Finite Group Databasebeta

← #10830129

Subgroups — #10830129

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5584C211yesmaximal, characteristic
2C3×#2581C211yesmaximal, characteristic, complemented
3C2×#5031C211yesmaximal, characteristic, complemented
4C2×#5026C211yesmaximal, characteristic
5#71635C211yesmaximal, characteristic, complemented
6#69690C212yesmaximal, complemented
7#69689C212yesmaximal
8#69631C212yesmaximal, complemented
9#69628C212yesmaximal
10#69556C211yesmaximal, characteristic
11#69552C211yesmaximal, characteristic, complemented
12#24201—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#946—21no—
16C22×#216C2212yes—
17C22×#215C2211yescharacteristic
18#5584—24no—
19#5584—24no—
20C2×#898—22no—
21C2×#892—21no—
22C2×#875C2212yes—
23C2×C4×D12C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C4×Dic6C2212yes—
26C3×#400—21no—
27C3×#395—21no—
28C3×#391C2212yes—
29C3×#389C2212yes—
30C3×#387—22no—
31C3×#382C2211yescharacteristic
32C2×#835C2211yescharacteristic
33#5031—24no—
34#5026—24no—
35C2×#810C2212yes—
36C2×#810C2212yes—
37C2×#810C2212yes—
38C2×#809C2211yescharacteristic
39C2×#809C2212yes—
40C2×#807C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#556—31no—
47C2×#462—31no—
48C2×#440—31no—
49#2581D311yescharacteristic, complemented
50#2042—32no—
51#2015—32no—
52#1985—31no—
53#1960—31no—
54#1878—32no—
55#1863—32no—
56#1804—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C22×Dic6C2312yes—
62C3×C4×Q8—22no—
63C3×C4×Q8—22no—
64C3×C4×Q8—22no—
65C3×C4×Q8—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4D411yescharacteristic
69C2×C3×C4⋊C4D412yes—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C42—21no—
73C2×C3×C42C2312yes—
74C2×C3×C42C2311yescharacteristic
75#946—28no—
76C2×#216—24no—
77C2×#216—24no—
78C2×#216—42no—
79C2×#215—22no—
80C2×#215—22no—
81C2×#215—22no—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3C2311yescharacteristic
88C2×Dic3⋊C4—22no—
89C2×C4×Dic3—22no—
90C22×C3⋊C8C2311yescharacteristic
91C22×C3⋊C8C2312yes—
92C22×C3⋊C8C2311yescharacteristic
93#898—216no—
94#892—28no—
95#875—24no—
96#875—24no—
97C4×D12—22no—
98C4×D12D414yes—
99#869—22no—
100#869D414yes—
101C4×Dic6—24no—
102C4×Dic6—24no—
103C3×#95—21no—
104C3×#95—21no—
105C3×#95—22no—
106C3×#95—22no—
107C3×#95—22no—
108#835—22no—
109#810—24no—
110#810—24no—
111#810—42no—
112#809—22no—
113#809—42no—
114#807—22no—
115C4⋊(C3⋊C8)—22no—
116#803—22no—
117C22×Q16—32no—
118C22×SD16—31no—
119#556—64no—
120#556—64no—
121C2×C4⋊Q8—31no—
122C2×#126—62no—
123C2×#124—32no—
124C2×C22⋊Q8—61no—
125C2×C4×Q8—32no—
126C2×C4×Q8C2×D311yescharacteristic
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4—31no—
129C2×#117—61no—
130#462—64no—
131#440—64no—
132C2×#107—31no—
133C2×C4⋊C8—31no—
134C2×#103—31no—
135C2×#103—32no—
136C2×#103—32no—
137C2×#103—32no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#97—31no—
141#400—21no—
142#395—21no—
143#391C2×D312yes—
144#389C2×D312yes—
145#387—22no—
146#382C2×D311yescharacteristic
147#340—32no—
148#336—32no—
149#336—32no—
150C23×D3—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—42no—
158C2×C3×Q8#4112yes—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2411yescharacteristic
167C22×Dic3—21no—
168C22×Dic3—22no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12C2×D412yes—
175C2×C4×D3—24no—
176C2×C4×D3—44no—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6C2×D414yes—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—44no—
190C3×C4⋊C4—42no—
191C3×C42—22no—
192C3×C42—22no—
193C3×C42—42no—
194C3×C42C2×D412yes—
195C3×C42C2×D412yes—
196#216—48no—
197#215—28no—
198(C2×D3)⋊C4—28no—
199(C2×D3)⋊C4—28no—
200(C2×D3)⋊C4—24no—
201C4⋊Dic3—24no—
202C4⋊Dic3—24no—
203C4⋊Dic3—21no—
204C4⋊Dic3C2×D412yes—
205Dic3⋊C4—28no—
206C4×Dic3—28no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8#4112yes—
216C22×Q8—32no—
217C22×Q8#6711yescharacteristic
218C22×Q8C22×D311yescharacteristic
219C22×D4—31no—
220C23×C4—61no—
221C2×Q16—64no—
222C2×Q16—64no—
223C2×Q16—122no—
224C2×SD16—62no—
225C2×SD16—62no—
226C2×SD16—62no—
227C22×C8—31no—
228C22×C8—32no—
229C22×C8—31no—
230C4⋊Q8—34no—
231C4⋊Q8—62no—
232#126—616no—
233#124—64no—
234#124—64no—
235C22⋊Q8—68no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—64no—
241C4×Q8—64no—
242C4×D4—34no—
243C4×D4—62no—
244#117—68no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4#6711yescharacteristic
252C2×C4⋊C4#6712yes—
253C2×C4⋊C4C22×D311yescharacteristic
254C2×C4⋊C4C22×D312yes—
255C2×C22⋊C4—62no—
256C2×C22⋊C4—61no—
257C2×C22⋊C4—62no—
258C2×C42—21no—
259C2×C42—62no—
260C2×C42C22×D312yes—
261C2×C42C22×D311yescharacteristic
262#107—62no—
263C4⋊C8—62no—
264#103—64no—
265#103—64no—
266#103—62no—
267#103—122no—
268#102—62no—
269#102—122no—
270#97—62no—
271#95—21no—
272#95—21no—
273#95—22no—
274#95—22no—
275#95—22no—
276C23×C3C22×D411yescharacteristic
277C22×D3—22no—
278C22×D3—41no—
279C22×D3—41no—
280C22×D3—42no—
281C22×D3—42no—
282C3×Q8—22no—
283C3×Q8—22no—
284C3×Q8—44no—
285C3×Q8—41no—
286C3×Q8—41no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—24no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4#12012yes—
303C2×C3×C4#12012yes—
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4H(5,𝔽2)11yescharacteristic
309C2×C3×C42−1+411yescharacteristic
310C2×Dic3—22no—
311C2×Dic3—24no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315D12—22no—
316D12—22no—
317D12—41no—
318D12—41no—
319C4×D3—48no—
320Dic6—24no—
321Dic6—24no—
322Dic6—42no—
323Dic6—42no—
324C3⋊C8—22no—
325C3⋊C8—41no—
326C3⋊C8—42no—
327C24—61no—
328C2×Q8—22no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—34no—
335C2×Q8—42no—
336C2×Q8—62no—
337C2×Q8—62no—
338C2×Q8—62no—
339C2×Q8—62no—
340C2×Q8—62no—
341C2×Q8#23912yes—
342C2×D4—32no—
343C2×D4—61no—
344C2×D4—61no—
345C2×D4—61no—
346C2×D4—61no—
347C2×D4—61no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—61no—
352C22×C4—62no—
353C22×C4—64no—
354C22×C4—124no—
355C22×C4C2×#6712yes—
356C22×C4C2×#6712yes—
357C22×C4C2×#6711yescharacteristic
358C22×C4C2×#6711yescharacteristic
359C22×C4C23×D311yescharacteristic
360Q16—128no—
361SD16—68no—
362C2×C8—32no—
363C2×C8—61no—
364C2×C8—61no—
365C2×C8—61no—
366C2×C8—62no—
367C2×C8—62no—
368C2×C8—61no—
369C2×C8—62no—
370C2×C8—61no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—32no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—68no—
381C4⋊C4—64no—
382C4⋊C4—64no—
383C4⋊C4—61no—
384C22⋊C4—68no—
385C22⋊C4—68no—
386C22⋊C4—64no—
387C42—22no—
388C42—22no—
389C42—42no—
390C42—68no—
391C42D3×D412yes—
392C42D3×D412yes—
393C22×C3C2×#12011yescharacteristic
394C22×C3#54811yescharacteristic
395C22×C3#55011yescharacteristic
396C22×C3#57512yes—
397C22×C3#57812yes—
398C2×D3—42no—
399C2×D3—42no—
400C2×D3—42no—
401C2×D3—82no—
402C2×D3—82no—
403C3×C4—22no—
404C3×C4—22no—
405C3×C4—42no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—82no—
409C3×C4—81no—
410C3×C4#52511yescharacteristic
411C3×C4#53711yescharacteristic
412C3×C4#54611yescharacteristic
413C3×C4#54711yescharacteristic
414Dic3—44no—
415Dic3—42no—
416C23—62no—
417C23—121no—
418C23—122no—
419C23—121no—
420C23—122no—
421C23C22×#6711yescharacteristic
422Q8—22no—
423Q8—22no—
424Q8—44no—
425Q8—41no—
426Q8—41no—
427Q8—64no—
428Q8—64no—
429Q8—122no—
430Q8—122no—
431D4—62no—
432D4—62no—
433D4—121no—
434D4—121no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—62no—
451C2×C4—64no—
452C2×C4—128no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4#93712yes—
457C2×C4#93712yes—
458C2×C4C2×D3×D411yescharacteristic
459C2×C4C2×D3×D411yescharacteristic
460C2×C4C2×#23911yescharacteristic
461C2×C4#100711yescharacteristic
462C2×C4D3×#4111yescharacteristic
463C2×C4#100911yescharacteristic
464C8—62no—
465C8—121no—
466C8—122no—
467C2×C3#240711yescharacteristic
468C2×C3#300511yescharacteristic
469C2×C3#301111yescharacteristic
470C2×C3#337712yes—
471C2×C3#340212yes—
472D3—82no—
473C22—122no—
474C22—122no—
475C22—122no—
476C22—242no—
477C22—242no—
478C22C2×#93711yescharacteristic
479C22#581711yescharacteristic
480C22#582212yes—
481C22#583411yescharacteristic
482C22#583812yes—
483C4—22no—
484C4—22no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—82no—
489C4—81no—
490C4—124no—
491C4—122no—
492C4#562711yescharacteristic
493C4#562911yescharacteristic
494C4#563011yescharacteristic
495C4#563111yescharacteristic
496C3#2420111yescharacteristic, complemented, Sylow
497C2—242no—
498C2#8281211yescharacteristic
499C2#8292811yescharacteristic
500C2#8341111yescharacteristic
501C2#8368112yes—
502C2#8370012yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2581C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2581D3
hypocenterC3#24201
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2581

Complements

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

NH ≅ G/Ntype
2. C3×#2581C2semidirect
3. C2×#5031C2semidirect
5. #71635C2semidirect
6. #69690C2semidirect
8. #69631C2semidirect
11. #69552C2semidirect
49. #2581D3semidirect
496. C3#24201semidirect

Maximal subgroups

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

subgroupindex
C2×#55842
C3×#25812
C2×#50312
C2×#50262
#716352
#696902
#696892
#696312
#696282
#695562
#695522
#242013