Finite Group Databasebeta

← #11824467

Subgroups — #11824467

Order 1152.

2028 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64105C212yesmaximal, complemented
2C2×#64097C211yesmaximal, characteristic, complemented
3C2×#64090C211yesmaximal, characteristic
4C2×#64087C212yesmaximal, complemented
5C32×#399C211yesmaximal, characteristic, complemented
6#10592692C211yesmaximal, characteristic, complemented
7#10592674C212yesmaximal, complemented
8#10592656C212yesmaximal
9#10592648C211yesmaximal, characteristic, complemented
10#10592376C211yesmaximal, characteristic, complemented
11#10592348C211yesmaximal, characteristic, complemented
12#78037—34nomaximal
13C22×#638C2211yescharacteristic
14C23×C3⋊Dic3C2211yescharacteristic
15C22×C4×C3⋊D3C2211yescharacteristic
16C22×#634C2212yes—
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C32×C4⋊C4C2212yes—
19C2×C32×C22⋊C4C2212yes—
20C2×C32×C22⋊C4C2211yescharacteristic
21C2×#3757C2212yes—
22C2×#3752C2211yescharacteristic
23C2×#3752C2212yes—
24C2×#3752C2212yes—
25C2×C4⋊(C3⋊Dic3)C2212yes—
26C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
27C2×#3750C2212yes—
28C2×#3750C2211yescharacteristic
29C2×#3750C2212yes—
30C2×#3750C2212yes—
31C2×#3750C2212yes—
32C2×C4×C3⋊Dic3C2212yes—
33C2×C4×C3⋊Dic3C2211yescharacteristic
34#64105—28no—
35#64097—24no—
36#64090—24no—
37#64087—28no—
38C32×#95C2211yescharacteristic
39#63658C2211yescharacteristic
40C2×#896—38no—
41C2×#888—34no—
42C2×#67⋊C4—34no—
43C2×#878—38no—
44C3×#399D314yescomplemented
45#4993—34no—
46#4975—38no—
47#4957—38no—
48#4949—34no—
49#4677—34no—
50#4649—34no—
51C24×C32C2311yescharacteristic
52C23×C3⋊D3C2311yescharacteristic
53C22×C32×C4C2312yes—
54C22×C32×C4C2311yescharacteristic
55C22×C32×C4C2312yes—
56C22×C32×C4C2311yescharacteristic
57C2×#638—22no—
58C2×#638—22no—
59C2×#638Q814yescomplemented
60C22×C3⋊Dic3—24no—
61C22×C3⋊Dic3—22no—
62C22×C3⋊Dic3C2312yes—
63C22×C3⋊Dic3C2311yescharacteristic
64C22×C3⋊Dic3C2312yes—
65C22×C3⋊Dic3C2311yescharacteristic
66C22×C3⋊Dic3C2311yescharacteristic
67C2×C4×C3⋊D3—22no—
68C2×C4×C3⋊D3—24no—
69C2×#634—24no—
70C2×#634—24no—
71C2×#634—24no—
72C32×C4⋊C4—22no—
73C32×C4⋊C4—24no—
74C32×C22⋊C4—24no—
75C32×C22⋊C4—22no—
76#3757—24no—
77#3752—22no—
78#3752—24no—
79#3752—24no—
80C4⋊(C3⋊Dic3)—24no—
81C4⋊(C3⋊Dic3)—22no—
82#3750—24no—
83#3750—22no—
84#3750—24no—
85#3750—24no—
86#3750D418yescomplemented
87C4×C3⋊Dic3—24no—
88C4×C3⋊Dic3D414yes—
89#2542—91noSylow
90C22×#67—34no—
91C23×Dic3—34no—
92C22×C4×D3—34no—
93C22×Dic6—38no—
94C2×C3×C4⋊C4C2×D318yes—
95C2×C3×C4⋊C4C2×D314yes—
96C2×C3×C22⋊C4C2×D314yes—
97C2×C3×C22⋊C4C2×D318yes—
98C2×#217—38no—
99C2×(C2×D3)⋊C4—38no—
100C2×(C2×D3)⋊C4—34no—
101C2×(C2×D3)⋊C4—38no—
102C2×C4⋊Dic3—34no—
103C2×C4⋊Dic3—38no—
104C2×Dic3⋊C4—38no—
105C2×Dic3⋊C4—38no—
106C2×Dic3⋊C4—38no—
107C2×Dic3⋊C4—38no—
108C2×Dic3⋊C4—34no—
109C2×C4×Dic3—34no—
110C2×C4×Dic3—38no—
111#896—632no—
112#888—616no—
113#67⋊C4—616no—
114#878—632no—
115C3×#95C2×D314yes—
116C3⋊#95—34no—
117C23×C32—22no—
118C23×C32—21no—
119C23×C32—21no—
120C23×C32—22no—
121C23×C32C2×Q812yes—
122C23×C32C2411yescharacteristic
123C22×C3⋊D3—22no—
124C22×C3⋊D3—21no—
125C22×C3⋊D3—21no—
126C22×C3⋊D3—22no—
127C22×C3⋊D3C2×Q812yes—
128C2×C32×C4—22no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—21no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—22no—
136C2×C32×C4C2×D412yes—
137C2×C32×C4C2×D414yes—
138#638—28no—
139C2×C3⋊Dic3—22no—
140C2×C3⋊Dic3—22no—
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3—22no—
143C2×C3⋊Dic3—22no—
144C2×C3⋊Dic3—21no—
145C2×C3⋊Dic3—22no—
146C2×C3⋊Dic3—44no—
147C2×C3⋊Dic3C2×D414yes—
148C2×C3⋊Dic3C2×D412yes—
149C2×C3⋊Dic3C2×D412yes—
150C2×C3⋊Dic3C2×D414yes—
151C2×C3⋊Dic3C2×Q812yes—
152C2×C3⋊Dic3#4112yes—
153C4×C3⋊D3—44no—
154#634—48no—
155C2×C4⋊Q8—91no—
156C2×C22⋊Q8—92no—
157C2×C22⋊Q8—92no—
158C2×C4×D4—91no—
159#399—91no—
160#399—91no—
161#399—91no—
162#399C3⋊D311yescharacteristic, complemented
163#390—92no—
164#384—92no—
165C4×C4⋊C4—91no—
166C24×C3C22×D314yes—
167C23×D3—34no—
168C22×C3×C4C22×D314yes—
169C22×C3×C4C22×D318yes—
170C22×C3×C4C22×D314yes—
171C22×C3×C4C22×D318yes—
172C2×#67—316no—
173C2×#67—68no—
174C2×#67—68no—
175C22×Dic3—34no—
176C22×Dic3—34no—
177C22×Dic3—38no—
178C22×Dic3—34no—
179C22×Dic3—38no—
180C22×Dic3—616no—
181C22×Dic3—68no—
182C2×C4×D3—68no—
183C2×C4×D3—616no—
184C2×Dic6—616no—
185C2×Dic6—616no—
186C2×Dic6—616no—
187C3×C4⋊C4—28no—
188C3×C4⋊C4—216no—
189C3×C22⋊C4—216no—
190C3×C22⋊C4—28no—
191#217—616no—
192(C2×D3)⋊C4—616no—
193(C2×D3)⋊C4—68no—
194(C2×D3)⋊C4—616no—
195C4⋊Dic3—616no—
196C4⋊Dic3—68no—
197Dic3⋊C4—332no—
198Dic3⋊C4—616no—
199Dic3⋊C4—68no—
200Dic3⋊C4—616no—
201Dic3⋊C4—616no—
202C4×Dic3—316no—
203C4×Dic3—616no—
204C22×C32—22no—
205C22×C32—24no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32—42no—
209C22×C32C22×D411yescharacteristic
210C22×C32C22×D412yes—
211C22×C32C22×Q811yescharacteristic
212C22×C32C2×#4111yescharacteristic
213C22×C322−1+412yes—
214C2×C3⋊D3—22no—
215C2×C3⋊D3—24no—
216C2×C3⋊D3—41no—
217C2×C3⋊D3—41no—
218C2×C3⋊D3—42no—
219C32×C4—22no—
220C32×C4—24no—
221C32×C4—42no—
222C32×C4—41no—
223C3⋊Dic3—24no—
224C3⋊Dic3—42no—
225C3⋊Dic3—41no—
226C3⋊Dic3C22⋊Q814yescomplemented
227C3⋊Dic3#12714yes—
228C22×Q8—92no—
229C22×D4—91no—
230C23×C4—91no—
231C23×C4—91no—
232C4⋊Q8—184no—
233C22⋊Q8—188no—
234C22⋊Q8—188no—
235C4×D4—184no—
236C2×C4⋊C4—92no—
237C2×C4⋊C4—91no—
238C2×C4⋊C4—92no—
239C2×C4⋊C4—92no—
240C2×C4⋊C4—92no—
241C2×C4⋊C4—92no—
242C2×C4⋊C4—91no—
243C2×C4⋊C4C2×C3⋊D312yes—
244C2×C4⋊C4C2×C3⋊D311yescharacteristic
245C2×C22⋊C4—92no—
246C2×C22⋊C4—91no—
247C2×C22⋊C4—92no—
248C2×C22⋊C4—92no—
249C2×C22⋊C4C2×C3⋊D311yescharacteristic
250C2×C22⋊C4C2×C3⋊D312yes—
251C2×C42—91no—
252C2×C42—92no—
253#95—91no—
254#95C2×C3⋊D311yescharacteristic
255C23×C3—28no—
256C23×C3—24no—
257C23×C3—24no—
258C23×C3—28no—
259C23×C3D3×Q818yes—
260C23×C3C23×D314yes—
261C22×D3—38no—
262C22×D3—68no—
263C22×D3—64no—
264C22×D3—64no—
265C22×D3—68no—
266C2×C3×C4—28no—
267C2×C3×C4—28no—
268C2×C3×C4—28no—
269C2×C3×C4—28no—
270C2×C3×C4—24no—
271C2×C3×C4—28no—
272C2×C3×C4—28no—
273C2×C3×C4—28no—
274C2×C3×C4D3×D418yes—
275C2×C3×C4D3×D4116yes—
276#67—632no—
277C2×Dic3—38no—
278C2×Dic3—38no—
279C2×Dic3—316no—
280C2×Dic3—38no—
281C2×Dic3—38no—
282C2×Dic3—316no—
283C2×Dic3—68no—
284C2×Dic3—68no—
285C2×Dic3—68no—
286C2×Dic3—68no—
287C2×Dic3—68no—
288C2×Dic3—64no—
289C2×Dic3—68no—
290C2×Dic3—1216no—
291C4×D3—1216no—
292Dic6—1232no—
293C2×C32—42no—
294C2×C32C2×C22⋊Q811yescharacteristic
295C2×C32C2×#12711yescharacteristic
296C2×C32#53611yescharacteristic
297C2×C32#54712yes—
298C2×C32D4×Q812yes—
299C3⋊D3—42no—
300C24—91no—
301C24C22×C3⋊D311yescharacteristic
302C2×Q8—184no—
303C2×Q8—184no—
304C2×Q8—184no—
305C2×D4—94no—
306C2×D4—182no—
307C2×D4—182no—
308C22×C4—91no—
309C22×C4—91no—
310C22×C4—92no—
311C22×C4—92no—
312C22×C4—91no—
313C22×C4—182no—
314C22×C4—184no—
315C22×C4—182no—
316C22×C4—184no—
317C22×C4C22×C3⋊D312yes—
318C22×C4C22×C3⋊D311yescharacteristic
319C22×C4C22×C3⋊D311yescharacteristic
320C22×C4C22×C3⋊D312yes—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—98no—
324C4⋊C4—182no—
325C4⋊C4—184no—
326C4⋊C4—184no—
327C4⋊C4—184no—
328C4⋊C4—184no—
329C4⋊C4—182no—
330C22⋊C4—22no—
331C22⋊C4—24no—
332C22⋊C4—184no—
333C22⋊C4—184no—
334C22⋊C4—182no—
335C22⋊C4—184no—
336C42—94no—
337C42—184no—
338C22×C3—216no—
339C22×C3—28no—
340C22×C3—44no—
341C22×C3—48no—
342C22×C3—44no—
343C22×C3C2×D3×D418yes—
344C22×C3C2×D3×D414yes—
345C22×C3C2×D3×Q814yes—
346C22×C3D3×#4114yes—
347C22×C3#101018yes—
348C2×D3—616no—
349C2×D3—68no—
350C2×D3—124no—
351C2×D3—128no—
352C2×D3—124no—
353C3×C4—28no—
354C3×C4—216no—
355C3×C4—48no—
356C3×C4—44no—
357Dic3—316no—
358Dic3—316no—
359Dic3—616no—
360Dic3—128no—
361Dic3—124no—
362C32#254211yescharacteristic, complemented, Sylow
363C23—22no—
364C23—21no—
365C23—21no—
366C23—22no—
367C23—92no—
368C23—182no—
369C23—181no—
370C23—181no—
371C23—182no—
372C23Q8×C3⋊D312yes—
373C23C23×C3⋊D311yescharacteristic
374Q8—368no—
375D4—188no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—92no—
385C2×C4—92no—
386C2×C4—94no—
387C2×C4—92no—
388C2×C4—92no—
389C2×C4—94no—
390C2×C4—182no—
391C2×C4—182no—
392C2×C4—181no—
393C2×C4—182no—
394C2×C4—182no—
395C2×C4—182no—
396C2×C4—182no—
397C2×C4—364no—
398C2×C4—364no—
399C2×C4D4×C3⋊D312yes—
400C2×C4D4×C3⋊D314yes—
401C2×C3—48no—
402C2×C3D3×C22⋊Q814yes—
403C2×C3#563218yes—
404C2×C3#564814yes—
405C2×C3#565518yes—
406C2×C3D3×#12714yes—
407D3—128no—
408C22—22no—
409C22—24no—
410C22—41no—
411C22—42no—
412C22—41no—
413C22—182no—
414C22—184no—
415C22—361no—
416C22—362no—
417C22—361no—
418C22C2×D4×C3⋊D311yescharacteristic
419C22C2×D4×C3⋊D312yes—
420C22C2×Q8×C3⋊D311yescharacteristic
421C22#41×C3⋊D311yescharacteristic
422C22#6436712yes—
423C4—22no—
424C4—24no—
425C4—41no—
426C4—42no—
427C4—94no—
428C4—94no—
429C4—184no—
430C4—361no—
431C4—362no—
432C3#7803714yescomplemented
433C2—42no—
434C2—362no—
435C2C3⋊D3×C22⋊Q811yescharacteristic
436C2#1059610312yes—
437C2#1059611911yescharacteristic
438C2#1059612612yes—
439C2C3⋊D3×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×C3⋊D3
commutator, socle, cosocleC23×C32C24
FittingC32×#399C2
prime radicalC22×#638C22
hypercenter#399C3⋊D3
hypocenterC32#2542
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#64105C2semidirect
2. C2×#64097C2semidirect
4. C2×#64087C2semidirect
5. C32×#399C2semidirect
6. #10592692C2semidirect
7. #10592674C2semidirect
9. #10592648C2semidirect
10. #10592376C2semidirect
11. #10592348C2semidirect
44. C3×#399D3semidirect
59. C2×#638Q8semidirect
86. #3750D4semidirect
162. #399C3⋊D3semidirect
226. C3⋊Dic3C22⋊Q8semidirect
362. C32#2542semidirect
432. C3#78037semidirect

Maximal subgroups

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

subgroupindex
C2×#641052
C2×#640972
C2×#640902
C2×#640872
C32×#3992
#105926922
#105926742
#105926562
#105926482
#105923762
#105923482
#780373