Finite Group Databasebeta

← #10830128

Subgroups — #10830128

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×#5018C211yesmaximal, characteristic
3C2×#4849C211yesmaximal, characteristic, complemented
4#74191C211yesmaximal, characteristic, complemented
5C3×#1804C211yesmaximal, characteristic
6#70473C211yesmaximal, characteristic, complemented
7#70215C212yesmaximal
8#70178C212yesmaximal
9#69552C211yesmaximal, characteristic, complemented
10#68908C212yesmaximal, complemented
11#68906C212yesmaximal, complemented
12#24201—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#936C2212yes—
15C22×#214C2211yescharacteristic
16C2×#926—21no—
17C2×#924—21no—
18C22×Dic12C2212yes—
19#5542—24no—
20#5542—24no—
21C2×#887C2212yes—
22C2×#879—22no—
23C2×#869C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#102C2212yes—
28C3×#382C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#816C2212yes—
31#5018—24no—
32#4984C2212yes—
33#4973C2212yes—
34C2×#810C2212yes—
35C2×#810C2211yescharacteristic
36C2×#810C2212yes—
37C2×#807C2211yescharacteristic
38C2×#803C2211yescharacteristic
39#4849—24no—
40#4659—21no—
41#4656—21no—
42#4653—22no—
43C3×#340C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#556—31no—
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#1804D311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C22×Dic6D411yescharacteristic
60C22×Dic6C2312yes—
61C22×Dic6C2311yescharacteristic
62C22×C3×C8C2312yes—
63C3×C4×D4—22no—
64C3×C4×D4D414yes—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C22⋊C4—21no—
68C2×C3×C42C2311yescharacteristic
69C2×#217—22no—
70C2×#217—22no—
71#936—24no—
72#936—24no—
73C2×#214—22no—
74C2×#214—22no—
75C2×#214—22no—
76#926—28no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3D411yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82#924—28no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—21no—
85C2×Dic3⋊C4D412yes—
86C2×C4×Dic3—22no—
87C2×C4×Dic3—21no—
88C2×C4×Dic3C2312yes—
89C22×C3⋊C8C2311yescharacteristic
90C22×C3⋊C8C2311yescharacteristic
91C2×Dic12—24no—
92C2×Dic12—24no—
93C2×Dic12—42no—
94#887—24no—
95#887—24no—
96#879—216no—
97#869—22no—
98#869D414yes—
99C4×Dic6—22no—
100C4×Dic6—22no—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C3×C4⋊C8—22no—
104C3×#102—42no—
105C3×#95—21no—
106C3×#95—21no—
107#832—22no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111#816—24no—
112#810—22no—
113#810—24no—
114#810—42no—
115#807—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—31no—
126C2×C4×Q8—32no—
127C2×C4×Q8—31no—
128C2×C4×D4C2×D311yescharacteristic
129C2×#117—61no—
130#462—64no—
131#440—64no—
132C2×#107—31no—
133C2×C4⋊C8C2×D311yescharacteristic
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—31no—
137C2×#103—32no—
138C2×#102—31no—
139C2×#102C2×D312yes—
140C2×#97—31no—
141#400—61no—
142#395—61no—
143#391—32no—
144#389—32no—
145#387—62no—
146#382C2×D311yescharacteristic
147#340C2×D312yes—
148#336—32no—
149#336—32no—
150C24×C3—21no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4C2×D412yes—
157C22×C3×C4—21no—
158C22×C3×C4—24no—
159C22×C3×C4—22no—
160C22×C3×C4—44no—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C22×Dic3—21no—
165C22×Dic3—22no—
166C22×Dic3—21no—
167C22×Dic3C2×D412yes—
168C22×Dic3C2×D412yes—
169C2×Dic6—21no—
170C2×Dic6—21no—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×Dic6—21no—
174C2×Dic6—22no—
175C2×Dic6—21no—
176C2×Dic6—22no—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—21no—
180C2×Dic6—42no—
181C2×Dic6C2×D414yes—
182C2×Dic6#4112yes—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4C2×D412yes—
190C3×C22⋊C4—24no—
191C3×C42C2×D412yes—
192C3×C42C2×D412yes—
193#217—28no—
194#217—28no—
195#214—28no—
196C4⋊Dic3—22no—
197C4⋊Dic3—24no—
198C4⋊Dic3—24no—
199C4⋊Dic3—21no—
200C4⋊Dic3—21no—
201C4⋊Dic3—44no—
202Dic3⋊C4—28no—
203Dic3⋊C4—24no—
204Dic3⋊C4—42no—
205C4×Dic3—28no—
206C4×Dic3—22no—
207C4×Dic3—22no—
208C4×Dic3—42no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8#4112yes—
215Dic12—48no—
216C22×Q8—31no—
217C22×Q8—32no—
218C22×Q8—31no—
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—31no—
229C22×C8C22×D312yes—
230C4⋊Q8—34no—
231C4⋊Q8—62no—
232#126—616no—
233#124—64no—
234#124—64no—
235C22⋊Q8—68no—
236C4×Q8—64no—
237C4×Q8—64no—
238C4×Q8—62no—
239C4×Q8—62no—
240C4×Q8—62no—
241C4×Q8—62no—
242C4×D4—22no—
243C4×D4D1214yes—
244#117—68no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—32no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4C22×D312yes—
254C2×C4⋊C4C22×D311yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—62no—
257C2×C22⋊C4—62no—
258C2×C42—32no—
259C2×C42—62no—
260C2×C42—61no—
261C2×C42C22×D311yescharacteristic
262#107—62no—
263C4⋊C8—22no—
264#103—64no—
265#103—64no—
266#103—62no—
267#103—122no—
268#102—42no—
269#102—62no—
270#97—62no—
271#95—21no—
272#95—21no—
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×D4—22no—
283C3×D4—22no—
284C3×D4—41no—
285C3×D4—41no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—48no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4#12012yes—
295C2×C3×C4#12012yes—
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C22×D411yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4C2×#4111yescharacteristic
300C2×C3×C4H(5,𝔽2)11yescharacteristic
301C2×C3×C42−1+411yescharacteristic
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—24no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—24no—
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—42no—
315Dic6—22no—
316Dic6—22no—
317Dic6—24no—
318Dic6—24no—
319Dic6—44no—
320Dic6—42no—
321Dic6—41no—
322Dic6—42no—
323Dic6—41no—
324C3×C8—42no—
325C3⋊C8—22no—
326C3⋊C8—41no—
327C24—21no—
328C2×Q8—34no—
329C2×Q8—32no—
330C2×Q8—62no—
331C2×Q8—62no—
332C2×Q8—61no—
333C2×Q8—61no—
334C2×Q8—61no—
335C2×Q8—62no—
336C2×Q8—61no—
337C2×Q8—62no—
338C2×Q8—62no—
339C2×Q8—61no—
340C2×Q8—62no—
341C2×Q8—122no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4C2×D1212yes—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—32no—
352C22×C4—32no—
353C22×C4—44no—
354C22×C4—61no—
355C22×C4—62no—
356C22×C4—61no—
357C22×C4D3×D411yescharacteristic
358C22×C4D3×D411yescharacteristic
359C22×C4C23×D311yescharacteristic
360Q16—128no—
361SD16—68no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—32no—
366C2×C8—61no—
367C2×C8—61no—
368C2×C8—61no—
369C2×C8—61no—
370C2×C8—61no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—68no—
375C4⋊C4—62no—
376C4⋊C4—64no—
377C4⋊C4—64no—
378C4⋊C4—64no—
379C4⋊C4—61no—
380C4⋊C4—61no—
381C4⋊C4—124no—
382C4⋊C4—122no—
383C4⋊C4C2×D1212yes—
384C22⋊C4—24no—
385C22⋊C4—68no—
386C22⋊C4—68no—
387C42—68no—
388C42—62no—
389C42—62no—
390C42—122no—
391C42C2×D1212yes—
392C42D3×D412yes—
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—82no—
407C3×C4#52511yescharacteristic
408C3×C4#53711yescharacteristic
409C3×C4#54611yescharacteristic
410C3×C4#54711yescharacteristic
411Dic3—42no—
412Dic3—44no—
413Dic3—42no—
414Dic3—42no—
415Dic3—81no—
416C23—22no—
417C23—41no—
418C23—42no—
419C23—42no—
420C23—41no—
421C23C2×D3×D411yescharacteristic
422Q8—62no—
423Q8—62no—
424Q8—64no—
425Q8—64no—
426Q8—124no—
427Q8—122no—
428Q8—121no—
429Q8—122no—
430Q8—121no—
431D4—22no—
432D4—22no—
433D4—41no—
434D4—41no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—48no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—64no—
447C2×C4—64no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—122no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—122no—
454C2×C4—121no—
455C2×C4—122no—
456C2×C4#88212yes—
457C2×C4#88212yes—
458C2×C4C22×D1211yescharacteristic
459C2×C4C2×D3×D411yescharacteristic
460C2×C4#100411yescharacteristic
461C2×C4#100711yescharacteristic
462C2×C4D3×#4111yescharacteristic
463C2×C4D3×#4111yescharacteristic
464C8—42no—
465C8—62no—
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—
478C22C2×#88211yescharacteristic
479C22#559911yescharacteristic
480C22#563311yescharacteristic
481C22#577112yes—
482C22#577312yes—
483C4—22no—
484C4—22no—
485C4—42no—
486C4—82no—
487C4—122no—
488C4—124no—
489C4—122no—
490C4—122no—
491C4—241no—
492C4#551011yescharacteristic
493C4#552411yescharacteristic
494C4#561211yescharacteristic
495C4#563111yescharacteristic
496C3#2420111yescharacteristic, complemented, Sylow
497C2—82no—
498C2#7401511yescharacteristic
499C2#7883711yescharacteristic
500C2#7892711yescharacteristic
501C2#8052912yes—
502C2#8115512yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1804C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1804D3
hypocenterC3#24201
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
3. C2×#4849C2semidirect
4. #74191C2semidirect
6. #70473C2semidirect
9. #69552C2semidirect
10. #68908C2semidirect
11. #68906C2semidirect
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×#50182
C2×#48492
#741912
C3×#18042
#704732
#702152
#701782
#695522
#689082
#689062
#242013