Finite Group Databasebeta

← #10829288

Subgroups — #10829288

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×C3×#554C211yesmaximal, characteristic
2C2×#5094C211yesmaximal, characteristic, complemented
3C2×#5030C211yesmaximal, characteristic, complemented
4#82375C211yesmaximal, characteristic
5#71788C211yesmaximal, characteristic, complemented
6#71700C212yesmaximal, complemented
7#71694C212yesmaximal, complemented
8#71627C211yesmaximal, characteristic, complemented
9#71615C211yesmaximal, characteristic
10#71559C212yesmaximal, complemented
11#71556C212yesmaximal, complemented
12#24134—31nomaximal, Sylow
13C3×#554—24no—
14C3×#554—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125—21no—
17C2×C3×C22⋊Q8—22no—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4—21no—
21C2×C3×C4×D4C2211yescharacteristic
22C22×C3×C4⋊C4C2212yes—
23C22×#216C2211yescharacteristic
24C2×#924C2212yes—
25C2×C4×Dic6C2211yescharacteristic
26C2×#835C2211yescharacteristic
27C2×#835C2212yes—
28#5222—21no—
29#5212C2212yes—
30C2×#830C2212yes—
31C2×#830C2212yes—
32#5094—24no—
33#5030—24no—
34#4983—21no—
35C2×#810C2212yes—
36C2×#810C2211yescharacteristic
37C2×#808C2212yes—
38C2×#808C2211yescharacteristic
39#4944—22no—
40#4936C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#554D311yescharacteristic
47C2×#475—31no—
48C2×#441—31no—
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×Q8—21no—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4D411yescharacteristic
60C23×C3×C4—22no—
61C23×C3×C4C2312yes—
62C22×Dic6C2311yescharacteristic
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8D414yes—
65C3×#125—28no—
66C3×C22⋊Q8—216no—
67C3×C22⋊Q8—24no—
68C3×C22⋊Q8—24no—
69C3×C4×Q8—22no—
70C3×C4×Q8D414yes—
71C3×C4×D4—28no—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4D412yes—
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
87C2×#217—22no—
88C2×#217—22no—
89C2×#216—22no—
90C2×#216—22no—
91C2×#216—22no—
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2311yescharacteristic
94#924—24no—
95#924—42no—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4—21no—
98C2×C4×Dic3—21no—
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2312yes—
101C22×C3⋊C8C2311yescharacteristic
102C4×Dic6—22no—
103C4×Dic6—22no—
104#835—22no—
105#835—42no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108C3⋊#95—22no—
109#830—24no—
110#830—42no—
111#810—22no—
112#810—24no—
113#808—22no—
114#808—24no—
115C4⋊(C3⋊C8)—22no—
116#803—22no—
117C22×Q16—31no—
118#554—24no—
119#554—24no—
120C2×C4⋊Q8C2×D311yescharacteristic
121C2×#125—21no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—32no—
124C2×C22⋊Q8C2×D312yes—
125C2×C4×Q8—31no—
126C2×C4×Q8C2×D311yescharacteristic
127C2×C4×D4—21no—
128C2×C4×D4C2×D311yescharacteristic
129C22×C4⋊C4C2×D312yes—
130#475—64no—
131#441—64no—
132C2×#106—32no—
133C2×#106—31no—
134C2×C4⋊C8—31no—
135C2×#103—31no—
136C2×#103—32no—
137C2×#103—31no—
138C2×#103—32no—
139C2×C22⋊C8—32no—
140C2×C22⋊C8—32no—
141C2×#97—31no—
142#398—61no—
143#394—61no—
144#388—62no—
145#382—31no—
146#380—32no—
147#340—32no—
148#340—32no—
149#340—32no—
150C24×C3C2×D412yes—
151C2×C3×Q8—24no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—82no—
158C2×C3×Q8C2×D412yes—
159C2×C3×D4—24no—
160C2×C3×D4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—28no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—82no—
172C22×C3×C4—82no—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2411yescharacteristic
177C22×Dic3—21no—
178C22×Dic3—22no—
179C22×Dic3—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6#4112yes—
186C3×C4⋊C4—28no—
187C3×C4⋊C4—28no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4C2×D412yes—
194C3×C4⋊C4C2×D414yes—
195C3×C22⋊C4—28no—
196C3×C22⋊C4—24no—
197C3×C42—24no—
198C3×C42C2×D412yes—
199C3×C42C2×D412yes—
200#217—44no—
201#217—44no—
202#216—28no—
203C4⋊Dic3—21no—
204C4⋊Dic3—21no—
205C4⋊Dic3—22no—
206C4⋊Dic3—22no—
207Dic3⋊C4—44no—
208Dic3⋊C4—42no—
209C4×Dic3—42no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8#4112yes—
219C22×Q8—21no—
220C22×Q8—31no—
221C22×Q8C22×D311yescharacteristic
222C22×D4#6711yescharacteristic
223C23×C4—22no—
224C23×C4C22×D312yes—
225C2×Q16—62no—
226C2×Q16—62no—
227C2×Q16—62no—
228C22×C8—31no—
229C22×C8—31no—
230C22×C8—32no—
231C4⋊Q8—22no—
232C4⋊Q8#6714yes—
233#125—28no—
234C22⋊Q8—216no—
235C22⋊Q8—24no—
236C22⋊Q8—24no—
237C22⋊Q8—64no—
238C22⋊Q8—122no—
239C4×Q8—22no—
240C4×Q8—62no—
241C4×Q8—62no—
242C4×Q8#6714yes—
243C4×D4—28no—
244C4×D4—22no—
245C4×D4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4#6711yescharacteristic
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C4⋊C4C22×D312yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4#6712yes—
263C2×C42—21no—
264C2×C42—61no—
265C2×C42C22×D311yescharacteristic
266#106—62no—
267#106—64no—
268C4⋊C8—62no—
269#103—62no—
270#103—62no—
271#103—64no—
272#103—122no—
273C22⋊C8—64no—
274C22⋊C8—122no—
275#97—62no—
276#95—62no—
277#95—61no—
278#95—61no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3#13714yes—
286C23×C3C22×D411yescharacteristic
287C3×Q8—22no—
288C3×Q8—22no—
289C3×Q8—41no—
290C3×Q8—41no—
291C3×Q8—84no—
292C3×D4—28no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—88no—
311C2×C3×C4#12012yes—
312C2×C3×C4#12012yes—
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4H(5,𝔽2)11yescharacteristic
318C2×C3×C42−1+411yescharacteristic
319C2×Dic3—22no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327Dic6—22no—
328Dic6—22no—
329Dic6—41no—
330Dic6—41no—
331C3⋊C8—22no—
332C3⋊C8—41no—
333C3⋊C8—42no—
334C24C2×#6712yes—
335C2×Q8—24no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—32no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—61no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—82no—
348C2×Q8C2×#6712yes—
349C2×D4—24no—
350C2×D4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—28no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—61no—
362C22×C4—62no—
363C22×C4—61no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4C2×#6712yes—
367C22×C4C2×#6711yescharacteristic
368C22×C4C2×#6711yescharacteristic
369C22×C4C23×D311yescharacteristic
370Q16—68no—
371C2×C8—32no—
372C2×C8—61no—
373C2×C8—61no—
374C2×C8—62no—
375C2×C8—61no—
376C2×C8—62no—
377C2×C8—61no—
378C2×C8—61no—
379C2×C8—62no—
380C4⋊C4—28no—
381C4⋊C4—28no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—62no—
388C4⋊C4—62no—
389C4⋊C4—61no—
390C4⋊C4—61no—
391C4⋊C4—124no—
392C4⋊C4—122no—
393C4⋊C4C2×#6712yes—
394C4⋊C4C2×#6714yes—
395C22⋊C4—28no—
396C22⋊C4—24no—
397C22⋊C4—124no—
398C22⋊C4—124no—
399C42—24no—
400C42—122no—
401C42C2×#6712yes—
402C42C2×#6712yes—
403C22×C3—24no—
404C22×C3—24no—
405C22×C3—24no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3C2×#12011yescharacteristic
411C22×C3#54611yescharacteristic
412C22×C3#54711yescharacteristic
413C22×C3C2×#13712yes—
414C22×C3#57712yes—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—42no—
418C3×C4—42no—
419C3×C4—42no—
420C3×C4—44no—
421C3×C4#52511yescharacteristic
422C3×C4#53711yescharacteristic
423C3×C4#54811yescharacteristic
424C3×C4#55011yescharacteristic
425Dic3—42no—
426Dic3—81no—
427Dic3—82no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23#94214yes—
435C23C22×#6711yescharacteristic
436Q8—22no—
437Q8—22no—
438Q8—41no—
439Q8—41no—
440Q8—62no—
441Q8—62no—
442Q8—84no—
443Q8—121no—
444Q8—121no—
445D4—28no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—62no—
464C2×C4—88no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—121no—
472C2×C4#95312yes—
473C2×C4#95312yes—
474C2×C4D3×#4111yescharacteristic
475C2×C4D3×#4111yescharacteristic
476C2×C4#100911yescharacteristic
477C2×C4#101011yescharacteristic
478C2×C4C22×#6711yescharacteristic
479C2×C4C22×#6711yescharacteristic
480C8—62no—
481C8—121no—
482C8—122no—
483C2×C3—44no—
484C2×C3#240711yescharacteristic
485C2×C3#300411yescharacteristic
486C2×C3#300911yescharacteristic
487C2×C3#331612yes—
488C2×C3#334412yes—
489C22—24no—
490C22—24no—
491C22—24no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22C2×#94212yes—
497C22#582411yescharacteristic
498C22#582911yescharacteristic
499C22#583612yes—
500C22C2×#95311yescharacteristic
501C4—22no—
502C4—22no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—44no—
507C4—122no—
508C4—241no—
509C4—242no—
510C4#582811yescharacteristic
511C4#583111yescharacteristic
512C4#583311yescharacteristic
513C4#583411yescharacteristic
514C3#2413411yescharacteristic, complemented, Sylow
515C2—44no—
516C2#8362412yes—
517C2#8364112yes—
518C2#8405911yescharacteristic
519C2#8409711yescharacteristic
520C2#8410511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#554C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenterC2×#554D3
hypocenterC3#24134
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
2. C2×#5094C2semidirect
3. C2×#5030C2semidirect
5. #71788C2semidirect
6. #71700C2semidirect
7. #71694C2semidirect
8. #71627C2semidirect
10. #71559C2semidirect
11. #71556C2semidirect
514. C3#24134semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5542
C2×#50942
C2×#50302
#823752
#717882
#717002
#716942
#716272
#716152
#715592
#715562
#241343