Finite Group Databasebeta

← #10844719

Subgroups — #10844719

Order 768.

886 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4848C211yesmaximal, characteristic
2C2×#4830C211yesmaximal, characteristic, complemented
3#74343C211yesmaximal, characteristic, complemented
4#73712C211yesmaximal, characteristic
5C3×#1948C211yesmaximal, characteristic, complemented
6#70585C212yesmaximal, complemented
7#70574C212yesmaximal, complemented
8#69580C211yesmaximal, characteristic
9C4×#810C211yesmaximal, characteristic, complemented
10#68863C212yesmaximal, complemented
11#68853C212yesmaximal, complemented
12#25295—31nomaximal, Sylow
13C22×#204C2212yes—
14C2×#875—21no—
15C2×#874C2211yescharacteristic
16C2×C4×Dic6C2211yescharacteristic
17C2×C4×Dic6C2211yescharacteristic
18C2×C3×#106C2212yes—
19C2×C3×C4⋊C8C2211yescharacteristic
20C3×#384C2211yescharacteristic
21C3×C4×C4⋊C4C2211yescharacteristic
22C2×#821C2212yes—
23C2×Dic3⋊C8C2212yes—
24C2×C8×Dic3C2212yes—
25#4988C2212yes—
26#4988C2212yes—
27#4984—21no—
28#4984C2212yes—
29#4981—22no—
30#4973—21no—
31#4973C2212yes—
32C2×#810C2211yescharacteristic
33C2×#810C2211yescharacteristic
34C2×#809C2212yes—
35C2×#809C2212yes—
36#4939—21no—
37C2×C4×C3⋊C8C2211yescharacteristic
38#4848—24no—
39#4830—24no—
40#4672—22no—
41C3×#340C2212yes—
42#4529C2211yescharacteristic
43#4529C2211yescharacteristic
44C2×#460—31no—
45C2×C8×Q8—31no—
46#2582—31no—
47#2548—31no—
48#2065—32no—
49#2055—32no—
50#1968—32no—
51#1957—31no—
52#1948D311yescharacteristic, complemented
53#1934—32no—
54C4×#103—31no—
55C22×D12—21no—
56C22×D12—21no—
57C22×D12C2312yes—
58C22×Dic6D411yescharacteristic
59C22×Dic6C2311yescharacteristic
60C22×C3×C8C2312yes—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C4⋊C4C2311yescharacteristic
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C42—21no—
67C2×C3×C42C2311yescharacteristic
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×C4⋊Dic3—22no—
73C2×C4⋊Dic3D411yescharacteristic
74C2×C4⋊Dic3C2311yescharacteristic
75C2×Dic3⋊C4—21no—
76C2×Dic3⋊C4D412yes—
77C2×C4×Dic3—21no—
78C2×C4×Dic3C2312yes—
79C22×C3⋊C8C2311yescharacteristic
80C22×C3⋊C8C2311yescharacteristic
81C2×#204—24no—
82C2×#204—24no—
83C2×#204—42no—
84#875—44no—
85#874—22no—
86#874—22no—
87C4×Dic6—22no—
88C4×Dic6—22no—
89C4×Dic6—22no—
90C4×Dic6—22no—
91C3×#106—42no—
92C3×C4⋊C8D414yes—
93C3×#95—21no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97#821—24no—
98Dic3⋊C8—24no—
99C8×Dic3—24no—
100#810—22no—
101#810—22no—
102#809—24no—
103#809—42no—
104C4×C3⋊C8—22no—
105C22×SD16—32no—
106C2×#127—31no—
107C2×#124—61no—
108C2×C4×Q8—31no—
109C2×C4×Q8—31no—
110#460—64no—
111C8×Q8—64no—
112C2×#106C2×D312yes—
113C2×C4⋊C8—32no—
114C2×C4⋊C8C2×D311yescharacteristic
115C2×#103—31no—
116C2×#103—31no—
117C2×#102—32no—
118C2×#102—32no—
119C2×#102—32no—
120C2×C4×C8—32no—
121C2×C4×C8—31no—
122#398—62no—
123#394—62no—
124#391—32no—
125#391—61no—
126#390—32no—
127#390—32no—
128#390—61no—
129#389—32no—
130#389—61no—
131#384C2×D311yescharacteristic
132C4×C4⋊C4C2×D311yescharacteristic
133#340—31no—
134#340—31no—
135#340C2×D312yes—
136C23×D3—22no—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2411yescharacteristic
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3C2×D412yes—
146C22×Dic3C2×D412yes—
147C2×D12—22no—
148C2×D12—22no—
149C2×D12—22no—
150C2×D12—22no—
151C2×D12—22no—
152C2×D12—22no—
153C2×D12—42no—
154C2×D12—42no—
155C2×D12—42no—
156C2×D12—42no—
157C2×D12—81no—
158C2×D12—81no—
159C2×Dic6—22no—
160C2×Dic6—22no—
161C2×Dic6—21no—
162C2×Dic6—21no—
163C2×Dic6—21no—
164C2×Dic6—21no—
165C2×Dic6—21no—
166C2×Dic6SD1614yescomplemented
167C2×Dic6#4112yes—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8C2×D414yes—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4—81no—
179C3×C42—42no—
180C3×C42C2×D412yes—
181C3×C42#4112yes—
182(C2×D3)⋊C4—44no—
183(C2×D3)⋊C4—44no—
184(C2×D3)⋊C4—82no—
185(C2×D3)⋊C4—82no—
186C4⋊Dic3—22no—
187C4⋊Dic3—21no—
188C4⋊Dic3—21no—
189C4⋊Dic3—82no—
190Dic3⋊C4—24no—
191Dic3⋊C4—42no—
192C4×Dic3—22no—
193C4×Dic3—22no—
194C4×Dic3—42no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—21no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8#4112yes—
201#204—48no—
202C22×Q8—31no—
203C22×Q8—31no—
204C22×D4—32no—
205C22×D4—61no—
206C22×D4—61no—
207C2×SD16—64no—
208C2×SD16—64no—
209C2×SD16—122no—
210C22×C8—31no—
211C22×C8—31no—
212C22×C8C22×D312yes—
213#127—62no—
214#127—62no—
215#124—124no—
216C4×Q8—62no—
217C4×Q8—62no—
218C4×Q8—62no—
219C4×Q8—62no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—31no—
223C2×C4⋊C4—32no—
224C2×C4⋊C4—31no—
225C2×C4⋊C4—61no—
226C2×C4⋊C4—62no—
227C2×C4⋊C4C22×D311yescharacteristic
228C2×C4⋊C4C22×D311yescharacteristic
229C2×C4⋊C4C22×D312yes—
230C2×C22⋊C4—62no—
231C2×C22⋊C4—62no—
232C2×C22⋊C4—62no—
233C2×C22⋊C4—62no—
234C2×C42—21no—
235C2×C42—32no—
236C2×C42—61no—
237C2×C42C22×D311yescharacteristic
238#106—42no—
239C4⋊C8—64no—
240C4⋊C8D1214yes—
241#103—62no—
242#103—62no—
243#102—64no—
244#102—64no—
245#102—122no—
246C4×C8—64no—
247C4×C8—62no—
248#95—21no—
249#95—62no—
250#95—62no—
251#95—62no—
252C23×C3C22×D411yescharacteristic
253C22×D3—42no—
254C22×D3—42no—
255C22×D3—42no—
256C22×D3—42no—
257C22×D3—42no—
258C22×D3—42no—
259C22×D3—42no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4#12112yes—
272C2×C3×C4#12112yes—
273C2×C3×C4C2×SD1612yes—
274C2×C3×C4#13512yes—
275C2×C3×C4C22×D411yescharacteristic
276C2×C3×C4C2×#4111yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278C2×C3×C4C2×#4111yescharacteristic
279C2×C3×C4H(5,𝔽2)11yescharacteristic
280C2×C3×C42−1+411yescharacteristic
281C2×Dic3—22no—
282C2×Dic3—22no—
283C2×Dic3—22no—
284C2×Dic3—22no—
285C2×Dic3—22no—
286C2×Dic3—22no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3C2×SD1614yes—
293D12—42no—
294D12—42no—
295D12—42no—
296D12—42no—
297D12—84no—
298D12—84no—
299Dic6—28no—
300Dic6—22no—
301Dic6—22no—
302Dic6—41no—
303Dic6—41no—
304C3×C8—24no—
305C3⋊C8—22no—
306C3⋊C8—41no—
307C24—62no—
308C2×Q8—34no—
309C2×Q8—32no—
310C2×Q8—62no—
311C2×Q8—62no—
312C2×Q8—61no—
313C2×Q8—61no—
314C2×Q8—61no—
315C2×Q8—61no—
316C2×Q8—61no—
317C2×D4—62no—
318C2×D4—62no—
319C2×D4—62no—
320C2×D4—62no—
321C2×D4—62no—
322C2×D4—62no—
323C2×D4—122no—
324C2×D4—122no—
325C2×D4—122no—
326C2×D4—122no—
327C2×D4—241no—
328C2×D4—241no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—32no—
333C22×C4—32no—
334C22×C4—61no—
335C22×C4—61no—
336C22×C4D3×D411yescharacteristic
337C22×C4D3×D411yescharacteristic
338C22×C4C23×D311yescharacteristic
339SD16—128no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—32no—
343C2×C8—61no—
344C2×C8—61no—
345C2×C8—61no—
346C2×C8—61no—
347C2×C8—61no—
348C2×C8C2×D1214yes—
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—42no—
356C4⋊C4—64no—
357C4⋊C4—62no—
358C4⋊C4—61no—
359C4⋊C4—61no—
360C4⋊C4—81no—
361C4⋊C4—122no—
362C4⋊C4—242no—
363C22⋊C4—124no—
364C22⋊C4—124no—
365C22⋊C4—242no—
366C22⋊C4—242no—
367C42—42no—
368C42—62no—
369C42—62no—
370C42—122no—
371C42C2×D1212yes—
372C42#23912yes—
373C22×C3C2×#12111yescharacteristic
374C22×C3#54811yescharacteristic
375C22×C3#55011yescharacteristic
376C22×C3C22×SD1611yescharacteristic
377C22×C3C2×#13511yescharacteristic
378C22×C3#57611yescharacteristic
379C22×C3#57711yescharacteristic
380C2×D3—82no—
381C2×D3—82no—
382C2×D3—82no—
383C2×D3—82no—
384C2×D3—82no—
385C2×D3—82no—
386C2×D3—82no—
387C3×C4—22no—
388C3×C4—81no—
389C3×C4—81no—
390C3×C4—82no—
391C3×C4#46514yes—
392C3×C4#52911yescharacteristic
393C3×C4#54211yescharacteristic
394C3×C4#54611yescharacteristic
395C3×C4#54811yescharacteristic
396Dic3—24no—
397Dic3—42no—
398Dic3—42no—
399Dic3—81no—
400C23—122no—
401C23—122no—
402C23—122no—
403C23—122no—
404C23—122no—
405C23—122no—
406C23—122no—
407C23C2×D3×D411yescharacteristic
408Q8—68no—
409Q8—62no—
410Q8—62no—
411Q8—121no—
412Q8—121no—
413D4—122no—
414D4—122no—
415D4—122no—
416D4—122no—
417D4—244no—
418D4—244no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—34no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—62no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—62no—
436C2×C4—62no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—121no—
441C2×C4—121no—
442C2×C4#89512yes—
443C2×C4#89512yes—
444C2×C4D3×SD1612yes—
445C2×C4#91612yes—
446C2×C4C22×D1211yescharacteristic
447C2×C4#100411yescharacteristic
448C2×C4C2×#23911yescharacteristic
449C2×C4#100711yescharacteristic
450C2×C4D3×#4111yescharacteristic
451C2×C4D3×#4111yescharacteristic
452C8—24no—
453C8—62no—
454C8—121no—
455C2×C3#242611yescharacteristic
456C2×C3C2×#46511yescharacteristic
457C2×C3#306211yescharacteristic
458C2×C3#336211yescharacteristic
459C2×C3#336511yescharacteristic
460C2×C3#339211yescharacteristic
461C2×C3#339611yescharacteristic
462D3—162no—
463C22—242no—
464C22—242no—
465C22—242no—
466C22—242no—
467C22—242no—
468C22—242no—
469C22—242no—
470C22C2×#89511yescharacteristic
471C22#559911yescharacteristic
472C22#563311yescharacteristic
473C22C2×D3×SD1611yescharacteristic
474C22C2×#91611yescharacteristic
475C22#577411yescharacteristic
476C22#577811yescharacteristic
477C4—22no—
478C4—64no—
479C4—81no—
480C4—81no—
481C4—82no—
482C4—122no—
483C4—122no—
484C4—241no—
485C4#486414yes—
486C4#555411yescharacteristic
487C4#557611yescharacteristic
488C4#569111yescharacteristic
489C4#573311yescharacteristic
490C3#2529511yescharacteristic, complemented, Sylow
491C2—482no—
492C2#7621811yescharacteristic
493C2C2×#486411yescharacteristic
494C2#7921311yescharacteristic
495C2#8037511yescharacteristic
496C2#8042011yescharacteristic
497C2#8104111yescharacteristic
498C2#8109011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1948C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#1948D3
hypocenterC3#25295
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
2. C2×#4830C2semidirect
3. #74343C2semidirect
5. C3×#1948C2semidirect
6. #70585C2semidirect
7. #70574C2semidirect
9. C4×#810C2semidirect
10. #68863C2semidirect
11. #68853C2semidirect
52. #1948D3semidirect
166. C2×Dic6SD16semidirect
490. C3#25295semidirect

Maximal subgroups

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

subgroupindex
C2×#48482
C2×#48302
#743432
#737122
C3×#19482
#705852
#705742
#695802
C4×#8102
#688632
#688532
#252953