Finite Group Databasebeta

← #10897680

Subgroups — #10897680

Order 768.

774 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
2#73299C211yesmaximal, characteristic
3#71792C211yesmaximal, characteristic
4#71782C211yesmaximal, characteristic, complemented
5#69625C212yesmaximal
6#69600C212yesmaximal, complemented
7#69585C211yesmaximal, characteristic
8#69569C211yesmaximal, characteristic, complemented
9#69462C212yesmaximal
10#69458C212yesmaximal, complemented
11#69393C211yesmaximal, characteristic, complemented
12#29388—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8—21no—
15C2×C4×D12C2211yescharacteristic
16C2×#872—21no—
17C3×#391—21no—
18C3×#389—21no—
19C3×#389C2212yes—
20C3×#384C2211yescharacteristic
21C3×C4×C4⋊C4—21no—
22C3×C4×C4⋊C4C2212yes—
23C3×C43C2211yescharacteristic
24C2×#809C2212yes—
25C2×#809C2212yes—
26C2×#807C2212yes—
27C2×#807C2212yes—
28#4940—21no—
29#4940—21no—
30#4939C2211yescharacteristic
31#4939C2211yescharacteristic
32C4×(C2×D3)⋊C4—21no—
33C4×C4⋊Dic3C2211yescharacteristic
34#4928C2211yescharacteristic
35#4928C2211yescharacteristic
36#4529C2212yes—
37#4529C2212yes—
38#4529C2212yes—
39#4529C2212yes—
40#4524C2212yes—
41#4524C2211yescharacteristic
42#4524C2211yescharacteristic
43#4524C2212yes—
44C4×C4⋊Q8D311yescharacteristic, complemented
45#2293—31no—
46#2083—31no—
47#2071—31no—
48#2057—31no—
49#2023—31no—
50#1925—32no—
51#1918—32no—
52#1918—32no—
53#1862—32no—
54#1839—31no—
55C22×C3×Q8—21no—
56C22×D12C2311yescharacteristic
57C22×C4×D3—21no—
58C3×C4⋊Q8—24no—
59C3×C4×Q8—44no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C42—22no—
67C2×C3×C42D411yescharacteristic
68C2×C3×C42D411yescharacteristic
69C2×C3×C42D411yescharacteristic
70C2×C3×C42D411yescharacteristic
71C2×C3×C42C2311yescharacteristic
72C2×C3×C42C2311yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—21no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—21no—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82C2×Dic3⋊C4—21no—
83C2×Dic3⋊C4—21no—
84C2×C4×Dic3—21no—
85C22×C3⋊C8C2312yes—
86C22×C3⋊C8C2312yes—
87C4×D12—22no—
88C4×D12—22no—
89#872—44no—
90C3×#95—22no—
91C3×#95—22no—
92C3⋊#95—21no—
93C3⋊#95—21no—
94C3⋊#95—21no—
95#809—24no—
96#809—24no—
97#807—24no—
98#807—24no—
99C2×C4⋊Q8C2×D311yescharacteristic
100C2×C4×Q8—21no—
101C2×C4×D4—31no—
102C2×#117—61no—
103C2×#107—32no—
104C2×#107—32no—
105C2×#102—32no—
106C2×#102—32no—
107#391—21no—
108#390—31no—
109#390—31no—
110#389—21no—
111#389—31no—
112#389—31no—
113#389C2×D312yes—
114#388—61no—
115#388—61no—
116#384C2×D311yescharacteristic
117C4×C4⋊C4—21no—
118C4×C4⋊C4—31no—
119C4×C4⋊C4C2×D312yes—
120C4×C22⋊C4—61no—
121C43C2×D311yescharacteristic
122#340—32no—
123#340—32no—
124#340—32no—
125#340—32no—
126#336—31no—
127#336—31no—
128#336—32no—
129#336—32no—
130C23×D3—21no—
131C2×C3×Q8—24no—
132C2×C3×Q8—42no—
133C2×C3×Q8—42no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2411yescharacteristic
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3—21no—
146C2×D12—22no—
147C2×D12—22no—
148C2×D12—21no—
149C2×D12#4112yes—
150C2×C4×D3—24no—
151C2×C4×D3—42no—
152C2×C4×D3—42no—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—42no—
158C3×C4⋊C4—44no—
159C3×C4⋊C4—42no—
160C3×C4⋊C4#4114yes—
161C3×C42—22no—
162C3×C42—21no—
163C3×C42—21no—
164C3×C42—22no—
165C3×C42—22no—
166C3×C42—41no—
167C3×C42—41no—
168C3×C42—44no—
169C3×C42D814yes—
170(C2×D3)⋊C4—24no—
171(C2×D3)⋊C4—42no—
172(C2×D3)⋊C4—42no—
173(C2×D3)⋊C4—42no—
174(C2×D3)⋊C4—42no—
175C4⋊Dic3—21no—
176C4⋊Dic3—22no—
177C4⋊Dic3—22no—
178C4⋊Dic3#4112yes—
179Dic3⋊C4—42no—
180Dic3⋊C4—42no—
181C4×Dic3—42no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—22no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—22no—
186C2×C3⋊C8—22no—
187C2×C3⋊C8—22no—
188C22×Q8—21no—
189C22×D4—31no—
190C23×C4—61no—
191C22×C8—32no—
192C22×C8—32no—
193C4⋊Q8—24no—
194C4×Q8—44no—
195C4×D4—62no—
196C4×D4—62no—
197#117—124no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—31no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4C22×D312yes—
208C2×C4⋊C4C22×D312yes—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C42—22no—
215C2×C42—61no—
216C2×C42#6711yescharacteristic
217C2×C42#6711yescharacteristic
218C2×C42#6711yescharacteristic
219C2×C42#6711yescharacteristic
220C2×C42C22×D311yescharacteristic
221C2×C42C22×D311yescharacteristic
222C2×C42C22×D311yescharacteristic
223#107—64no—
224#107—64no—
225#102—64no—
226#102—64no—
227#95—22no—
228#95—22no—
229#95—61no—
230#95—61no—
231#95—61no—
232C23×C3C22×D411yescharacteristic
233C22×D3—22no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—41no—
237C22×D3—41no—
238C22×D3—41no—
239C22×D3—41no—
240C3×Q8—48no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—24no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4C2×D812yes—
262C2×C3×C4C2×D812yes—
263C2×C3×C4C2×D812yes—
264C2×C3×C4#13712yes—
265C2×C3×C4C2×#4112yes—
266C2×C3×C4C2×#4111yescharacteristic
267C2×C3×C4C2×#4111yescharacteristic
268C2×C3×C42−1+412yes—
269C2×Dic3—22no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C2×Dic3—41no—
276C2×Dic3—41no—
277C2×Dic3—41no—
278D12—24no—
279D12—42no—
280C4×D3—48no—
281C3⋊C8—42no—
282C3⋊C8—42no—
283C24—61no—
284C2×Q8—24no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×D4—32no—
288C2×D4—62no—
289C2×D4—62no—
290C2×D4—61no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—64no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—122no—
298C22×C4—122no—
299C22×C4C2×#6711yescharacteristic
300C22×C4C2×#6711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302C22×C4C2×#6711yescharacteristic
303C22×C4C2×#6711yescharacteristic
304C22×C4C2×#6711yescharacteristic
305C22×C4C23×D311yescharacteristic
306C2×C8—62no—
307C2×C8—62no—
308C2×C8—62no—
309C2×C8—62no—
310C2×C8—62no—
311C2×C8—62no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—32no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—61no—
321C4⋊C4—62no—
322C4⋊C4—62no—
323C4⋊C4—122no—
324C4⋊C4—122no—
325C4⋊C4#23514yes—
326C22⋊C4—64no—
327C22⋊C4—122no—
328C22⋊C4—122no—
329C22⋊C4—122no—
330C22⋊C4—122no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—21no—
335C42—21no—
336C42—41no—
337C42—41no—
338C42—44no—
339C42—122no—
340C42#21314yes—
341C22×C3#52511yescharacteristic
342C22×C3#52511yescharacteristic
343C22×C3#53611yescharacteristic
344C22×C3C22×D811yescharacteristic
345C22×C3C2×#13711yescharacteristic
346C22×C3#57511yescharacteristic
347C22×C3#57511yescharacteristic
348C2×D3—42no—
349C2×D3—42no—
350C2×D3—42no—
351C2×D3—81no—
352C2×D3—81no—
353C2×D3—81no—
354C2×D3—81no—
355C3×C4—22no—
356C3×C4—22no—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—44no—
360C3×C4—41no—
361C3×C4—41no—
362C3×C4—82no—
363C3×C4#55212yes—
364C3×C4#55612yes—
365Dic3—42no—
366Dic3—81no—
367Dic3—81no—
368C23—62no—
369C23—121no—
370C23—121no—
371C23—121no—
372C23—121no—
373C23—121no—
374C23—121no—
375C23C22×#6711yescharacteristic
376Q8—48no—
377D4—64no—
378D4—122no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—24no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—62no—
400C2×C4—128no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4—121no—
408C2×C4—121no—
409C2×C4C2×#21312yes—
410C2×C4C2×#21312yes—
411C2×C4C2×#21312yes—
412C2×C4#94212yes—
413C2×C4C2×#23512yes—
414C2×C4#100712yes—
415C2×C4D3×#4111yescharacteristic
416C2×C4D3×#4111yescharacteristic
417C8—122no—
418C8—122no—
419C2×C3#259911yescharacteristic
420C2×C3#310011yescharacteristic
421C2×C3#310011yescharacteristic
422C2×C3#311611yescharacteristic
423C2×C3#311711yescharacteristic
424C2×C3#322611yescharacteristic
425C2×C3#324011yescharacteristic
426D3—82no—
427C22—122no—
428C22—122no—
429C22—122no—
430C22—241no—
431C22—241no—
432C22—241no—
433C22—241no—
434C22#579111yescharacteristic
435C22C22×#21311yescharacteristic
436C22C2×#94211yescharacteristic
437C22#581611yescharacteristic
438C22#582211yescharacteristic
439C22#582211yescharacteristic
440C22#582811yescharacteristic
441C4—22no—
442C4—22no—
443C4—22no—
444C4—22no—
445C4—44no—
446C4—41no—
447C4—41no—
448C4—82no—
449C4—122no—
450C4—241no—
451C4—241no—
452C4#556812yes—
453C4#558112yes—
454C3#2938811yescharacteristic, complemented, Sylow
455C2—242no—
456C2#8230811yescharacteristic
457C2#8244511yescharacteristic
458C2#8250211yescharacteristic
459C2#8321511yescharacteristic
460C2#8329811yescharacteristic
461C2#8335411yescharacteristic
462C2#8336211yescharacteristic
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#29388
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
4. #71782C2semidirect
6. #69600C2semidirect
8. #69569C2semidirect
10. #69458C2semidirect
11. #69393C2semidirect
44. C4×C4⋊Q8D3semidirect
454. C3#29388semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C4⋊Q82
#732992
#717922
#717822
#696252
#696002
#695852
#695692
#694622
#694582
#693932
#293883