Finite Group Databasebeta

← #10829294

Subgroups — #10829294

Order 768.

1120 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5579C211yesmaximal, characteristic
2C2×C3×#441C211yesmaximal, characteristic, complemented
3#82499C211yesmaximal, characteristic
4C2×#5020C211yesmaximal, characteristic, complemented
5#72071C212yesmaximal, complemented
6#71627C211yesmaximal, characteristic, complemented
7#70579C212yesmaximal, complemented
8#70274C212yesmaximal, complemented
9#70268C212yesmaximal, complemented
10#69837C211yesmaximal, characteristic
11#69828C211yesmaximal, characteristic, complemented
12#24134—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×#946C2212yes—
17C2×#946C2212yes—
18#5579—24no—
19#5579—24no—
20C2×#896—22no—
21C2×D3×C4⋊C4C2212yes—
22C2×C4×D12—21no—
23C2×C4×D12C2211yescharacteristic
24C2×#870—21no—
25C2×#869C2211yescharacteristic
26C3×#441—24no—
27C2×C3×#106C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31C2×#835C2212yes—
32C2×#835C2212yes—
33C2×(C2×D3)⋊C8C2212yes—
34C2×(C2×D3)⋊C8C2212yes—
35#5020—24no—
36#4991—21no—
37#4987C2212yes—
38#4981—21no—
39C2×#808C2212yes—
40#4940—22no—
41#4936C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4551C2212yes—
44#4551C2212yes—
45#4529C2212yes—
46C2×#554—31no—
47C2×#475—31no—
48C2×#441D311yescharacteristic, complemented
49#2536—31no—
50#2035—32no—
51#2001—32no—
52#1993—31no—
53#1955—31no—
54#1872—32no—
55#1853—32no—
56#1805—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×D12D411yescharacteristic
60C22×C4×D3—22no—
61C22×C4×D3C2312yes—
62C22×Dic6—21no—
63C2×C3×Q16—22no—
64C2×C3×Q16—22no—
65C2×C3×Q16—22no—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×C4×Q8—22no—
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C3×C4×Q8D414yes—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C42—21no—
77C2×C3×C42—21no—
78C2×C3×C42C2311yescharacteristic
79#946—24no—
80#946—24no—
81#946—24no—
82#946—42no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4D412yes—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3D411yescharacteristic
89C2×C4⋊Dic3C2312yes—
90C2×C4⋊Dic3C2312yes—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4—22no—
94C22×C3⋊C8C2312yes—
95#896—216no—
96D3×C4⋊C4—24no—
97D3×C4⋊C4—24no—
98C4×D12—28no—
99C4×D12—22no—
100C4×D12—22no—
101#870—28no—
102#869—22no—
103#869D414yes—
104C3×#106—22no—
105C3×#103—22no—
106C3×#103—22no—
107C3×#97—22no—
108#835—24no—
109#835—42no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—22no—
113(C2×D3)⋊C8—24no—
114(C2×D3)⋊C8—42no—
115#808—24no—
116C4⋊(C3⋊C8)—22no—
117C22×Q16C2×D311yescharacteristic
118#554—64no—
119#554—64no—
120C2×C4⋊Q8—31no—
121C2×#125—61no—
122C2×C22⋊Q8—32no—
123C2×C22⋊Q8—32no—
124C2×C22⋊Q8—62no—
125C2×C4×Q8C2×D311yescharacteristic
126C2×C4×Q8C2×D311yescharacteristic
127C2×C4×D4—31no—
128C2×C4×D4—61no—
129C22×C4⋊C4—32no—
130#475—64no—
131#441—24no—
132C2×#106—32no—
133C2×#106C2×D311yescharacteristic
134C2×C4⋊C8—31no—
135C2×#103—32no—
136C2×#103—32no—
137C2×#103C2×D311yescharacteristic
138C2×#103C2×D311yescharacteristic
139C2×C22⋊C8—32no—
140C2×C22⋊C8—32no—
141C2×#97C2×D311yescharacteristic
142#398—61no—
143#394—61no—
144#388—62no—
145#382—31no—
146#380—32no—
147#340—32no—
148#340—32no—
149#340—32no—
150C23×D3C2×D412yes—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8C2×D412yes—
162C2×C3×Q8#4112yes—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2411yescharacteristic
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3C2×D412yes—
173C2×D12—24no—
174C2×D12—22no—
175C2×C4×D3—22no—
176C2×C4×D3—22no—
177C2×C4×D3—22no—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—28no—
182C2×C4×D3—82no—
183C2×C4×D3—82no—
184C2×Dic6—24no—
185C2×Dic6—82no—
186C3×Q16—28no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8#4112yes—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—42no—
198C3×C4⋊C4C2×D412yes—
199C3×C42—24no—
200C3×C42—42no—
201C3×C42C2×D412yes—
202C3×C42C2×D412yes—
203(C2×D3)⋊C4—28no—
204(C2×D3)⋊C4—24no—
205(C2×D3)⋊C4—44no—
206(C2×D3)⋊C4—44no—
207C4⋊Dic3—24no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3—22no—
211C4⋊Dic3—22no—
212C4⋊Dic3C2×D414yes—
213Dic3⋊C4—28no—
214Dic3⋊C4—28no—
215Dic3⋊C4—44no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C22×Q8—61no—
220C22×Q8C22×D311yescharacteristic
221C22×Q8C22×D311yescharacteristic
222C22×D4—31no—
223C23×C4—32no—
224C23×C4—62no—
225C2×Q16—22no—
226C2×Q16—22no—
227C2×Q16—22no—
228C22×C8—32no—
229C22×C8C22×D311yescharacteristic
230C22×C8C22×D311yescharacteristic
231C4⋊Q8—34no—
232C4⋊Q8—62no—
233#125—68no—
234C22⋊Q8—616no—
235C22⋊Q8—64no—
236C22⋊Q8—64no—
237C22⋊Q8—64no—
238C22⋊Q8—122no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8#6714yes—
243C4×D4—68no—
244C4×D4—62no—
245C4×D4—62no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C4⋊C4C22×D311yescharacteristic
259C2×C22⋊C4—32no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42C22×D311yescharacteristic
266#106—22no—
267#106—64no—
268C4⋊C8—62no—
269#103—22no—
270#103—22no—
271#103—64no—
272#103—122no—
273C22⋊C8—64no—
274C22⋊C8—122no—
275#97—22no—
276#95—62no—
277#95—61no—
278#95—61no—
279C23×C3C22×D411yescharacteristic
280C22×D3—22no—
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—22no—
284C22×D3—22no—
285C22×D3—22no—
286C22×D3#13714yes—
287C3×Q8—22no—
288C3×Q8—22no—
289C3×Q8—22no—
290C3×Q8—22no—
291C3×Q8—41no—
292C3×Q8—41no—
293C3×Q8—41no—
294C3×Q8—41no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4#12012yes—
308C2×C3×C4#12012yes—
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4H(5,𝔽2)11yescharacteristic
314C2×C3×C42−1+411yescharacteristic
315C2×Dic3—22no—
316C2×Dic3—24no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324D12—28no—
325C4×D3—42no—
326C4×D3—42no—
327C4×D3—42no—
328C4×D3—42no—
329C4×D3—88no—
330Dic6—84no—
331C3×C8—22no—
332C3×C8—41no—
333C3⋊C8—42no—
334C24—32no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—64no—
346C2×Q8—242no—
347C2×Q8#23512yes—
348C2×Q8C2×#6712yes—
349C2×D4—64no—
350C2×D4—62no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—32no—
356C22×C4—68no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—242no—
366C22×C4—242no—
367C22×C4D3×D411yescharacteristic
368C22×C4D3×D411yescharacteristic
369C22×C4C23×D311yescharacteristic
370Q16—28no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—62no—
377C2×C8—62no—
378C2×C8—62no—
379C2×C8#23512yes—
380C4⋊C4—24no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—34no—
385C4⋊C4—42no—
386C4⋊C4—68no—
387C4⋊C4—68no—
388C4⋊C4—62no—
389C4⋊C4—64no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4—62no—
393C4⋊C4—124no—
394C4⋊C4C2×#6712yes—
395C22⋊C4—68no—
396C22⋊C4—64no—
397C22⋊C4—124no—
398C22⋊C4—124no—
399C42—24no—
400C42—42no—
401C42D3×D412yes—
402C42C2×#6712yes—
403C22×C3C2×#12011yescharacteristic
404C22×C3#54611yescharacteristic
405C22×C3#54711yescharacteristic
406C22×C3C2×#13712yes—
407C22×C3#57712yes—
408C2×D3—24no—
409C2×D3—24no—
410C2×D3—24no—
411C2×D3—42no—
412C2×D3—42no—
413C2×D3—42no—
414C2×D3—42no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—42no—
418C3×C4—42no—
419C3×C4—42no—
420C3×C4—81no—
421C3×C4#52511yescharacteristic
422C3×C4#53711yescharacteristic
423C3×C4#54811yescharacteristic
424C3×C4#55011yescharacteristic
425Dic3—42no—
426Dic3—44no—
427Dic3—82no—
428C23—34no—
429C23—62no—
430C23—62no—
431C23—62no—
432C23—62no—
433C23—62no—
434C23—62no—
435C23C2×D3×D411yescharacteristic
436Q8—22no—
437Q8—22no—
438Q8—22no—
439Q8—22no—
440Q8—41no—
441Q8—41no—
442Q8—41no—
443Q8—41no—
444Q8—244no—
445D4—68no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—64no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—248no—
472C2×C4#93712yes—
473C2×C4#93712yes—
474C2×C4C2×#23511yescharacteristic
475C2×C4C2×#23511yescharacteristic
476C2×C4C2×D3×D411yescharacteristic
477C2×C4#100911yescharacteristic
478C2×C4#101011yescharacteristic
479C2×C4C22×#6711yescharacteristic
480C8—22no—
481C8—41no—
482C8—122no—
483C2×C3#240711yescharacteristic
484C2×C3#300411yescharacteristic
485C2×C3#300911yescharacteristic
486C2×C3#331612yes—
487C2×C3#334412yes—
488D3—44no—
489C22—64no—
490C22—64no—
491C22—64no—
492C22—122no—
493C22—122no—
494C22—122no—
495C22—122no—
496C22#555011yescharacteristic
497C22#555111yescharacteristic
498C22D3×#13712yes—
499C22#577812yes—
500C22C2×#93711yescharacteristic
501C4—22no—
502C4—22no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—81no—
507C4—124no—
508C4—122no—
509C4—242no—
510C4#555211yescharacteristic
511C4#555311yescharacteristic
512C4#579111yescharacteristic
513C4#583311yescharacteristic
514C3#2413411yescharacteristic, complemented, Sylow
515C2—124no—
516C2#7984312yes—
517C2#7984512yes—
518C2#8240211yescharacteristic
519C2#8397911yescharacteristic
520C2#8398211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#441C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenterC2×#441D3
hypocenterC3#24134
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
2. C2×C3×#441C2semidirect
4. C2×#5020C2semidirect
5. #72071C2semidirect
6. #71627C2semidirect
7. #70579C2semidirect
8. #70274C2semidirect
9. #70268C2semidirect
11. #69828C2semidirect
48. C2×#441D3semidirect
514. C3#24134semidirect

Maximal subgroups

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

subgroupindex
C2×#55792
C2×C3×#4412
#824992
C2×#50202
#720712
#716272
#705792
#702742
#702682
#698372
#698282
#241343