Finite Group Databasebeta

← #11824495

Subgroups — #11824495

Order 1152.

1565 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64090C211yesmaximal, characteristic
2C2×#63982C212yesmaximal, complemented
3C2×#63919C211yesmaximal, characteristic, complemented
4C2×#63865C212yesmaximal, complemented
5C3×#4949C212yesmaximal, complemented
6#10591704C212yesmaximal
7#10591350C212yesmaximal, complemented
8#10591257C211yesmaximal, characteristic, complemented
9#10591244C212yesmaximal, complemented
10#78037—32nomaximal
11C22×#638C2211yescharacteristic
12C23×C3⋊Dic3C2211yescharacteristic
13C22×C4×C3⋊D3C2211yescharacteristic
14C22×#627C2212yes—
15C2×C32×C22⋊C4C2211yescharacteristic
16C2×#3752C2211yescharacteristic
17C2×#3750C2211yescharacteristic
18C2×C4×C3⋊Dic3C2211yescharacteristic
19#64090—24no—
20C2×C3×#217C2214yes—
21C2×C3×C4⋊Dic3C2212yes—
22C2×C3×Dic3⋊C4C2214yes—
23#63982—28no—
24C2×#3724C2212yes—
25C2×#3724C2212yes—
26C2×Dic3⋊Dic3C2214yes—
27C2×#3722C2214yes—
28C2×Dic32C2212yes—
29#63919—24no—
30#63865—28no—
31C3×C3⋊#95C2212yes—
32C2×#896—34no—
33C2×#888—32no—
34C2×#67⋊C4—32no—
35C2×#67⋊C4—61no—
36C2×#878—34no—
37C3×#399—32no—
38#4993—32no—
39#4975—34no—
40#4957—34no—
41#4949D312yescomplemented
42#4677—32no—
43#4649—32no—
44C24×C32C2311yescharacteristic
45C23×C3⋊D3C2311yescharacteristic
46C22×C32×C4C2311yescharacteristic
47C22×C32×C4C2311yescharacteristic
48C2×#638—22no—
49C2×#638—22no—
50C2×#638Q814yescomplemented
51C22×C3⋊Dic3—22no—
52C22×C3⋊Dic3—24no—
53C22×C3⋊Dic3C2311yescharacteristic
54C22×C3⋊Dic3C2311yescharacteristic
55C22×C3⋊Dic3C2311yescharacteristic
56C2×C4×C3⋊D3—22no—
57C2×C4×C3⋊D3—24no—
58C22×C3×Dic3C2314yes—
59C22×C3×Dic3C2314yes—
60C2×#627—24no—
61C2×#627—24no—
62C2×#627—24no—
63C32×C22⋊C4—22no—
64#3752—22no—
65#3750—22no—
66C4×C3⋊Dic3D414yes—
67C3×#217—28no—
68C3×C4⋊Dic3—24no—
69C3×Dic3⋊C4—28no—
70#3724—24no—
71#3724D418yescomplemented
72Dic3⋊Dic3—28no—
73#3722—28no—
74Dic32—24no—
75#2542—91noSylow
76C22×#67—32no—
77C22×#67—61no—
78C23×Dic3—32no—
79C23×Dic3—61no—
80C22×C4×D3—32no—
81C22×C4×D3—61no—
82C22×Dic6—34no—
83C2×C3×C4⋊C4—34no—
84C2×C3×C4⋊C4—32no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—34no—
87C2×C3×C22⋊C4C2×D312yes—
88C2×#217C2×D314yes—
89C2×(C2×D3)⋊C4—32no—
90C2×(C2×D3)⋊C4—34no—
91C2×(C2×D3)⋊C4—34no—
92C2×(C2×D3)⋊C4—61no—
93C2×C4⋊Dic3—34no—
94C2×C4⋊Dic3C2×D312yes—
95C2×Dic3⋊C4—34no—
96C2×Dic3⋊C4—32no—
97C2×Dic3⋊C4—34no—
98C2×Dic3⋊C4—34no—
99C2×Dic3⋊C4—61no—
100C2×Dic3⋊C4C2×D314yes—
101C2×C4×Dic3—32no—
102C2×C4×Dic3—34no—
103C2×C4×Dic3—61no—
104#896—616no—
105#888—68no—
106#67⋊C4—68no—
107#67⋊C4—68no—
108#878—616no—
109C3×#95—32no—
110C3⋊#95C2×D312yes—
111C23×C32—21no—
112C23×C32—21no—
113C23×C32—22no—
114C23×C32—22no—
115C23×C32C2×Q812yes—
116C23×C32C2411yescharacteristic
117C22×C3⋊D3—21no—
118C22×C3⋊D3—21no—
119C22×C3⋊D3—22no—
120C22×C3⋊D3—22no—
121C22×C3⋊D3C2×Q812yes—
122C2×C32×C4—21no—
123C2×C32×C4—22no—
124C2×C32×C4—22no—
125C2×C32×C4C2×D412yes—
126#638—28no—
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—22no—
129C2×C3⋊Dic3—44no—
130C2×C3⋊Dic3C2×D412yes—
131C2×C3⋊Dic3C2×D414yes—
132C2×C3⋊Dic3C2×D412yes—
133C2×C3⋊Dic3C2×Q812yes—
134C2×C3⋊Dic3#4112yes—
135C4×C3⋊D3—44no—
136C2×C3×Dic3—24no—
137C2×C3×Dic3—24no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3C2×D418yes—
142#627—48no—
143C2×C4⋊Q8—91no—
144C2×C22⋊Q8—92no—
145C2×C22⋊Q8—92no—
146C2×C4×D4—91no—
147#399—32no—
148#399—92no—
149#390—92no—
150#384—92no—
151C4×C4⋊C4—91no—
152C24×C3—21no—
153C24×C3C22×D312yes—
154C23×D3—32no—
155C23×D3—61no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—34no—
159C22×C3×C4—34no—
160C22×C3×C4C22×D312yes—
161C22×C3×C4C22×D312yes—
162C2×#67—38no—
163C2×#67—64no—
164C2×#67—64no—
165C2×#67—64no—
166C2×#67—64no—
167C2×#67—64no—
168C22×Dic3—32no—
169C22×Dic3—32no—
170C22×Dic3—32no—
171C22×Dic3—61no—
172C22×Dic3—61no—
173C22×Dic3—61no—
174C22×Dic3—68no—
175C22×Dic3—64no—
176C22×Dic3—64no—
177C22×Dic3—124no—
178C22×Dic3C22×D314yes—
179C22×Dic3C22×D314yes—
180C2×C4×D3—64no—
181C2×C4×D3—68no—
182C2×C4×D3—64no—
183C2×C4×D3—124no—
184C2×Dic6—68no—
185C2×Dic6—68no—
186C2×Dic6—68no—
187C3×C4⋊C4—68no—
188C3×C4⋊C4—64no—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—68no—
192#217—28no—
193(C2×D3)⋊C4—68no—
194(C2×D3)⋊C4—64no—
195(C2×D3)⋊C4—64no—
196(C2×D3)⋊C4—68no—
197C4⋊Dic3—24no—
198C4⋊Dic3—68no—
199Dic3⋊C4—28no—
200Dic3⋊C4—316no—
201Dic3⋊C4—68no—
202Dic3⋊C4—64no—
203Dic3⋊C4—64no—
204Dic3⋊C4—68no—
205C4×Dic3—38no—
206C4×Dic3—64no—
207C4×Dic3—68no—
208C22×C32—22no—
209C22×C32—24no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—42no—
213C22×C32C22×D411yescharacteristic
214C22×C32C22×D412yes—
215C22×C32C22×Q811yescharacteristic
216C22×C32C2×#4111yescharacteristic
217C22×C322−1+412yes—
218C2×C3⋊D3—22no—
219C2×C3⋊D3—24no—
220C2×C3⋊D3—41no—
221C2×C3⋊D3—41no—
222C2×C3⋊D3—42no—
223C32×C4—22no—
224C32×C4—41no—
225C3⋊Dic3—41no—
226C3⋊Dic3C22⋊Q814yescomplemented
227C3⋊Dic3#12714yes—
228C3×Dic3—28no—
229C3×Dic3—44no—
230C22×Q8—92no—
231C22×D4—91no—
232C23×C4—91no—
233C23×C4—91no—
234C4⋊Q8—184no—
235C22⋊Q8—188no—
236C22⋊Q8—188no—
237C4×D4—184no—
238C2×C4⋊C4—34no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—92no—
241C2×C4⋊C4—94no—
242C2×C4⋊C4—92no—
243C2×C4⋊C4—91no—
244C2×C22⋊C4—34no—
245C2×C22⋊C4—94no—
246C2×C22⋊C4—91no—
247C2×C22⋊C4D3211yescharacteristic
248C2×C42—91no—
249C2×C42—92no—
250#95—32no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—24no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—24no—
257C23×C3—21no—
258C23×C3—22no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3D3×Q814yes—
262C23×C3C23×D312yes—
263C22×D3—34no—
264C22×D3—62no—
265C22×D3—62no—
266C22×D3—62no—
267C22×D3—64no—
268C22×D3—62no—
269C22×D3—62no—
270C22×D3—64no—
271C22×D3—122no—
272C22×D3—122no—
273C2×C3×C4—24no—
274C2×C3×C4—22no—
275C2×C3×C4—24no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—38no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—64no—
282C2×C3×C4—64no—
283C2×C3×C4—64no—
284C2×C3×C4—64no—
285C2×C3×C4—64no—
286C2×C3×C4D3×D414yes—
287#67—616no—
288#67—616no—
289C2×Dic3—24no—
290C2×Dic3—24no—
291C2×Dic3—24no—
292C2×Dic3—24no—
293C2×Dic3—24no—
294C2×Dic3—34no—
295C2×Dic3—34no—
296C2×Dic3—38no—
297C2×Dic3—34no—
298C2×Dic3—34no—
299C2×Dic3—62no—
300C2×Dic3—62no—
301C2×Dic3—64no—
302C2×Dic3—62no—
303C2×Dic3—62no—
304C2×Dic3—62no—
305C2×Dic3—64no—
306C2×Dic3—62no—
307C2×Dic3—122no—
308C2×Dic3—128no—
309C2×Dic3—128no—
310C2×Dic3D3×D418yes—
311C4×D3—128no—
312C4×D3—128no—
313Dic6—1216no—
314C2×C32—42no—
315C2×C32C2×C22⋊Q811yescharacteristic
316C2×C32C2×#12711yescharacteristic
317C2×C32#53611yescharacteristic
318C2×C32#54712yes—
319C2×C32D4×Q812yes—
320C3⋊D3—42no—
321C24—91no—
322C24C2×D3211yescharacteristic
323C2×Q8—184no—
324C2×Q8—184no—
325C2×Q8—184no—
326C2×D4—94no—
327C2×D4—182no—
328C2×D4—182no—
329C22×C4—34no—
330C22×C4—34no—
331C22×C4—91no—
332C22×C4—91no—
333C22×C4—91no—
334C22×C4—182no—
335C22×C4—184no—
336C22×C4—182no—
337C22×C4—184no—
338C22×C4C2×D3211yescharacteristic
339C22×C4C2×D3211yescharacteristic
340C4⋊C4—68no—
341C4⋊C4—64no—
342C4⋊C4—98no—
343C4⋊C4—182no—
344C4⋊C4—188no—
345C4⋊C4—184no—
346C22⋊C4—22no—
347C22⋊C4—68no—
348C22⋊C4—188no—
349C22⋊C4—182no—
350C42—94no—
351C42—184no—
352C22×C3—24no—
353C22×C3—28no—
354C22×C3—24no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—21no—
360C22×C3—44no—
361C22×C3—42no—
362C22×C3—44no—
363C22×C3—42no—
364C22×C3—42no—
365C22×C3—44no—
366C22×C3—42no—
367C22×C3C2×D3×D414yes—
368C22×C3C2×D3×D412yes—
369C22×C3C2×D3×Q812yes—
370C22×C3D3×#4112yes—
371C22×C3#101014yes—
372C2×D3—64no—
373C2×D3—68no—
374C2×D3—64no—
375C2×D3—124no—
376C2×D3—122no—
377C2×D3—124no—
378C2×D3—122no—
379C2×D3—122no—
380C2×D3—124no—
381C2×D3—122no—
382C3×C4—24no—
383C3×C4—42no—
384C3×C4—42no—
385C3×C4—42no—
386C3×C4—68no—
387C3×C4—124no—
388Dic3—28no—
389Dic3—38no—
390Dic3—38no—
391Dic3—44no—
392Dic3—64no—
393Dic3—64no—
394Dic3—122no—
395Dic3—122no—
396C32#254211yescharacteristic, complemented, Sylow
397C23—21no—
398C23—22no—
399C23—22no—
400C23—21no—
401C23—92no—
402C23—181no—
403C23—182no—
404C23—182no—
405C23—181no—
406C23#379712yes—
407C23C22×D3211yescharacteristic
408Q8—368no—
409D4—188no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—38no—
414C2×C4—64no—
415C2×C4—64no—
416C2×C4—64no—
417C2×C4—64no—
418C2×C4—64no—
419C2×C4—92no—
420C2×C4—92no—
421C2×C4—94no—
422C2×C4—92no—
423C2×C4—92no—
424C2×C4—182no—
425C2×C4—181no—
426C2×C4—364no—
427C2×C4—364no—
428C2×C4#381112yes—
429C2×C3—22no—
430C2×C3—21no—
431C2×C3—22no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—44no—
435C2×C3—44no—
436C2×C3D3×C22⋊Q812yes—
437C2×C3#563214yes—
438C2×C3#564812yes—
439C2×C3#565514yes—
440C2×C3D3×#12712yes—
441D3—124no—
442D3—124no—
443C22—24no—
444C22—22no—
445C22—42no—
446C22—41no—
447C22—41no—
448C22—184no—
449C22—182no—
450C22—362no—
451C22—361no—
452C22—361no—
453C22C2×#379711yescharacteristic
454C22#6430611yescharacteristic
455C22#6430912yes—
456C22D32×D412yes—
457C22C2×#381111yescharacteristic
458C4—22no—
459C4—41no—
460C4—68no—
461C4—94no—
462C4—94no—
463C4—124no—
464C4—361no—
465C3—21no—
466C3#7803712yescomplemented
467C2—42no—
468C2—362no—
469C2#1059407212yes—
470C2#1059418512yes—
471C2#1059492511yescharacteristic
472C2#1059527111yescharacteristic
473C2#1059530011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutator, socle, cosocleC23×C32C24
FittingC2×C32×C22⋊C4C22
prime radicalC22×#638C22
hypercenterC2×C22⋊C4D32
hypocenterC32#2542
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#63982C2semidirect
3. C2×#63919C2semidirect
4. C2×#63865C2semidirect
5. C3×#4949C2semidirect
7. #10591350C2semidirect
8. #10591257C2semidirect
9. #10591244C2semidirect
41. #4949D3semidirect
50. C2×#638Q8semidirect
71. #3724D4semidirect
226. C3⋊Dic3C22⋊Q8semidirect
396. C32#2542semidirect
466. C3#78037semidirect

Maximal subgroups

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

subgroupindex
C2×#640902
C2×#639822
C2×#639192
C2×#638652
C3×#49492
#105917042
#105913502
#105912572
#105912442
#780373