Finite Group Databasebeta

← #10830127

Subgroups — #10830127

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×#5542C211yesmaximal, characteristic
2C2×C3×#440C211yesmaximal, characteristic
3#82481C211yesmaximal, characteristic, complemented
4C2×#5031C211yesmaximal, characteristic, complemented
5#72058C212yesmaximal
6#71605C211yesmaximal, characteristic
7#70642C212yesmaximal, complemented
8#70281C212yesmaximal, complemented
9#70216C212yesmaximal
10#69838C211yesmaximal, characteristic, complemented
11#69822C211yesmaximal, characteristic, complemented
12#24201—31nomaximal, Sylow
13C22×C3×SD16C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C22×#216C2212yes—
17C2×#936C2212yes—
18C2×#926—21no—
19C2×#924—21no—
20#5542—24no—
21#5542—24no—
22C2×#887C2212yes—
23C2×#879—22no—
24C2×#869C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C3×#440—24no—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31#5229C2212yes—
32C2×#832C2212yes—
33C2×#816C2212yes—
34C2×#816C2212yes—
35#5031—24no—
36#4986—21no—
37#4979—21no—
38#4972C2212yes—
39C2×#810C2212yes—
40#4936C2211yescharacteristic
41#4933—22no—
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#556—31no—
47C2×#462—31no—
48C2×#440D311yescharacteristic
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×Q8C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×Dic6D411yescharacteristic
61C22×Dic6C2312yes—
62C2×C3×SD16—22no—
63C2×C3×SD16—22no—
64C2×C3×SD16—22no—
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C3×C4×D4—22no—
70C3×C4×D4D414yes—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×C3×C42—21no—
76C2×C3×C42C2311yescharacteristic
77C2×#217—22no—
78C2×#217—22no—
79C2×#216—24no—
80C2×#216—24no—
81C2×#216—42no—
82#936—24no—
83#936—24no—
84#926—28no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3D411yescharacteristic
88C2×C4⋊Dic3C2312yes—
89#924—28no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4D412yes—
93C2×C4×Dic3—22no—
94C2×C4×Dic3C2312yes—
95C22×C3⋊C8C2312yes—
96#887—24no—
97#887—24no—
98#879—216no—
99#869—22no—
100#869D414yes—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C3×#107—22no—
104C3×#103—22no—
105C3×#102—22no—
106C3×#97—22no—
107#832—42no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113#816—24no—
114#816—42no—
115#810—24no—
116C4⋊(C3⋊C8)—22no—
117C22×Q16—32no—
118C22×SD16C2×D311yescharacteristic
119#556—64no—
120#556—64no—
121C2×C4⋊Q8—31no—
122C2×#126—62no—
123C2×#124—32no—
124C2×C22⋊Q8—61no—
125C2×C4×Q8—31no—
126C2×C4×Q8—32no—
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4C2×D311yescharacteristic
129C2×#117—61no—
130#462—64no—
131#440—24no—
132C2×#107C2×D311yescharacteristic
133C2×C4⋊C8—31no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—32no—
137C2×#103C2×D311yescharacteristic
138C2×#102—32no—
139C2×#102C2×D311yescharacteristic
140C2×#97C2×D311yescharacteristic
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—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8#4112yes—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4C2×D412yes—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—24no—
166C22×C3×C4—21no—
167C22×C3×C4—44no—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2411yescharacteristic
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3C2×D412yes—
174C22×Dic3C2×D412yes—
175C2×Dic6—22no—
176C2×Dic6—22no—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—42no—
182C2×Dic6C2×D414yes—
183C3×SD16—28no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8#4112yes—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—42no—
194C3×C4⋊C4C2×D412yes—
195C3×C22⋊C4—24no—
196C3×C42—42no—
197C3×C42C2×D412yes—
198C3×C42C2×D412yes—
199#217—28no—
200#217—28no—
201#216—48no—
202C4⋊Dic3—22no—
203C4⋊Dic3—24no—
204C4⋊Dic3—24no—
205C4⋊Dic3—22no—
206C4⋊Dic3—22no—
207Dic3⋊C4—28no—
208Dic3⋊C4—24no—
209Dic3⋊C4—44no—
210C4×Dic3—28no—
211C4×Dic3—22no—
212C4×Dic3—22no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C22×Q8—31no—
217C22×Q8—32no—
218C22×Q8C22×D311yescharacteristic
219C22×D4C22×D311yescharacteristic
220C23×C4—21no—
221C2×Q16—64no—
222C2×Q16—64no—
223C2×Q16—122no—
224C2×SD16—22no—
225C2×SD16—22no—
226C2×SD16—22no—
227C22×C8—32no—
228C22×C8C22×D311yescharacteristic
229C22×C8C22×D311yescharacteristic
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—64no—
239C4×Q8—64no—
240C4×Q8—62no—
241C4×Q8—62no—
242C4×D4—22no—
243C4×D4#6714yes—
244#117—68no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—31no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4C22×D311yescharacteristic
254C2×C4⋊C4C22×D311yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—62no—
257C2×C22⋊C4—62no—
258C2×C42—21no—
259C2×C42—32no—
260C2×C42—62no—
261C2×C42C22×D311yescharacteristic
262#107—22no—
263C4⋊C8—62no—
264#103—22no—
265#103—64no—
266#103—64no—
267#103—122no—
268#102—22no—
269#102—122no—
270#97—22no—
271#95—62no—
272#95—62no—
273#95—62no—
274#95—61no—
275#95—61no—
276C23×C3—22no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3C22×D411yescharacteristic
282C3×Q8—22no—
283C3×Q8—22no—
284C3×Q8—41no—
285C3×Q8—41no—
286C3×D4—22no—
287C3×D4—22no—
288C3×D4—41no—
289C3×D4—41no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—48no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4#12012yes—
301C2×C3×C4#12012yes—
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C4C2×#4111yescharacteristic
306C2×C3×C4H(5,𝔽2)11yescharacteristic
307C2×C3×C42−1+411yescharacteristic
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—24no—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—24no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319Dic6—24no—
320Dic6—24no—
321Dic6—44no—
322Dic6—42no—
323Dic6—42no—
324C3×C8—22no—
325C3×C8—41no—
326C3⋊C8—42no—
327C24—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—34no—
334C2×Q8—62no—
335C2×Q8—62no—
336C2×Q8—62no—
337C2×Q8—62no—
338C2×Q8—62no—
339C2×Q8—62no—
340C2×Q8—122no—
341C2×Q8#23512yes—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4C2×#6712yes—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—32no—
353C22×C4—32no—
354C22×C4—44no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4D3×D411yescharacteristic
358C22×C4D3×D411yescharacteristic
359C22×C4C23×D311yescharacteristic
360Q16—128no—
361SD16—28no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—62no—
368C2×C8—62no—
369C2×C8—62no—
370C2×C8#23512yes—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—42no—
375C4⋊C4—68no—
376C4⋊C4—62no—
377C4⋊C4—64no—
378C4⋊C4—64no—
379C4⋊C4—64no—
380C4⋊C4—62no—
381C4⋊C4—62no—
382C4⋊C4—124no—
383C4⋊C4C2×#6712yes—
384C22⋊C4—24no—
385C22⋊C4—68no—
386C22⋊C4—68no—
387C42—42no—
388C42—68no—
389C42—62no—
390C42—62no—
391C42D3×D412yes—
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—42no—
406C3×C4—42no—
407C3×C4—81no—
408C3×C4#52511yescharacteristic
409C3×C4#53711yescharacteristic
410C3×C4#54611yescharacteristic
411C3×C4#54711yescharacteristic
412Dic3—44no—
413Dic3—42no—
414Dic3—42no—
415Dic3—82no—
416C23—22no—
417C23—42no—
418C23—42no—
419C23—41no—
420C23—41no—
421C23C2×D3×D411yescharacteristic
422Q8—22no—
423Q8—22no—
424Q8—41no—
425Q8—41no—
426Q8—64no—
427Q8—64no—
428Q8—124no—
429Q8—122no—
430Q8—122no—
431D4—22no—
432D4—22no—
433D4—41no—
434D4—41no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—48no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—64no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—64no—
451C2×C4—122no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4#93712yes—
457C2×C4#93712yes—
458C2×C4C2×#23511yescharacteristic
459C2×C4C2×#23511yescharacteristic
460C2×C4C2×D3×D411yescharacteristic
461C2×C4#100911yescharacteristic
462C2×C4#101011yescharacteristic
463C2×C4C22×#6711yescharacteristic
464C8—22no—
465C8—41no—
466C8—122no—
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#555211yescharacteristic
479C22#555311yescharacteristic
480C22#577312yes—
481C22#577912yes—
482C22C2×#93711yescharacteristic
483C4—22no—
484C4—22no—
485C4—42no—
486C4—42no—
487C4—81no—
488C4—122no—
489C4—124no—
490C4—122no—
491C4—242no—
492C4#555011yescharacteristic
493C4#555111yescharacteristic
494C4#579111yescharacteristic
495C4#583311yescharacteristic
496C3#2420111yescharacteristic, complemented, Sylow
497C2—82no—
498C2#7975812yes—
499C2#7976512yes—
500C2#8240211yescharacteristic
501C2#8394511yescharacteristic
502C2#8398311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#440C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#440D3
hypocenterC3#24201
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
3. #82481C2semidirect
4. C2×#5031C2semidirect
7. #70642C2semidirect
8. #70281C2semidirect
10. #69838C2semidirect
11. #69822C2semidirect
496. C3#24201semidirect

Maximal subgroups

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

subgroupindex
C2×#55422
C2×C3×#4402
#824812
C2×#50312
#720582
#716052
#706422
#702812
#702162
#698382
#698222
#242013