×Dic31—22no—
110C4×Dic31—22no—
111C23×C31—21no—
112C23×C31—21no—
113C23×C31—21no—
114C23×C31—21no—
115C23×C31—21no—
116C23×C31—21no—
117C23×C31—21no—
118C23×C31C2411yescharacteristic
119C22×D31—21no—
120C22×D31—21no—
121C22×D31—21no—
122C22×D31—21no—
123C22×D31—21no—
124C22×D31—21no—
125C22×D31—21no—
126C22×D31—21no—
127C22×D31—21no—
128C22×D31—21no—
129C22×D31—21no—
130C22×D31C2×D412yes—
131C22×D31C2×D412yes—
132C22×D31#4112yes—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—21no—
140C2×C4×C31—21no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—21no—
145C2×C4×C31—21no—
146C2×C4×C31—21no—
147C2×C4×C31—21no—
148C2×C4×C31C2×D412yes—
149C2×C4×C31C2×D412yes—
150C2×C4×C31#4112yes—
151C22⋊D31—44no—
152C22⋊D31—44no—
153C2×Dic31—21no—
154C2×Dic31—21no—
155C2×Dic31—21no—
156C2×Dic31—21no—
157C2×Dic31—21no—
158C2×Dic31—21no—
159C2×Dic31—21no—
160C2×Dic31—21no—
161C2×Dic31—21no—
162C2×Dic31—21no—
163C2×Dic31—21no—
164C2×Dic31—21no—
165C2×Dic31—21no—
166C2×Dic31C2×D412yes—
167C2×Dic31C2×D412yes—
168C2×Dic31#4112yes—
169C2×Dic31#4112yes—
170C2×Dic31#4112yes—
171D124—44no—
172C4×D31—28no—
173C4×D31—28no—
174C4×D31—28no—
175#2854—311nomaximal, Sylow
176C22×C31—41no—
177C22×C31—41no—
178C22×C31—41no—
179C22×C31—41no—
180C22×C31—41no—
181C22×C31—41no—
182C22×C31—41no—
183C22×C31C22×D411yescharacteristic
184C22×C31C22×D411yescharacteristic
185C22×C31C2×#4111yescharacteristic
186C22×C31C2×#4111yescharacteristic
187C22×C31C2×#4111yescharacteristic
188C22×C31C2×#4111yescharacteristic
189C22×C31C2×#4111yescharacteristic
190C2×D31—22no—
191C2×D31—22no—
192C2×D31—22no—
193C2×D31—22no—
194C2×D31—22no—
195C2×D31—22no—
196C2×D31—41no—
197C2×D31—41no—
198C2×D31—41no—
199C2×D31—41no—
200C2×D31—41no—
201C2×D31—41no—
202C2×D31—41no—
203C2×D31#12114yes—
204C4×C31—22no—
205C4×C31—22no—
206C4×C31—22no—
207C4×C31—41no—
208C4×C31—41no—
209C4×C31—41no—
210Dic31—22no—
211Dic31—22no—
212Dic31—41no—
213Dic31—41no—
214Dic31—41no—
215Dic31#12314yescomplemented
216C2×#123—311no—
217C2×#123—311no—
218C2×#121—311no—
219C22×C42—311no—
220#390—311no—
221#390—311no—
222#389—311no—
223#389—311no—
224#388—311no—
225#388—311no—
226#388—311no—
227#388D3111yescharacteristic, complemented
228#386—311no—
229#386—311no—
230#380—311no—
231C2×C31—81no—
232C2×C31C2×#12111yescharacteristic
233C2×C31C2×#12311yescharacteristic
234C2×C31#52511yescharacteristic
235C2×C31#52911yescharacteristic
236C2×C31#52911yescharacteristic
237C2×C31#53211yescharacteristic
238C2×C31#53211yescharacteristic
239D31—24no—
240D31—81no—
241C22×D4—311no—
242C22×D4—311no—
243C22×D4—311no—
244C23×C4—311no—
245C23×C4—311no—
246C23×C4—311no—
247#123—624no—
248#123—624no—
249#121—624no—
250C2×C4⋊C4—311no—
251C2×C4⋊C4—311no—
252C2×C4⋊C4—311no—
253C2×C4⋊C4—311no—
254C2×C4⋊C4—311no—
255C2×C4⋊C4—311no—
256C2×C4⋊C4C2×D3111yescharacteristic
257C2×C22⋊C4—311no—
258C2×C22⋊C4—311no—
259C2×C22⋊C4—311no—
260C2×C22⋊C4—311no—
261C2×C22⋊C4—311no—
262C2×C22⋊C4—311no—
263C2×C22⋊C4—311no—
264C2×C22⋊C4—311no—
265C2×C22⋊C4—311no—
266C2×C22⋊C4C2×D3111yescharacteristic
267C2×C22⋊C4C2×D3111yescharacteristic
268C2×C22⋊C4C2×D3111yescharacteristic
269C2×C42—311no—
270C2×C42—311no—
271C2×C42—311no—
272C2×C42—624no—
273C2×C42C2×D3111yescharacteristic
274#95—311no—
275#95—311no—
276#95—311no—
277#95—311no—
278#95C2×D3111yescharacteristic
279#95C2×D3111yescharacteristic
280C31#285411yescharacteristic, complemented, Sylow
281C24—311no—
282C24—311no—
283C24C22×D3111yescharacteristic
284C2×D4—622no—
285C2×D4—622no—
286C2×D4—622no—
287C2×D4—622no—
288C2×D4—622no—
289C2×D4—622no—
290C2×D4—622no—
291C2×D4—622no—
292C2×D4—622no—
293C22×C4—311no—
294C22×C4—311no—
295C22×C4—311no—
296C22×C4—314no—
297C22×C4—314no—
298C22×C4—311no—
299C22×C4—311no—
300C22×C4—311no—
301C22×C4—622no—
302C22×C4—622no—
303C22×C4—622no—
304C22×C4—622no—
305C22×C4—622no—
306C22×C4—622no—
307C22×C4—622no—
308C22×C4C22×D3111yescharacteristic
309C22×C4C22×D3111yescharacteristic
310C22×C4C22×D3111yescharacteristic
311C22×C4C22×D3111yescharacteristic
312C22×C4C22×D3111yescharacteristic
313C22×C4C22×D3111yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—622no—
316C4⋊C4—622no—
317C4⋊C4—622no—
318C4⋊C4—622no—
319C4⋊C4—622no—
320C4⋊C4—622no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—622no—
325C22⋊C4—622no—
326C22⋊C4—622no—
327C22⋊C4—622no—
328C22⋊C4—622no—
329C22⋊C4—622no—
330C22⋊C4—622no—
331C22⋊C4—622no—
332C22⋊C4—622no—
333C42—22no—
334C42—622no—
335C42—622no—
336C42—622no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—312no—
345C23—312no—
346C23—312no—
347C23—621no—
348C23—621no—
349C23—621no—
350C23—621no—
351C23—621no—
352C23—621no—
353C23—621no—
354C23—621no—
355C23—621no—
356C23—621no—
357C23—621no—
358C23C23×D3111yescharacteristic
359D4—1244no—
360D4—1244no—
361D4—1244no—
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—21no—
376C2×C4—21no—
377C2×C4—312no—
378C2×C4—312no—
379C2×C4—312no—
380C2×C4—312no—
381C2×C4—312no—
382C2×C4—628no—
383C2×C4—628no—
384C2×C4—628no—
385C2×C4—621no—
386C2×C4—621no—
387C2×C4—621no—
388C2×C4—621no—
389C2×C4—621no—
390C2×C4—621no—
391C2×C4—621no—
392C2×C4—621no—
393C2×C4—621no—
394C2×C4—621no—
395C2×C4—621no—
396C2×C4—621no—
397C2×C4—621no—
398C2×C4#9249512yes—
399C2×C4D4×D3112yes—
400C2×C4D4×D3112yes—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—314no—
409C22—622no—
410C22—622no—
411C22—622no—
412C22—622no—
413C22—622no—
414C22—622no—
415C22—1241no—
416C22—1241no—
417C22—1241no—
418C22—1241no—
419C22—1241no—
420C22—1241no—
421C22—1241no—
422C22C2×#9249511yescharacteristic
423C22C2×D4×D3111yescharacteristic
424C22C2×D4×D3111yescharacteristic
425C22#41×D3111yescharacteristic
426C22#41×D3111yescharacteristic
427C22#41×D3111yescharacteristic
428C22#41×D3111yescharacteristic
429C4—22no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—314no—
436C4—622no—
437C4—622no—
438C4—1241no—
439C4—1241no—
440C4—1241no—
441C2—81no—
442C2—624no—
443C2—2481no—
444C2#121×D3111yescharacteristic
445C2#123×D3111yescharacteristic
446C2#1452155811yescharacteristic
447C2#1452157711yescharacteristic
448C2#1452158211yescharacteristic
449C2#1452159611yescharacteristic
450C2#1452159811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC31×#388C2
prime radicalC2×#11758461C2
hypercenter#388D31
hypocenterC31#2854
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#11758471C2semidirect
2. C2×#11758460C2semidirect
4. C2×C42×D31C2semidirect
5. #14520949C2semidirect
6. #14520949C2semidirect
7. #14520946C2semidirect
8. #14520944C2semidirect
9. #14520941C2semidirect
10. #14520940C2semidirect
11. #14520937C2semidirect
12. #14520927C2semidirect
13. #14520925C2semidirect
14. #14520908C2semidirect
15. C31×#388C2semidirect
87. C2×C4×D31D4semidirect
215. Dic31#123semidirect
227. #388D31semidirect
280. C31#2854semidirect

Maximal subgroups

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

subgroupindex
C2×#117584712
C2×#117584602
C2×#117584612
C2×C42×D312
#145209492
#145209462
#145209442
#145209412
#145209402
#145209372
#145209272
#145209252
#145209082
C31×#3882
#285431