Finite Group Databasebeta

← #10896356

Subgroups — #10896356

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
1C3×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
2C4×#875C211yesmaximal, characteristic
3#71794C211yesmaximal, characteristic, complemented
4#71792C211yesmaximal, characteristic
5#69631C212yesmaximal, complemented
6#69626C212yesmaximal
7#69591C211yesmaximal, characteristic, complemented
8#69586C211yesmaximal, characteristic
9#69488C212yesmaximal
10#69468C212yesmaximal, complemented
11#69392C211yesmaximal, characteristic, complemented
12#29262—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8—21no—
15C2×#875C2211yescharacteristic
16C2×C4×D12C2211yescharacteristic
17C2×C4×Dic6C2211yescharacteristic
18C3×#391—21no—
19C3×#389—21no—
20C3×#389C2212yes—
21C3×#384C2211yescharacteristic
22C3×C4×C4⋊C4—21no—
23C3×C4×C4⋊C4C2212yes—
24C3×C43C2211yescharacteristic
25C2×#810C2212yes—
26C2×#810C2212yes—
27C2×#809C2212yes—
28C2×#809C2212yes—
29C2×#808C2212yes—
30C2×#807C2212yes—
31#4940—21no—
32#4940—21no—
33#4939C2211yescharacteristic
34C4×(C2×D3)⋊C4—21no—
35C4×(C2×D3)⋊C4—21no—
36#4936C2211yescharacteristic
37#4929C2211yescharacteristic
38C2×C4⋊(C3⋊C8)C2212yes—
39#4529C2212yes—
40#4529C2212yes—
41#4524C2211yescharacteristic
42#4524C2211yescharacteristic
43#4524C2212yes—
44C4×C4⋊Q8D311yescharacteristic, complemented
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#1831—31no—
55C22×C3×Q8—21no—
56C22×D12C2311yescharacteristic
57C22×C4×D3—21no—
58C22×Dic6C2311yescharacteristic
59C3×C4⋊Q8—24no—
60C3×C4×Q8—44no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C42—22no—
68C2×C3×C42D411yescharacteristic
69C2×C3×C42D411yescharacteristic
70C2×C3×C42D411yescharacteristic
71C2×C3×C42D411yescharacteristic
72C2×C3×C42C2311yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×(C2×D3)⋊C4—21no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—21no—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82C2×Dic3⋊C4—21no—
83C2×C4×Dic3—21no—
84C22×C3⋊C8C2312yes—
85C22×C3⋊C8C2312yes—
86#875—22no—
87#875Q814yescomplemented
88C4×D12—42no—
89C4×Dic6—42no—
90C3×#95—22no—
91C3×#95—22no—
92C3⋊#95—21no—
93C3⋊#95—21no—
94C3⋊#95—21no—
95C3⋊#95—21no—
96#810—24no—
97#810—42no—
98#809—24no—
99#809—42no—
100#808—42no—
101#807—42no—
102C4⋊(C3⋊C8)—24no—
103C2×C4⋊Q8C2×D311yescharacteristic
104C2×#124—31no—
105C2×C4×Q8—21no—
106C2×C4×Q8—31no—
107C2×C4×D4—31no—
108C2×#107—32no—
109C2×#106—32no—
110C2×C4⋊C8—32no—
111C2×#103—32no—
112C2×#103—32no—
113C2×#102—32no—
114C2×#102—32no—
115#391—21no—
116#391—31no—
117#390—31no—
118#389—21no—
119#389C2×D312yes—
120#388—61no—
121#388—61no—
122#384C2×D311yescharacteristic
123#382—31no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4C2×D312yes—
126C4×C22⋊C4—61no—
127C4×C22⋊C4—61no—
128C43C2×D311yescharacteristic
129#340—32no—
130#340—32no—
131#336—31no—
132#336—31no—
133#336—32no—
134C23×D3—21no—
135C2×C3×Q8—24no—
136C2×C3×Q8—42no—
137C2×C3×Q8—42no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2411yescharacteristic
147C22×Dic3—21no—
148C22×Dic3—21no—
149C22×Dic3—21no—
150C2×D12—22no—
151C2×D12—22no—
152C2×D12—21no—
153C2×D12C2×Q812yes—
154C2×C4×D3—42no—
155C2×C4×D3—42no—
156C2×C4×D3—42no—
157C2×Dic6—22no—
158C2×Dic6—22no—
159C2×Dic6—21no—
160C2×Dic6C2×Q812yes—
161C3×C4⋊C4—24no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—44no—
166C3×C4⋊C4—42no—
167C3×C4⋊C4—42no—
168C3×C4⋊C4#4114yes—
169C3×C42—22no—
170C3×C42—22no—
171C3×C42—22no—
172C3×C42—22no—
173C3×C42—22no—
174C3×C42—41no—
175C3×C42—44no—
176C3×C42C2×Q812yes—
177C3×C42#4112yes—
178(C2×D3)⋊C4—24no—
179(C2×D3)⋊C4—24no—
180(C2×D3)⋊C4—42no—
181(C2×D3)⋊C4—42no—
182(C2×D3)⋊C4—42no—
183C4⋊Dic3—22no—
184C4⋊Dic3—22no—
185Dic3⋊C4—42no—
186C4×Dic3—42no—
187C2×C3⋊C8—22no—
188C2×C3⋊C8—22no—
189C2×C3⋊C8—22no—
190C2×C3⋊C8—22no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—22no—
193C22×Q8—21no—
194C22×Q8—31no—
195C22×D4—31no—
196C23×C4—61no—
197C22×C8—32no—
198C22×C8—32no—
199C4⋊Q8—24no—
200#124—34no—
201#124—62no—
202C4×Q8—44no—
203C4×Q8—122no—
204C4×D4—122no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—31no—
210C2×C4⋊C4—31no—
211C2×C4⋊C4—61no—
212C2×C4⋊C4C22×D312yes—
213C2×C4⋊C4C22×D312yes—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—61no—
217C2×C22⋊C4—61no—
218C2×C22⋊C4—61no—
219C2×C42—22no—
220C2×C42—61no—
221C2×C42#6711yescharacteristic
222C2×C42#6711yescharacteristic
223C2×C42#6711yescharacteristic
224C2×C42#6711yescharacteristic
225C2×C42C22×D311yescharacteristic
226C2×C42C22×D311yescharacteristic
227C2×C42C22×D311yescharacteristic
228#107—122no—
229#106—122no—
230C4⋊C8—64no—
231#103—64no—
232#103—122no—
233#102—64no—
234#102—122no—
235#95—22no—
236#95—22no—
237#95—61no—
238#95—61no—
239#95—61no—
240#95—61no—
241C23×C3C22×D411yescharacteristic
242C22×D3—22no—
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—41no—
248C22×D3—41no—
249C3×Q8—48no—
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—24no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4C22⋊Q812yes—
270C2×C3×C4C22⋊Q812yes—
271C2×C3×C4#13512yes—
272C2×C3×C4AGL(1,ℤ/8)12yes—
273C2×C3×C4#13712yes—
274C2×C3×C4C22×Q811yescharacteristic
275C2×C3×C4C2×#4112yes—
276C2×C3×C4C2×#4111yescharacteristic
277C2×C3×C4H(5,𝔽2)12yes—
278C2×Dic3—22no—
279C2×Dic3—41no—
280C2×Dic3—41no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287D12—24no—
288D12—42no—
289C4×D3—84no—
290Dic6—24no—
291Dic6—42no—
292C3⋊C8—42no—
293C3⋊C8—42no—
294C24—61no—
295C2×Q8—24no—
296C2×Q8—32no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—61no—
300C2×Q8—62no—
301C2×Q8—62no—
302C2×D4—32no—
303C2×D4—62no—
304C2×D4—61no—
305C2×D4—62no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—122no—
312C22×C4—122no—
313C22×C4—122no—
314C22×C4C2×#6711yescharacteristic
315C22×C4C2×#6711yescharacteristic
316C22×C4C2×#6711yescharacteristic
317C22×C4C2×#6711yescharacteristic
318C22×C4C2×#6711yescharacteristic
319C22×C4C2×#6711yescharacteristic
320C22×C4C23×D311yescharacteristic
321C2×C8—62no—
322C2×C8—62no—
323C2×C8—62no—
324C2×C8—62no—
325C2×C8—62no—
326C2×C8—62no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—44no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—62no—
335C4⋊C4—62no—
336C4⋊C4—122no—
337C4⋊C4#23514yes—
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—44no—
350C42—122no—
351C42D3×Q812yes—
352C42#23912yes—
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—44no—
373C3×C4—41no—
374C3×C4—82no—
375C3×C4#55412yes—
376C3×C4#55912yes—
377Dic3—42no—
378Dic3—81no—
379Dic3—81no—
380C23—62no—
381C23—121no—
382C23—121no—
383C23—121no—
384C23—121no—
385C23—121no—
386C23—121no—
387C23C22×#6711yescharacteristic
388Q8—48no—
389Q8—64no—
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—24no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—62no—
413C2×C4—121no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—244no—
422C2×C4#93212yes—
423C2×C4#94212yes—
424C2×C4#94612yes—
425C2×C4#94612yes—
426C2×C4#95012yes—
427C2×C4C2×#23512yes—
428C2×C4C2×D3×Q811yescharacteristic
429C2×C4C2×#23911yescharacteristic
430C2×C4#100912yes—
431C8—122no—
432C8—122no—
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—
448C22#579111yescharacteristic
449C22C2×#93211yescharacteristic
450C22C2×#94211yescharacteristic
451C22C2×#94611yescharacteristic
452C22C2×#95011yescharacteristic
453C22#582211yescharacteristic
454C22#583011yescharacteristic
455C4—22no—
456C4—22no—
457C4—22no—
458C4—22no—
459C4—22no—
460C4—41no—
461C4—44no—
462C4—82no—
463C4—122no—
464C4—241no—
465C4—241no—
466C4#557912yes—
467C4#558512yes—
468C3#2926211yescharacteristic, complemented, Sylow
469C2—242no—
470C2#8231111yescharacteristic
471C2#8244311yescharacteristic
472C2#8250111yescharacteristic
473C2#8321211yescharacteristic
474C2#8330411yescharacteristic
475C2#8334911yescharacteristic
476C2#8335711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×C4⋊Q8C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC4×C4⋊Q8D3
hypocenterC3#29262
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C43 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×C4×C4⋊Q8C2semidirect
3. #71794C2semidirect
5. #69631C2semidirect
7. #69591C2semidirect
10. #69468C2semidirect
11. #69392C2semidirect
44. C4×C4⋊Q8D3semidirect
87. #875Q8semidirect
468. C3#29262semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C4⋊Q82
C4×#8752
#717942
#717922
#696312
#696262
#695912
#695862
#694882
#694682
#693922
#292623