Finite Group Databasebeta

← #10830121

Subgroups — #10830121

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×C3×#556C211yesmaximal, characteristic
2C2×#5090C211yesmaximal, characteristic, complemented
3#82480C211yesmaximal, characteristic, complemented
4C2×#5018C211yesmaximal, characteristic
5#71786C211yesmaximal, characteristic, complemented
6#71729C212yesmaximal
7#71722C212yesmaximal
8#71635C211yesmaximal, characteristic, complemented
9#71632C212yesmaximal, complemented
10#71628C212yesmaximal, complemented
11#71605C211yesmaximal, characteristic
12#24201—31nomaximal, Sylow
13C3×#556—24no—
14C3×#556—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—22no—
17C2×C3×#124C2212yes—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C4×Q8C2212yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×#117—21no—
23C22×#216C2212yes—
24C22×#214C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26#5230—21no—
27C2×#835C2212yes—
28C2×#835C2212yes—
29C2×#835C2212yes—
30C2×#832C2211yescharacteristic
31C2×#832C2212yes—
32#5090—24no—
33#5018—24no—
34#4985—21no—
35#4978—22no—
36C2×#810C2211yescharacteristic
37C2×#807C2211yescharacteristic
38#4936C2211yescharacteristic
39#4929C2212yes—
40#4928C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#556D311yescharacteristic
47C2×#462—31no—
48C2×#440—31no—
49#2581—31no—
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×Q8C2312yes—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C22×Dic6C2311yescharacteristic
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8D414yes—
64C3×#126—216no—
65C3×#124—24no—
66C3×#124—24no—
67C3×C22⋊Q8—28no—
68C3×C4×Q8—24no—
69C3×C4×Q8—24no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×D4—22no—
73C3×C4×D4D414yes—
74C3×#117—28no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4D412yes—
79C2×C3×C4⋊C4D411yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C42—22no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C2×#216—24no—
88C2×#216—24no—
89C2×#216—42no—
90C2×#214—22no—
91C2×#214—22no—
92C2×#214—22no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2312yes—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—21no—
97C2×C4×Dic3—21no—
98C22×C3⋊C8C2311yescharacteristic
99C22×C3⋊C8C2312yes—
100C22×C3⋊C8C2311yescharacteristic
101C4×Dic6—22no—
102C4×Dic6—22no—
103#835—24no—
104#835—24no—
105#835—42no—
106#832—22no—
107#832—42no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113#810—22no—
114#807—22no—
115C4⋊(C3⋊C8)—22no—
116#803—22no—
117C22×Q16—32no—
118C22×SD16—31no—
119#556—24no—
120#556—24no—
121C2×C4⋊Q8C2×D311yescharacteristic
122C2×#126—22no—
123C2×#124C2×D312yes—
124C2×C22⋊Q8—21no—
125C2×C4×Q8—31no—
126C2×C4×Q8C2×D312yes—
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4C2×D311yescharacteristic
129C2×#117—21no—
130#462—64no—
131#440—64no—
132C2×#107—31no—
133C2×C4⋊C8—31no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—31no—
137C2×#103—32no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#97—31no—
141#400—61no—
142#395—61no—
143#391—32no—
144#389—32no—
145#387—62no—
146#382—31no—
147#340—32no—
148#336—32no—
149#336—32no—
150C24×C3—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—42no—
158C2×C3×Q8C2×D414yes—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4C2×D412yes—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—24no—
168C22×C3×C4—44no—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C22×Dic3—21no—
175C22×Dic3—22no—
176C22×Dic3—21no—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6#4112yes—
183C3×C4⋊C4—28no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4C2×D412yes—
190C3×C22⋊C4—28no—
191C3×C22⋊C4—28no—
192C3×C22⋊C4—24no—
193C3×C42—28no—
194C3×C42—22no—
195C3×C42—22no—
196C3×C42C2×D412yes—
197C3×C42C2×D412yes—
198#216—48no—
199#214—28no—
200C4⋊Dic3—21no—
201C4⋊Dic3—21no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204Dic3⋊C4—44no—
205Dic3⋊C4—42no—
206C4×Dic3—42no—
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—31no—
217C22×Q8#6711yescharacteristic
218C22×Q8C22×D312yes—
219C22×D4C22×D311yescharacteristic
220C23×C4—21no—
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—22no—
231C4⋊Q8#6714yes—
232#126—216no—
233#124—24no—
234#124—24no—
235C22⋊Q8—28no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—62no—
241C4×Q8—62no—
242C4×D4—22no—
243C4×D4#6714yes—
244#117—28no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4#6711yescharacteristic
253C2×C4⋊C4#6712yes—
254C2×C4⋊C4C22×D311yescharacteristic
255C2×C22⋊C4—22no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—22no—
258C2×C42—22no—
259C2×C42—61no—
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—61no—
272#95—61no—
273#95—62no—
274#95—62no—
275#95—62no—
276C23×C3—22no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3C22×D411yescharacteristic
282C3×Q8—24no—
283C3×Q8—24no—
284C3×Q8—44no—
285C3×Q8—42no—
286C3×Q8—42no—
287C3×D4—22no—
288C3×D4—22no—
289C3×D4—41no—
290C3×D4—41no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—24no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—48no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4#12012yes—
305C2×C3×C4#12012yes—
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4C2×#4111yescharacteristic
310C2×C3×C4H(5,𝔽2)11yescharacteristic
311C2×C3×C42−1+411yescharacteristic
312C2×Dic3—22no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320Dic6—22no—
321Dic6—22no—
322Dic6—41no—
323Dic6—41no—
324C3⋊C8—22no—
325C3⋊C8—41no—
326C3⋊C8—42no—
327C24—21no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—32no—
335C2×Q8—42no—
336C2×Q8—61no—
337C2×Q8—61no—
338C2×Q8—61no—
339C2×Q8—61no—
340C2×Q8—61no—
341C2×Q8C2×#6714yes—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4C2×#6712yes—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—44no—
352C22×C4—61no—
353C22×C4—62no—
354C22×C4—61no—
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—62no—
366C2×C8—62no—
367C2×C8—61no—
368C2×C8—61no—
369C2×C8—61no—
370C2×C8—62no—
371C4⋊C4—28no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—21no—
377C4⋊C4—62no—
378C4⋊C4—62no—
379C4⋊C4—61no—
380C4⋊C4—61no—
381C4⋊C4—124no—
382C4⋊C4—122no—
383C4⋊C4C2×#6712yes—
384C22⋊C4—28no—
385C22⋊C4—28no—
386C22⋊C4—24no—
387C42—28no—
388C42—22no—
389C42—22no—
390C42—122no—
391C42C2×#6712yes—
392C42C2×#6712yes—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—82no—
397C22×C3—82no—
398C22×C3C2×#12011yescharacteristic
399C22×C3#54811yescharacteristic
400C22×C3#55011yescharacteristic
401C22×C3#57512yes—
402C22×C3#57812yes—
403C3×C4—22no—
404C3×C4—22no—
405C3×C4—44no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—42no—
409C3×C4#52511yescharacteristic
410C3×C4#53711yescharacteristic
411C3×C4#54611yescharacteristic
412C3×C4#54711yescharacteristic
413Dic3—42no—
414Dic3—82no—
415Dic3—81no—
416C23—22no—
417C23—42no—
418C23—41no—
419C23—41no—
420C23—42no—
421C23C22×#6711yescharacteristic
422Q8—24no—
423Q8—24no—
424Q8—44no—
425Q8—42no—
426Q8—42no—
427Q8—62no—
428Q8—62no—
429Q8—121no—
430Q8—121no—
431D4—22no—
432D4—22no—
433D4—41no—
434D4—41no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—48no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—62no—
449C2×C4—121no—
450C2×C4—121no—
451C2×C4—122no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—121no—
456C2×C4#95312yes—
457C2×C4#95312yes—
458C2×C4D3×#4111yescharacteristic
459C2×C4D3×#4111yescharacteristic
460C2×C4#100911yescharacteristic
461C2×C4#101011yescharacteristic
462C2×C4C22×#6711yescharacteristic
463C2×C4C22×#6711yescharacteristic
464C8—62no—
465C8—122no—
466C8—121no—
467C2×C3—82no—
468C2×C3#240711yescharacteristic
469C2×C3#300511yescharacteristic
470C2×C3#301111yescharacteristic
471C2×C3#337712yes—
472C2×C3#340212yes—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—82no—
477C22—82no—
478C22#582212yes—
479C22#583111yescharacteristic
480C22#583411yescharacteristic
481C22#583812yes—
482C22C2×#95311yescharacteristic
483C4—22no—
484C4—22no—
485C4—42no—
486C4—42no—
487C4—44no—
488C4—42no—
489C4—122no—
490C4—241no—
491C4—242no—
492C4#582411yescharacteristic
493C4#582811yescharacteristic
494C4#582911yescharacteristic
495C4#583311yescharacteristic
496C3#2420111yescharacteristic, complemented, Sylow
497C2—82no—
498C2#8367812yes—
499C2#8368112yes—
500C2#8408111yescharacteristic
501C2#8409711yescharacteristic
502C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#556C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#556D3
hypocenterC3#24201
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. C2×#5090C2semidirect
3. #82480C2semidirect
5. #71786C2semidirect
8. #71635C2semidirect
9. #71632C2semidirect
10. #71628C2semidirect
496. C3#24201semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×#50902
#824802
C2×#50182
#717862
#717292
#717222
#716352
#716322
#716282
#716052
#242013