Finite Group Databasebeta

← #10845325

Subgroups — #10845325

Order 768.

894 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C8×D4C211yesmaximal, characteristic
2#82382C211yesmaximal, characteristic
3#82376C211yesmaximal, characteristic
4C2×#4851C211yesmaximal, characteristic, complemented
5#71972C212yesmaximal, complemented
6#71970C212yesmaximal, complemented
7#70454C211yesmaximal, characteristic, complemented
8#69940C212yesmaximal, complemented
9#69936C212yesmaximal, complemented
10#69777C211yesmaximal, characteristic, complemented
11C4×#816C211yesmaximal, characteristic, complemented
12#25341—31nomaximal, Sylow
13C23×C3×C8C2212yes—
14C2×C3×C4×D4C2211yescharacteristic
15C2×#936—21no—
16C2×#924C2212yes—
17C2×#869C2211yescharacteristic
18C2×C4×Dic6C2211yescharacteristic
19C3×C8×D4—24no—
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C22⋊C8C2212yes—
22C2×C3×C4×C8C2211yescharacteristic
23#5221—21no—
24#5212C2212yes—
25#5212C2212yes—
26#5212C2212yes—
27C2×#817C2212yes—
28C2×#817C2212yes—
29C2×#816C2211yescharacteristic
30C2×#816C2212yes—
31C2×#816C2211yescharacteristic
32#4973—21no—
33#4972—21no—
34#4953—22no—
35#4948—22no—
36#4935C2211yescharacteristic
37C4×C4⋊Dic3C2211yescharacteristic
38#4851—24no—
39#4551C2211yescharacteristic
40#4551C2212yes—
41#4551C2212yes—
42#4551C2212yes—
43#4551C2211yescharacteristic
44C2×#475—31no—
45C2×C8×D4D311yescharacteristic
46#2547—31no—
47#2538—31no—
48#2077—32no—
49#2035—32no—
50#1990—31no—
51#1948—31no—
52#1892—32no—
53#1807—32no—
54C4×#103—31no—
55C22×C3×D4D411yescharacteristic
56C23×C3×C4C2312yes—
57C22×Dic6—21no—
58C22×Dic6C2311yescharacteristic
59C22×C3×C8—24no—
60C22×C3×C8—24no—
61C22×C3×C8—42no—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2312yes—
64C22×C3×C8C2311yescharacteristic
65C3×C4×D4—22no—
66C3×C4×D4—22no—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C22⋊C4D412yes—
69C2×C3×C42C2311yescharacteristic
70C2×#217—22no—
71C2×#217—22no—
72C2×#217—22no—
73C2×#217—22no—
74#936—44no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3C2311yescharacteristic
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4⋊Dic3C2312yes—
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2312yes—
83#924—24no—
84#924—42no—
85C2×Dic3⋊C4—21no—
86C2×Dic3⋊C4—22no—
87C2×C4×Dic3—21no—
88C2×C4×Dic3—21no—
89#869—22no—
90#869—22no—
91C4×Dic6—22no—
92C4×Dic6—22no—
93C3×C4⋊C8D414yes—
94C3×C22⋊C8—24no—
95C3×C4×C8—22no—
96C3⋊#95—21no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100#817—24no—
101#817—42no—
102#816—22no—
103#816—24no—
104#816—22no—
105C23×C8C2×D312yes—
106C2×C4⋊Q8—31no—
107C2×#124—61no—
108C2×C22⋊Q8—32no—
109C2×C4×Q8—31no—
110C2×C4×D4C2×D311yescharacteristic
111#475—64no—
112C8×D4—24no—
113C2×#106—32no—
114C2×#106—32no—
115C2×C4⋊C8C2×D311yescharacteristic
116C2×#103—32no—
117C2×#103—31no—
118C2×#103—31no—
119C2×C22⋊C8C2×D312yes—
120C2×C4×C8C2×D311yescharacteristic
121#397—62no—
122#396—62no—
123#391—61no—
124#390—61no—
125#389—61no—
126#384—31no—
127#380—32no—
128#380—32no—
129#380—32no—
130C4×C4⋊C4—31no—
131#340—31no—
132#340—32no—
133#340—31no—
134#340—32no—
135#340—32no—
136C24×C3C2×D412yes—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4SD1614yes—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2411yescharacteristic
150C22×Dic3—21no—
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—21no—
154C22×Dic3—22no—
155C22×Dic3—22no—
156C2×Dic6—21no—
157C2×Dic6—21no—
158C2×Dic6—21no—
159C2×Dic6—21no—
160C2×Dic6—21no—
161C2×Dic6—42no—
162C2×Dic6—42no—
163C2×Dic6—42no—
164C2×Dic6#4112yes—
165C2×C3×C8—21no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—21no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8—21no—
172C2×C3×C8—48no—
173C2×C3×C8C2×D414yes—
174C2×C3×C8#4112yes—
175C3×C4⋊C4—22no—
176C3×C22⋊C4—24no—
177C3×C42C2×D412yes—
178C3×C42#4112yes—
179#217—44no—
180#217—44no—
181#217—44no—
182#217—82no—
183C4⋊Dic3—22no—
184C4⋊Dic3—21no—
185C4⋊Dic3—21no—
186C4⋊Dic3—21no—
187C4⋊Dic3—21no—
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3—21no—
191C4⋊Dic3—22no—
192C4⋊Dic3—21no—
193C4⋊Dic3—42no—
194C4⋊Dic3—42no—
195C4⋊Dic3—81no—
196Dic3⋊C4—44no—
197Dic3⋊C4—42no—
198C4×Dic3—42no—
199C4×Dic3—42no—
200C22×Q8—31no—
201C22×Q8—61no—
202C22×D4D1211yescharacteristic
203C23×C4C22×D312yes—
204C22×C8—24no—
205C22×C8—24no—
206C22×C8—42no—
207C22×C8C22×D311yescharacteristic
208C22×C8C22×D311yescharacteristic
209C22×C8C22×D312yes—
210C4⋊Q8—62no—
211C4⋊Q8—62no—
212#124—124no—
213C22⋊Q8—64no—
214C22⋊Q8—122no—
215C4×Q8—62no—
216C4×Q8—62no—
217C4×D4—22no—
218C4×D4—22no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—32no—
221C2×C4⋊C4—32no—
222C2×C4⋊C4—31no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—62no—
226C2×C4⋊C4—61no—
227C2×C4⋊C4—61no—
228C2×C4⋊C4—61no—
229C2×C4⋊C4D1211yescharacteristic
230C2×C22⋊C4—62no—
231C2×C22⋊C4—62no—
232C2×C22⋊C4—62no—
233C2×C22⋊C4—62no—
234C2×C22⋊C4D1212yes—
235C2×C42—61no—
236C2×C42—61no—
237C2×C42C22×D311yescharacteristic
238#106—64no—
239#106—122no—
240C4⋊C8#6714yes—
241#103—62no—
242#103—62no—
243#103—64no—
244C22⋊C8—24no—
245C4×C8—22no—
246#95—61no—
247#95—62no—
248#95—62no—
249#95—62no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3C2×SD1614yes—
257C23×C3C22×D411yescharacteristic
258C3×D4—28no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4#12112yes—
269C2×C3×C4#12112yes—
270C2×C3×C4C2×SD1612yes—
271C2×C3×C4#13512yes—
272C2×C3×C4C22×D411yescharacteristic
273C2×C3×C4C2×#4111yescharacteristic
274C2×C3×C4C2×#4111yescharacteristic
275C2×C3×C4C2×#4111yescharacteristic
276C2×C3×C4H(5,𝔽2)11yescharacteristic
277C2×C3×C42−1+411yescharacteristic
278C2×Dic3—22no—
279C2×Dic3—41no—
280C2×Dic3—41no—
281C2×Dic3—41no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296Dic6—22no—
297Dic6—22no—
298Dic6—41no—
299Dic6—41no—
300Dic6—84no—
301C3×C8—22no—
302C3×C8—24no—
303C3×C8—41no—
304C24C2×D1212yes—
305C2×Q8—32no—
306C2×Q8—61no—
307C2×Q8—61no—
308C2×Q8—61no—
309C2×Q8—61no—
310C2×Q8—61no—
311C2×Q8—122no—
312C2×Q8—122no—
313C2×Q8—122no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4#20414yes—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—61no—
324C22×C4—61no—
325C22×C4—61no—
326C22×C4—61no—
327C22×C4—62no—
328C22×C4—62no—
329C22×C4C2×D1212yes—
330C22×C4C2×D1211yescharacteristic
331C22×C4C2×D1211yescharacteristic
332C22×C4C23×D311yescharacteristic
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—48no—
341C2×C8#23712yes—
342C2×C8C2×#6714yes—
343C4⋊C4—22no—
344C4⋊C4—61no—
345C4⋊C4—62no—
346C4⋊C4—61no—
347C4⋊C4—61no—
348C4⋊C4—62no—
349C4⋊C4—61no—
350C4⋊C4—62no—
351C4⋊C4—61no—
352C4⋊C4—62no—
353C4⋊C4—61no—
354C4⋊C4—124no—
355C4⋊C4—122no—
356C4⋊C4—122no—
357C4⋊C4—122no—
358C4⋊C4—241no—
359C22⋊C4—24no—
360C22⋊C4—124no—
361C22⋊C4—124no—
362C22⋊C4—124no—
363C22⋊C4—242no—
364C42—122no—
365C42—122no—
366C42#23512yes—
367C42C2×#6712yes—
368C22×C3—24no—
369C22×C3—24no—
370C22×C3—24no—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—42no—
375C22×C3C2×#12111yescharacteristic
376C22×C3#54611yescharacteristic
377C22×C3#54711yescharacteristic
378C22×C3C22×SD1611yescharacteristic
379C22×C3C2×#13511yescharacteristic
380C22×C3#57711yescharacteristic
381C22×C3#57811yescharacteristic
382C3×C4—22no—
383C3×C4—42no—
384C3×C4#46514yes—
385C3×C4#52911yescharacteristic
386C3×C4#54211yescharacteristic
387C3×C4#54611yescharacteristic
388C3×C4#54811yescharacteristic
389Dic3—42no—
390Dic3—81no—
391Dic3—81no—
392Dic3—82no—
393Dic3—82no—
394Dic3—81no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23C2×#20414yes—
402C23C22×D1211yescharacteristic
403Q8—62no—
404Q8—62no—
405Q8—121no—
406Q8—121no—
407Q8—244no—
408D4—28no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—62no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—121no—
427C2×C4—122no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4C2×#20412yes—
437C2×C4#90412yes—
438C2×C4#93012yes—
439C2×C4#93012yes—
440C2×C4C2×#23511yescharacteristic
441C2×C4C2×#23711yescharacteristic
442C2×C4#100411yescharacteristic
443C2×C4D3×#4111yescharacteristic
444C2×C4#101011yescharacteristic
445C2×C4C22×#6711yescharacteristic
446C8—22no—
447C8—24no—
448C8—41no—
449C2×C3—44no—
450C2×C3#242711yescharacteristic
451C2×C3C2×#46511yescharacteristic
452C2×C3#306311yescharacteristic
453C2×C3#329811yescharacteristic
454C2×C3#330411yescharacteristic
455C2×C3#333511yescharacteristic
456C2×C3#334211yescharacteristic
457C22—24no—
458C22—24no—
459C22—24no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22#555411yescharacteristic
465C22#555511yescharacteristic
466C22C22×#20411yescharacteristic
467C22C2×#90411yescharacteristic
468C22#575111yescharacteristic
469C22#575311yescharacteristic
470C22C2×#93011yescharacteristic
471C4—22no—
472C4—42no—
473C4—122no—
474C4—242no—
475C4—241no—
476C4—241no—
477C4—242no—
478C4—241no—
479C4#511414yes—
480C4#555711yescharacteristic
481C4#556111yescharacteristic
482C4#580611yescharacteristic
483C4#583111yescharacteristic
484C3#2534111yescharacteristic, complemented, Sylow
485C2—44no—
486C2#7941211yescharacteristic
487C2#7941911yescharacteristic
488C2#7944711yescharacteristic
489C2#7945111yescharacteristic
490C2#8240811yescharacteristic
491C2C2×#511411yescharacteristic
492C2#8380811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C8×D4C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenterC2×C8×D4D3
hypocenterC3#25341
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
4. C2×#4851C2semidirect
5. #71972C2semidirect
6. #71970C2semidirect
7. #70454C2semidirect
8. #69940C2semidirect
9. #69936C2semidirect
10. #69777C2semidirect
11. C4×#816C2semidirect
484. C3#25341semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C8×D42
#823822
#823762
C2×#48512
#719722
#719702
#704542
#699402
#699362
#697772
C4×#8162
#253413