Finite Group Databasebeta

← #10896358

Subgroups — #10896358

Order 768.

800 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#869C211yesmaximal, characteristic, complemented
2C4×#875C211yesmaximal, characteristic
3C3×#1831C211yesmaximal, characteristic, complemented
4#70458C212yesmaximal
5#70455C212yesmaximal, complemented
6#68692C211yesmaximal, characteristic, complemented
7#68689C211yesmaximal, characteristic, complemented
8#68683C212yesmaximal, complemented
9#68681C211yesmaximal, characteristic
10#68678C211yesmaximal, characteristic
11#68671C212yesmaximal
12#29262—31nomaximal, Sylow
13C2×#875C2211yescharacteristic
14C2×C4×D12C2211yescharacteristic
15C2×#869C2211yescharacteristic
16C2×C4×Dic6—21no—
17C2×C4×Dic6C2211yescharacteristic
18C2×C3×C4⋊C8C2212yes—
19C3×C43C2211yescharacteristic
20C2×#821C2212yes—
21C2×#821C2212yes—
22C2×#818C2212yes—
23C2×#817C2212yes—
24C2×#816C2212yes—
25C2×#816C2212yes—
26#4940—21no—
27#4940—21no—
28#4939C2211yescharacteristic
29C4×(C2×D3)⋊C4—21no—
30C4×(C2×D3)⋊C4—21no—
31#4936C2211yescharacteristic
32#4935C2211yescharacteristic
33C4×C4⋊Dic3—21no—
34C4×C4⋊Dic3C2212yes—
35#4929—21no—
36#4929C2211yescharacteristic
37#4928—21no—
38#4928C2212yes—
39C3×#336C2212yes—
40C3×#336C2211yescharacteristic
41C3×#336C2211yescharacteristic
42#4551C2212yes—
43#4551C2212yes—
44C4×C4⋊Q8—31no—
45C4×#124—31no—
46#2096—31no—
47#2093—31no—
48#2024—31no—
49#2023—31no—
50#1983—32no—
51#1949—32no—
52#1864—32no—
53#1863—32no—
54#1831D311yescharacteristic, complemented
55C22×D12C2311yescharacteristic
56C22×C4×D3—21no—
57C22×Dic6—21no—
58C22×Dic6C2311yescharacteristic
59C22×C3×C8C2312yes—
60C22×C3×C8C2312yes—
61C2×C3×C42D411yescharacteristic
62C2×C3×C42D411yescharacteristic
63C2×C3×C42D411yescharacteristic
64C2×C3×C42D411yescharacteristic
65C2×C3×C42C2311yescharacteristic
66C2×C3×C42C2311yescharacteristic
67C2×C3×C42C2311yescharacteristic
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—21no—
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—21no—
72C2×(C2×D3)⋊C4—21no—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3C2312yes—
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2311yescharacteristic
79C2×C4⋊Dic3C2312yes—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—21no—
82C2×C4×Dic3—22no—
83C2×C4×Dic3—21no—
84#875—22no—
85#875Q814yescomplemented
86C4×D12—42no—
87#869—24no—
88C4×Dic6—42no—
89C4×Dic6—44no—
90C3×C4⋊C8—24no—
91C3⋊#95—22no—
92C3⋊#95—22no—
93C3⋊#95—21no—
94C3⋊#95—21no—
95C3⋊#95—21no—
96C3⋊#95—21no—
97#821—24no—
98#821—42no—
99#818—42no—
100#817—42no—
101#816—24no—
102#816—42no—
103C2×C4⋊Q8—31no—
104C2×#124—31no—
105C2×C4×Q8—31no—
106C2×C4×Q8—61no—
107C2×C4×D4—31no—
108C2×#107—32no—
109C2×#106—32no—
110C2×C4⋊C8C2×D312yes—
111C2×#103—32no—
112C2×#103—32no—
113C2×#102—32no—
114C2×#102—32no—
115#391—31no—
116#391—61no—
117#390—31no—
118#389—32no—
119#389—61no—
120#388—61no—
121#388—61no—
122#384—31no—
123#382—31no—
124C4×C4⋊C4—32no—
125C4×C4⋊C4—61no—
126C4×C22⋊C4—61no—
127C4×C22⋊C4—61no—
128C43C2×D311yescharacteristic
129#340—32no—
130#340—32no—
131#336C2×D311yescharacteristic
132#336C2×D311yescharacteristic
133#336C2×D312yes—
134C23×D3—21no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2411yescharacteristic
142C22×Dic3—22no—
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3—22no—
146C22×Dic3—21no—
147C2×D12—22no—
148C2×D12—22no—
149C2×D12—21no—
150C2×D12C2×Q812yes—
151C2×C4×D3—42no—
152C2×C4×D3—42no—
153C2×C4×D3—42no—
154C2×Dic6—22no—
155C2×Dic6—22no—
156C2×Dic6—24no—
157C2×Dic6—21no—
158C2×Dic6—42no—
159C2×Dic6—42no—
160C2×Dic6C2×Q812yes—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C3×C42—22no—
168C3×C42—22no—
169C3×C42—22no—
170C3×C42—22no—
171C3×C42—22no—
172C3×C42—41no—
173C3×C42C2×Q812yes—
174C3×C42#4112yes—
175(C2×D3)⋊C4—24no—
176(C2×D3)⋊C4—24no—
177(C2×D3)⋊C4—42no—
178(C2×D3)⋊C4—42no—
179(C2×D3)⋊C4—42no—
180C4⋊Dic3—24no—
181C4⋊Dic3—22no—
182C4⋊Dic3—22no—
183C4⋊Dic3—22no—
184C4⋊Dic3—22no—
185C4⋊Dic3—22no—
186C4⋊Dic3—42no—
187C4⋊Dic3—42no—
188C4⋊Dic3#4114yes—
189Dic3⋊C4—42no—
190Dic3⋊C4—44no—
191C4×Dic3—42no—
192C4×Dic3—44no—
193C22×Q8—31no—
194C22×Q8—61no—
195C22×D4—31no—
196C23×C4—61no—
197C22×C8C22×D312yes—
198C22×C8C22×D312yes—
199C4⋊Q8—64no—
200#124—34no—
201#124—62no—
202C4×Q8—122no—
203C4×Q8—124no—
204C4×D4—122no—
205C2×C4⋊C4—32no—
206C2×C4⋊C4—31no—
207C2×C4⋊C4—31no—
208C2×C4⋊C4—32no—
209C2×C4⋊C4—61no—
210C2×C4⋊C4—61no—
211C2×C4⋊C4—61no—
212C2×C4⋊C4—62no—
213C2×C4⋊C4—61no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—61no—
217C2×C22⋊C4—61no—
218C2×C22⋊C4—61no—
219C2×C42—61no—
220C2×C42—62no—
221C2×C42D1211yescharacteristic
222C2×C42D1211yescharacteristic
223C2×C42D1211yescharacteristic
224C2×C42D1211yescharacteristic
225C2×C42C22×D311yescharacteristic
226C2×C42C22×D311yescharacteristic
227C2×C42C22×D311yescharacteristic
228#107—122no—
229#106—122no—
230C4⋊C8—24no—
231#103—64no—
232#103—122no—
233#102—64no—
234#102—122no—
235#95—61no—
236#95—61no—
237#95—61no—
238#95—61no—
239#95—62no—
240#95—62no—
241C23×C3C22×D411yescharacteristic
242C22×D3—22no—
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—41no—
248C22×D3—41no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4C22⋊Q812yes—
263C2×C3×C4C22⋊Q812yes—
264C2×C3×C4#13512yes—
265C2×C3×C4AGL(1,ℤ/8)12yes—
266C2×C3×C4#13712yes—
267C2×C3×C4C22×Q811yescharacteristic
268C2×C3×C4C2×#4111yescharacteristic
269C2×C3×C4C2×#4112yes—
270C2×C3×C4H(5,𝔽2)12yes—
271C2×Dic3—22no—
272C2×Dic3—24no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C2×Dic3—42no—
276C2×Dic3—42no—
277C2×Dic3—41no—
278C2×Dic3—41no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—41no—
282C2×Dic3—42no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286D12—24no—
287D12—42no—
288C4×D3—84no—
289Dic6—24no—
290Dic6—48no—
291Dic6—42no—
292C3×C8—42no—
293C3×C8—42no—
294C24—61no—
295C2×Q8—32no—
296C2×Q8—64no—
297C2×Q8—61no—
298C2×Q8—62no—
299C2×Q8—62no—
300C2×Q8—122no—
301C2×Q8—122no—
302C2×D4—32no—
303C2×D4—61no—
304C2×D4—62no—
305C2×D4—62no—
306C22×C4—61no—
307C22×C4—62no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—62no—
311C22×C4—122no—
312C22×C4—122no—
313C22×C4—122no—
314C22×C4C2×D1211yescharacteristic
315C22×C4C2×D1211yescharacteristic
316C22×C4C2×D1211yescharacteristic
317C22×C4C2×D1211yescharacteristic
318C22×C4C2×D1211yescharacteristic
319C22×C4C2×D1211yescharacteristic
320C22×C4C23×D311yescharacteristic
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C4⋊C4—34no—
328C4⋊C4—64no—
329C4⋊C4—62no—
330C4⋊C4—62no—
331C4⋊C4—62no—
332C4⋊C4—62no—
333C4⋊C4—62no—
334C4⋊C4—124no—
335C4⋊C4—122no—
336C4⋊C4—122no—
337C4⋊C4—122no—
338C22⋊C4—64no—
339C22⋊C4—64no—
340C22⋊C4—122no—
341C22⋊C4—122no—
342C22⋊C4—122no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—41no—
349C42—124no—
350C42—122no—
351C42#23712yes—
352C42D3×Q812yes—
353C22×C3C2×C22⋊Q811yescharacteristic
354C22×C3#52511yescharacteristic
355C22×C3#53511yescharacteristic
356C22×C3C2×#13511yescharacteristic
357C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
358C22×C3C2×#13711yescharacteristic
359C22×C3#57511yescharacteristic
360C2×D3—42no—
361C2×D3—42no—
362C2×D3—42no—
363C2×D3—81no—
364C2×D3—81no—
365C2×D3—81no—
366C2×D3—81no—
367C3×C4—22no—
368C3×C4—22no—
369C3×C4—22no—
370C3×C4—22no—
371C3×C4—22no—
372C3×C4—41no—
373C3×C4#55412yes—
374C3×C4#55912yes—
375Dic3—42no—
376Dic3—44no—
377Dic3—81no—
378Dic3—82no—
379Dic3—81no—
380C23—62no—
381C23—121no—
382C23—121no—
383C23—121no—
384C23—121no—
385C23—121no—
386C23—121no—
387C23C22×D1211yescharacteristic
388Q8—64no—
389Q8—128no—
390Q8—122no—
391D4—64no—
392D4—122no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—62no—
407C2×C4—64no—
408C2×C4—122no—
409C2×C4—122no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—121no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—122no—
421C2×C4—244no—
422C2×C4#89712yes—
423C2×C4#89712yes—
424C2×C4#90412yes—
425C2×C4#90812yes—
426C2×C4#90912yes—
427C2×C4C2×#23711yescharacteristic
428C2×C4C2×D3×Q811yescharacteristic
429C2×C4D3×#4112yes—
430C2×C4#100912yes—
431C8—42no—
432C8—42no—
433C2×C3#258011yescharacteristic
434C2×C3#307611yescharacteristic
435C2×C3#307911yescharacteristic
436C2×C3#310811yescharacteristic
437C2×C3#311411yescharacteristic
438C2×C3#320711yescharacteristic
439C2×C3#321511yescharacteristic
440D3—82no—
441C22—122no—
442C22—122no—
443C22—122no—
444C22—241no—
445C22—241no—
446C22—241no—
447C22—241no—
448C22C2×#89711yescharacteristic
449C22#552411yescharacteristic
450C22#552511yescharacteristic
451C22C2×#90411yescharacteristic
452C22C2×#90811yescharacteristic
453C22C2×#90911yescharacteristic
454C22#574811yescharacteristic
455C4—22no—
456C4—22no—
457C4—22no—
458C4—22no—
459C4—22no—
460C4—41no—
461C4—122no—
462C4—124no—
463C4—241no—
464C4—242no—
465C4—241no—
466C4#568112yes—
467C4#569012yes—
468C3#2926211yescharacteristic, complemented, Sylow
469C2—242no—
470C2#7419011yescharacteristic
471C2#7902511yescharacteristic
472C2#7906011yescharacteristic
473C2#7930411yescharacteristic
474C2#7931111yescharacteristic
475C2#7934611yescharacteristic
476C2#7935111yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1831C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1831D3
hypocenterC3#29262
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C43 ◁ #1831

Complements

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

NH ≅ G/Ntype
1. C4×#869C2semidirect
3. C3×#1831C2semidirect
5. #70455C2semidirect
6. #68692C2semidirect
7. #68689C2semidirect
8. #68683C2semidirect
54. #1831D3semidirect
85. #875Q8semidirect
468. C3#29262semidirect

Maximal subgroups

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

subgroupindex
C4×#8692
C4×#8752
C3×#18312
#704582
#704552
#686922
#686892
#686832
#686812
#686782
#686712
#292623