Finite Group Databasebeta

← #10824336

Subgroups — #10824336

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C4×#215C212yesmaximal
3#73299C211yesmaximal, characteristic
4#71791C211yesmaximal, characteristic
5#71781C211yesmaximal, characteristic, complemented
6C4×#835C212yesmaximal, complemented
7#69682C212yesmaximal, complemented
8#69680C212yesmaximal
9#69585C211yesmaximal, characteristic
10#69570C211yesmaximal, characteristic, complemented
11C4×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
12#23677—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2212yes—
15C22×#215C2212yes—
16C2×C4×D12C2211yescharacteristic
17C2×#872—21no—
18C3×#391C2212yes—
19C3×#389—21no—
20C3×#389—21no—
21C3×#384C2211yescharacteristic
22C3×C4×C4⋊C4—21no—
23C3×C4×C4⋊C4—21no—
24C3×C43C2211yescharacteristic
25C2×#835C2212yes—
26C2×#835C2212yes—
27C2×#835C2212yes—
28C4×#215—28no—
29C2×#809C2212yes—
30C2×#808C2212yes—
31#4940—21no—
32#4940—21no—
33#4939C2211yescharacteristic
34#4939C2211yescharacteristic
35C4×(C2×D3)⋊C4—21no—
36C4×C4⋊Dic3C2211yescharacteristic
37#4928C2211yescharacteristic
38#4928C2211yescharacteristic
39C2×C4⋊(C3⋊C8)C2212yes—
40C2×C4×C3⋊C8C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44#4524C2211yescharacteristic
45C2×C4×SD16—32no—
46C4×C4⋊Q8D311yescharacteristic, complemented
47#2293—31no—
48#2084—31no—
49#2071—31no—
50#2066—32no—
51#2055—32no—
52#2043—31no—
53#2013—31no—
54C4×C4⋊C8—31no—
55C4×#103—32no—
56C22×C3×Q8C2312yes—
57C22×D12C2311yescharacteristic
58C22×C4×D3—21no—
59C3×C4⋊Q8—24no—
60C3×C4×Q8—24no—
61C3×C4×Q8—24no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C42—22no—
69C2×C3×C42D411yescharacteristic
70C2×C3×C42D411yescharacteristic
71C2×C3×C42D411yescharacteristic
72C2×C3×C42D411yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×#215—24no—
77C2×#215—24no—
78C2×#215—24no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—21no—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×Dic3⋊C4—21no—
88C2×Dic3⋊C4—21no—
89C2×C4×Dic3—21no—
90C22×C3⋊C8C2312yes—
91C22×C3⋊C8C2312yes—
92C4×D12—22no—
93C4×D12D414yes—
94#872—44no—
95C3×#95—22no—
96C3×#95—22no—
97#835—24no—
98#835—24no—
99#835—24no—
100C3⋊#95—21no—
101C3⋊#95—21no—
102C3⋊#95—21no—
103#809—24no—
104#808—24no—
105C4⋊(C3⋊C8)—24no—
106C4×C3⋊C8—24no—
107C22×SD16—32no—
108C2×C4⋊Q8C2×D311yescharacteristic
109C2×C4×Q8C2×D312yes—
110C2×C4×D4—31no—
111C2×#117—61no—
112C4×SD16—68no—
113C2×#106—32no—
114C2×C4⋊C8—32no—
115C2×#103—32no—
116C2×#103—32no—
117C2×#103—32no—
118C2×#102—32no—
119C2×C4×C8—32no—
120#391C2×D312yes—
121#390—31no—
122#390—31no—
123#389—21no—
124#389—21no—
125#389—31no—
126#389—31no—
127#388—61no—
128#388—61no—
129#384C2×D311yescharacteristic
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—31no—
133C4×C22⋊C4—61no—
134C43C2×D311yescharacteristic
135#340—32no—
136#340—32no—
137#336—31no—
138#336—31no—
139C23×D3—21no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8#4114yes—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2411yescharacteristic
155C22×Dic3—21no—
156C22×Dic3—21no—
157C22×Dic3—21no—
158C2×D12—22no—
159C2×D12—22no—
160C2×D12—21no—
161C2×D12C2×D412yes—
162C2×C4×D3—24no—
163C2×C4×D3—42no—
164C2×C4×D3—42no—
165C3×C4⋊C4—24no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—42no—
170C3×C4⋊C4—44no—
171C3×C4⋊C4—42no—
172C3×C42—22no—
173C3×C42—22no—
174C3×C42—22no—
175C3×C42—22no—
176C3×C42—41no—
177C3×C42—44no—
178C3×C42SD1614yescomplemented
179C3×C42C2×D412yes—
180C3×C42#4112yes—
181#215—216no—
182(C2×D3)⋊C4—24no—
183(C2×D3)⋊C4—42no—
184(C2×D3)⋊C4—42no—
185(C2×D3)⋊C4—42no—
186(C2×D3)⋊C4—42no—
187C4⋊Dic3—21no—
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3C2×D412yes—
191Dic3⋊C4—42no—
192Dic3⋊C4—42no—
193C4×Dic3—42no—
194C2×C3⋊C8—22no—
195C2×C3⋊C8—22no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8#4114yes—
200C22×Q8C22×D312yes—
201C22×D4—31no—
202C23×C4—61no—
203C2×SD16—64no—
204C2×SD16—64no—
205C2×SD16—64no—
206C22×C8—32no—
207C22×C8—32no—
208C4⋊Q8—24no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×D4—34no—
212C4×D4—62no—
213#117—124no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—31no—
221C2×C4⋊C4—31no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4C22×D312yes—
225C2×C22⋊C4—61no—
226C2×C22⋊C4—61no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—61no—
229C2×C22⋊C4—61no—
230C2×C42—22no—
231C2×C42—61no—
232C2×C42#6711yescharacteristic
233C2×C42#6711yescharacteristic
234C2×C42#6711yescharacteristic
235C2×C42#6711yescharacteristic
236C2×C42C22×D311yescharacteristic
237C2×C42C22×D311yescharacteristic
238C2×C42C22×D311yescharacteristic
239#106—64no—
240C4⋊C8—64no—
241#103—64no—
242#103—64no—
243#103—64no—
244#102—64no—
245C4×C8—64no—
246#95—22no—
247#95—22no—
248#95—61no—
249#95—61no—
250#95—61no—
251C23×C3C22×D411yescharacteristic
252C22×D3—22no—
253C22×D3—41no—
254C22×D3—41no—
255C22×D3—41no—
256C22×D3—41no—
257C22×D3—41no—
258C22×D3—41no—
259C3×Q8—24no—
260C3×Q8—24no—
261C3×Q8—42no—
262C3×Q8—42no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—24no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4#12112yes—
281C2×C3×C4#12112yes—
282C2×C3×C4C2×SD1612yes—
283C2×C3×C4C2×SD1612yes—
284C2×C3×C4C2×SD1612yes—
285C2×C3×C4#13512yes—
286C2×C3×C4#13712yes—
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C2×#4112yes—
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C4C2×#4112yes—
291C2×Dic3—22no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300D12—24no—
301D12—42no—
302C4×D3—48no—
303C3⋊C8—24no—
304C3⋊C8—42no—
305C24—61no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8#23514yes—
312C2×D4—32no—
313C2×D4—62no—
314C2×D4—61no—
315C2×D4—62no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—61no—
319C22×C4—64no—
320C22×C4—61no—
321C22×C4—61no—
322C22×C4—122no—
323C22×C4—122no—
324C22×C4C2×#6711yescharacteristic
325C22×C4C2×#6711yescharacteristic
326C22×C4C2×#6711yescharacteristic
327C22×C4C2×#6711yescharacteristic
328C22×C4C2×#6711yescharacteristic
329C22×C4C2×#6711yescharacteristic
330C22×C4C23×D311yescharacteristic
331SD16—616no—
332C2×C8—34no—
333C2×C8—62no—
334C2×C8—62no—
335C2×C8—62no—
336C2×C8—62no—
337C2×C8—62no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—32no—
343C4⋊C4—42no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—61no—
347C4⋊C4—62no—
348C4⋊C4—62no—
349C4⋊C4—122no—
350C4⋊C4—122no—
351C22⋊C4—64no—
352C22⋊C4—122no—
353C22⋊C4—122no—
354C22⋊C4—122no—
355C22⋊C4—122no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—41no—
361C42—44no—
362C42—122no—
363C42#21514yescomplemented
364C42D3×D412yes—
365C42#23712yes—
366C22×C3C2×#12111yescharacteristic
367C22×C3#52511yescharacteristic
368C22×C3#53211yescharacteristic
369C22×C3C22×SD1611yescharacteristic
370C22×C3C2×#13511yescharacteristic
371C22×C3C2×#13711yescharacteristic
372C22×C3#57511yescharacteristic
373C2×D3—42no—
374C2×D3—42no—
375C2×D3—42no—
376C2×D3—81no—
377C2×D3—81no—
378C2×D3—81no—
379C2×D3—81no—
380C3×C4—22no—
381C3×C4—22no—
382C3×C4—22no—
383C3×C4—22no—
384C3×C4—44no—
385C3×C4—41no—
386C3×C4—82no—
387C3×C4#46114yes—
388C3×C4#52512yes—
389C3×C4#52912yes—
390Dic3—42no—
391Dic3—81no—
392Dic3—81no—
393C23—62no—
394C23—121no—
395C23—121no—
396C23—121no—
397C23—121no—
398C23—121no—
399C23—121no—
400C23C22×#6711yescharacteristic
401Q8—24no—
402Q8—24no—
403Q8—42no—
404Q8—42no—
405D4—64no—
406D4—122no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—24no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—62no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—128no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4#93812yes—
435C2×C4#93812yes—
436C2×C4C2×#21512yes—
437C2×C4C2×#21512yes—
438C2×C4C2×#21512yes—
439C2×C4#94212yes—
440C2×C4#95012yes—
441C2×C4C2×#23512yes—
442C2×C4C2×D3×D411yescharacteristic
443C2×C4C2×#23711yescharacteristic
444C2×C4D3×#4112yes—
445C8—64no—
446C8—122no—
447C2×C3#239211yescharacteristic
448C2×C3C2×#46111yescharacteristic
449C2×C3#303511yescharacteristic
450C2×C3#310311yescharacteristic
451C2×C3#310811yescharacteristic
452C2×C3#316711yescharacteristic
453C2×C3#317211yescharacteristic
454D3—82no—
455C22—122no—
456C22—122no—
457C22—122no—
458C22—241no—
459C22—241no—
460C22—241no—
461C22—241no—
462C22#579111yescharacteristic
463C22C2×#93811yescharacteristic
464C22C22×#21511yescharacteristic
465C22C2×#94211yescharacteristic
466C22C2×#95011yescharacteristic
467C22#582211yescharacteristic
468C22#583211yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—22no—
473C4—41no—
474C4—44no—
475C4—82no—
476C4—122no—
477C4—241no—
478C4—241no—
479C4#508314yes—
480C4#553812yes—
481C4#554712yes—
482C3#2367711yescharacteristic, complemented, Sylow
483C2—242no—
484C2#8240111yescharacteristic
485C2#8244211yescharacteristic
486C2#8244311yescharacteristic
487C2C2×#508311yescharacteristic
488C2#8329611yescharacteristic
489C2#8334311yescharacteristic
490C2#8334511yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×C4⋊Q8C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC4×C4⋊Q8D3
hypocenterC3#23677
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×C4×C4⋊Q8C2semidirect
5. #71781C2semidirect
6. C4×#835C2semidirect
7. #69682C2semidirect
10. #69570C2semidirect
11. C4×C4⋊(C3⋊C8)C2semidirect
46. C4×C4⋊Q8D3semidirect
178. C3×C42SD16semidirect
363. C42#215semidirect
482. C3#23677semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C4⋊Q82
C2×C4×#2152
#732992
#717912
#717812
C4×#8352
#696822
#696802
#695852
#695702
C4×C4⋊(C3⋊C8)2
#236773