Finite Group Databasebeta

← #10813477

Subgroups — #10813477

Order 768.

1018 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
1C3×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#5026C212yesmaximal
3C2×C4×#215C211yesmaximal, characteristic
4C4⋊C4×D12C211yesmaximal, characteristic
5#71628C212yesmaximal, complemented
6#71626C211yesmaximal, characteristic, complemented
7#69625C212yesmaximal
8#69621C212yesmaximal, complemented
9#69584C211yesmaximal, characteristic
10#69577C211yesmaximal, characteristic, complemented
11#69477C211yesmaximal, characteristic, complemented
12#22671—31nomaximal, Sylow
13C2×C3×C4×Q8C2212yes—
14C2×C3×C4×Q8C2211yescharacteristic
15C22×#215C2211yescharacteristic
16C2×D3×C4⋊C4—21no—
17C2×C4×D12C2212yes—
18C2×C4×D12C2211yescharacteristic
19C3×#389—22no—
20C3×#389C2212yes—
21C3×#384—21no—
22C3×#384C2211yescharacteristic
23C3×C4×C4⋊C4—21no—
24C3×C4×C4⋊C4C2211yescharacteristic
25C2×#835C2212yes—
26C2×#835C2211yescharacteristic
27#5026—28no—
28C4×#215C418yescomplemented
29#4987—21no—
30C2×#809C2212yes—
31C2×#809C2211yescharacteristic
32C2×#808C2211yescharacteristic
33C2×#807C2212yes—
34C2×#807C2212yes—
35#4935C2211yescharacteristic
36#4928C2212yes—
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38C2×#803C2212yes—
39C2×C4×C3⋊C8C2211yescharacteristic
40#4668—22no—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2212yes—
44C2×#440—32no—
45C2×C4×SD16—31no—
46Q8×C4⋊C4D311yescharacteristic, complemented
47D4×C4⋊C4—31no—
48#1944—32no—
49#1863—32no—
50#1862—32no—
51#1845—31no—
52#1803—31no—
53#1802—31no—
54#1770—31no—
55C22×C3×Q8C2311yescharacteristic
56C22×D12C2311yescharacteristic
57C22×C4×D3—21no—
58C22×C4×D3—22no—
59C3×C4×Q8—22no—
60C3×C4×Q8—24no—
61C3×C4×Q8—42no—
62C3×C4×Q8C2×C414yescomplemented
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4D411yescharacteristic
69C2×C3×C4⋊C4D411yescharacteristic
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C42—21no—
73C2×C3×C42—22no—
74C2×C3×C42C2312yes—
75C2×C3×C42C2311yescharacteristic
76C2×#215C2×C414yes—
77C2×#215D414yes—
78C2×#215Q814yes—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—22no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3C2312yes—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×Dic3⋊C4—22no—
86C22×C3⋊C8C2312yes—
87C22×C3⋊C8C2311yescharacteristic
88C22×C3⋊C8C2311yescharacteristic
89D3×C4⋊C4—44no—
90C4×D12—22no—
91C4×D12—24no—
92C4×D12—42no—
93C4×D12C2×C414yes—
94C3×#95—22no—
95C3×#95D412yes—
96#835—24no—
97#835C2×C414yescomplemented
98C3⋊#95—22no—
99#809—24no—
100#809C2×C414yes—
101#808C2×C414yescomplemented
102#807—24no—
103#807—42no—
104C4⋊(C3⋊C8)—22no—
105#803—24no—
106C4×C3⋊C8C2×C414yescomplemented
107C22×SD16—31no—
108C2×C4×Q8C2×D312yes—
109C2×C4×Q8C2×D311yescharacteristic
110C2×C4×D4—31no—
111C2×C4×D4—32no—
112C22×C4⋊C4—61no—
113#440—68no—
114C4×SD16—38no—
115C2×#107—32no—
116C2×#107—32no—
117C2×#106—31no—
118C2×C4⋊C8—31no—
119C2×#103—31no—
120C2×#103—32no—
121C2×#102—31no—
122C2×#102—32no—
123C2×#97—32no—
124C2×C4×C8—31no—
125#389—22no—
126#389—32no—
127#389C2×D312yes—
128#385—62no—
129#384—21no—
130#384—31no—
131#384C2×D311yescharacteristic
132#380—61no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2×D311yescharacteristic
135#340—32no—
136#340—32no—
137#336—32no—
138C23×D3—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8—21no—
142C2×C3×Q8C22×C412yes—
143C2×C3×Q8C2×D412yes—
144C2×C3×Q8C2×Q812yes—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2411yescharacteristic
153C22×Dic3—21no—
154C22×Dic3—22no—
155C2×D12—21no—
156C2×D12—21no—
157C2×D12—21no—
158C2×D12C22×C412yes—
159C2×D12C2×D412yes—
160C2×D12C2×Q812yes—
161C2×C4×D3—24no—
162C2×C4×D3—42no—
163C2×C4×D3—42no—
164C2×C4×D3—44no—
165C2×C4×D3—44no—
166C2×C4×D3—82no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—44no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4—81no—
176C3×C4⋊C4—82no—
177C3×C4⋊C4C22×C412yes—
178C3×C42—24no—
179C3×C42—22no—
180C3×C42—22no—
181C3×C42—44no—
182C3×C42C22×C412yes—
183C3×C42#4112yes—
184#215C4⋊C4116yescomplemented
185(C2×D3)⋊C4—24no—
186(C2×D3)⋊C4—44no—
187C4⋊Dic3—21no—
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3—42no—
191C4⋊Dic3—81no—
192C4⋊Dic3C22×C412yes—
193Dic3⋊C4—44no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8C22×C412yes—
200C2×C3⋊C8C22×C412yes—
201C2×C3⋊C8C2×D412yes—
202C2×C3⋊C8C2×Q812yes—
203C22×Q8C22×D311yescharacteristic
204C22×D4—31no—
205C23×C4—61no—
206C23×C4—62no—
207C2×SD16—34no—
208C2×SD16—34no—
209C2×SD16—34no—
210C22×C8—31no—
211C22×C8—31no—
212C22×C8—32no—
213C4×Q8—22no—
214C4×Q8—24no—
215C4×Q8—42no—
216C4×Q8C4×D314yescomplemented
217C4×D4—34no—
218C4×D4—64no—
219C4×D4—62no—
220C4×D4—122no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—31no—
227C2×C4⋊C4—32no—
228C2×C4⋊C4—61no—
229C2×C4⋊C4—62no—
230C2×C4⋊C4—61no—
231C2×C4⋊C4—124no—
232C2×C4⋊C4#6711yescharacteristic
233C2×C4⋊C4#6711yescharacteristic
234C2×C4⋊C4C22×D312yes—
235C2×C4⋊C4C22×D311yescharacteristic
236C2×C22⋊C4—61no—
237C2×C22⋊C4—62no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42C22×D312yes—
241C2×C42C22×D311yescharacteristic
242#107—64no—
243#107—122no—
244#106—34no—
245C4⋊C8—62no—
246#103—34no—
247#103—64no—
248#102—34no—
249#102—64no—
250#97—64no—
251C4×C8—34no—
252#95—22no—
253#95—62no—
254#95#6712yes—
255C23×C3C22×D411yescharacteristic
256C22×D3—22no—
257C22×D3—22no—
258C22×D3—22no—
259C22×D3—41no—
260C22×D3—41no—
261C22×D3—41no—
262C22×D3—41no—
263C3×Q8—22no—
264C3×Q8—22no—
265C3×Q8—22no—
266C3×Q8C2×C4⋊C414yescomplemented
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4C4×D412yes—
285C2×C3×C4C4×D412yes—
286C2×C3×C4AGL(1,ℤ/8)12yes—
287C2×C3×C4#13712yes—
288C2×C3×C4C23×C411yescharacteristic
289C2×C3×C4C22×D411yescharacteristic
290C2×C3×C4C22×Q811yescharacteristic
291C2×C3×C4C2×#4111yescharacteristic
292C2×C3×C4H(5,𝔽2)11yescharacteristic
293C2×C3×C42−1+411yescharacteristic
294C2×Dic3—22no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300D12—22no—
301D12—22no—
302D12—22no—
303D12C2×C4⋊C414yes—
304C4×D3—48no—
305C4×D3—88no—
306C3⋊C8—22no—
307C3⋊C8—42no—
308C3⋊C8C2×C4⋊C414yescomplemented
309C24—61no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8C2×C4×D312yes—
314C2×Q8D3×D412yes—
315C2×Q8D3×Q812yes—
316C2×D4—32no—
317C2×D4—32no—
318C2×D4—32no—
319C2×D4—61no—
320C2×D4—61no—
321C2×D4—61no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—61no—
326C22×C4—64no—
327C22×C4—62no—
328C22×C4—124no—
329C22×C4—124no—
330C22×C4—122no—
331C22×C4—122no—
332C22×C4—242no—
333C22×C4C2×#6712yes—
334C22×C4C2×#6712yes—
335C22×C4C2×#6711yescharacteristic
336C22×C4C2×#6711yescharacteristic
337C22×C4C23×D311yescharacteristic
338SD16—316no—
339C2×C8—32no—
340C2×C8—32no—
341C2×C8—32no—
342C2×C8—32no—
343C2×C8—61no—
344C2×C8—61no—
345C2×C8—62no—
346C2×C8—62no—
347C2×C8—62no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—32no—
355C4⋊C4—44no—
356C4⋊C4—42no—
357C4⋊C4—61no—
358C4⋊C4—62no—
359C4⋊C4—62no—
360C4⋊C4—81no—
361C4⋊C4—82no—
362C4⋊C4—124no—
363C4⋊C4—122no—
364C4⋊C4—241no—
365C4⋊C4C2×C4×D312yes—
366C22⋊C4—64no—
367C22⋊C4—124no—
368C42—24no—
369C42—22no—
370C42—22no—
371C42—44no—
372C42C2×C4×D312yes—
373C42#23512yes—
374C22×C3C2×C4×D411yescharacteristic
375C22×C3D4211yescharacteristic
376C22×C3D4×Q811yescharacteristic
377C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
378C22×C3C2×#13711yescharacteristic
379C22×C3#57611yescharacteristic
380C22×C3#57811yescharacteristic
381C2×D3—24no—
382C2×D3—42no—
383C2×D3—42no—
384C2×D3—42no—
385C2×D3—42no—
386C2×D3—42no—
387C2×D3—42no—
388C3×C4—24no—
389C3×C4—22no—
390C3×C4—42no—
391C3×C4—42no—
392C3×C4—42no—
393C3×C4—82no—
394C3×C4#44014yescomplemented
395C3×C4C22×C4⋊C411yescharacteristic
396C3×C4#52011yescharacteristic
397C3×C4#54711yescharacteristic
398C3×C4#55411yescharacteristic
399Dic3—42no—
400Dic3—82no—
401C23—62no—
402C23—62no—
403C23—62no—
404C23—121no—
405C23—121no—
406C23—121no—
407C23—121no—
408C23C22×#6711yescharacteristic
409Q8—22no—
410Q8—22no—
411Q8—22no—
412Q8D3×C4⋊C414yescomplemented
413D4—34no—
414D4—62no—
415D4—62no—
416D4—62no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—62no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—128no—
438C2×C4—122no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4—248no—
442C2×C4C4×#6712yes—
443C2×C4C4×#6712yes—
444C2×C4#94212yes—
445C2×C4#94912yes—
446C2×C4C22×C4×D311yescharacteristic
447C2×C4C2×#23511yescharacteristic
448C2×C4C2×D3×D411yescharacteristic
449C2×C4#100411yescharacteristic
450C2×C4C2×D3×Q811yescharacteristic
451C2×C4#100711yescharacteristic
452C8—34no—
453C8—62no—
454C8—122no—
455C2×C3D4×C4⋊C411yescharacteristic
456C2×C3C2×#44011yescharacteristic
457C2×C3#295111yescharacteristic
458C2×C3#327411yescharacteristic
459C2×C3#327611yescharacteristic
460C2×C3#338511yescharacteristic
461C2×C3#338611yescharacteristic
462D3—44no—
463C22—64no—
464C22—122no—
465C22—122no—
466C22—122no—
467C22—122no—
468C22—122no—
469C22—122no—
470C22C2×C4×#6711yescharacteristic
471C22D4×#6711yescharacteristic
472C22C2×#94211yescharacteristic
473C22Q8×#6711yescharacteristic
474C22C2×#94911yescharacteristic
475C22#583511yescharacteristic
476C22#583811yescharacteristic
477C4—24no—
478C4—22no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—82no—
483C4—122no—
484C4—242no—
485C4#502614yescomplemented
486C4C2×D3×C4⋊C411yescharacteristic
487C4#550411yescharacteristic
488C4#550711yescharacteristic
489C4#551111yescharacteristic
490C3#2267111yescharacteristic, complemented, Sylow
491C2—124no—
492C2C4⋊C4×#6711yescharacteristic
493C2C2×#502611yescharacteristic
494C2#8255711yescharacteristic
495C2#8356011yescharacteristic
496C2#8357811yescharacteristic
497C2#8369411yescharacteristic
498C2#8370611yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×Q8×C4⋊C4C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenterQ8×C4⋊C4D3
hypocenterC3#22671
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×Q8×C4⋊C4C2semidirect
5. #71628C2semidirect
6. #71626C2semidirect
8. #69621C2semidirect
10. #69577C2semidirect
11. #69477C2semidirect
28. C4×#215C4semidirect
46. Q8×C4⋊C4D3semidirect
62. C3×C4×Q8C2×C4semidirect
97. #835C2×C4semidirect
101. #808C2×C4semidirect
106. C4×C3⋊C8C2×C4semidirect
184. #215C4⋊C4semidirect
216. C4×Q8C4×D3semidirect
266. C3×Q8C2×C4⋊C4semidirect
308. C3⋊C8C2×C4⋊C4semidirect
394. C3×C4#440semidirect
412. Q8D3×C4⋊C4semidirect
485. C4#5026semidirect
490. C3#22671semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C4⋊C42
C2×#50262
C2×C4×#2152
C4⋊C4×D122
#716282
#716262
#696252
#696212
#695842
#695772
#694772
#226713