Finite Group Databasebeta

← #10829290

Subgroups — #10829290

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×#5543C211yesmaximal, characteristic
2C3×#2536C211yesmaximal, characteristic
3C2×#5020C211yesmaximal, characteristic, complemented
4C2×#5030C211yesmaximal, characteristic, complemented
5#71700C212yesmaximal, complemented
6#71692C212yesmaximal, complemented
7#71615C211yesmaximal, characteristic
8#69559C211yesmaximal, characteristic, complemented
9#69555C211yesmaximal, characteristic, complemented
10#69526C212yesmaximal, complemented
11#69505C212yesmaximal, complemented
12#24134—31nomaximal, Sylow
13C2×C3×C22⋊Q8C2212yes—
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C22×#216C2211yescharacteristic
17C2×D4×Dic3—21no—
18C22×C4⋊Dic3C2212yes—
19C2×#924C2212yes—
20#5543—24no—
21#5543—24no—
22C2×#890—21no—
23C2×#878—22no—
24C2×#869C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C3×#398—21no—
27C3×#394—21no—
28C3×#388—22no—
29C3×#382C2211yescharacteristic
30C3×#380C2212yes—
31C2×#835C2211yescharacteristic
32C2×#835C2212yes—
33C2×#830C2212yes—
34C2×#830C2212yes—
35#5030—24no—
36#5020—24no—
37C2×#810C2211yescharacteristic
38C2×#810C2212yes—
39C2×#808C2211yescharacteristic
40C2×#808C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#554—31no—
47C2×#475—31no—
48C2×#441—31no—
49#2536D311yescharacteristic
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×D4D411yescharacteristic
59C23×C3×C4C2312yes—
60C23×Dic3—22no—
61C22×Dic6—21no—
62C22×Dic6C2311yescharacteristic
63C3×C22⋊Q8—24no—
64C3×C22⋊Q8—42no—
65C3×C4×Q8—22no—
66C3×C4×Q8—22no—
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4D412yes—
77C2×C3×C42—21no—
78C2×C3×C42C2311yescharacteristic
79C2×#217—22no—
80C2×#216—22no—
81C2×#216—22no—
82C2×#216—22no—
83D4×Dic3—28no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—24no—
87C2×C4⋊Dic3—24no—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2312yes—
90#924—24no—
91#924—24no—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4—22no—
94C2×C4×Dic3—21no—
95C22×C3⋊C8C2311yescharacteristic
96C22×C3⋊C8C2312yes—
97C22×C3⋊C8C2311yescharacteristic
98#890—28no—
99#878—216no—
100#869—22no—
101#869D414yes—
102C4×Dic6—22no—
103C4×Dic6D414yes—
104C3×#95—22no—
105C3×#95—21no—
106C3×#95—21no—
107#835—24no—
108#835—22no—
109#830—24no—
110#830—42no—
111#810—22no—
112#810—42no—
113#808—24no—
114#808—22no—
115C4⋊(C3⋊C8)—22no—
116#803—22no—
117C22×Q16—31no—
118#554—64no—
119#554—64no—
120C2×C4⋊Q8—31no—
121C2×#125—61no—
122C2×C22⋊Q8—32no—
123C2×C22⋊Q8—62no—
124C2×C22⋊Q8C2×D312yes—
125C2×C4×Q8—31no—
126C2×C4×Q8C2×D311yescharacteristic
127C2×C4×D4—61no—
128C2×C4×D4C2×D311yescharacteristic
129C22×C4⋊C4—32no—
130#475—64no—
131#441—64no—
132C2×#106—32no—
133C2×#106—31no—
134C2×C4⋊C8—31no—
135C2×#103—32no—
136C2×#103—31no—
137C2×#103—31no—
138C2×#103—32no—
139C2×C22⋊C8—32no—
140C2×C22⋊C8—32no—
141C2×#97—31no—
142#398—21no—
143#394—21no—
144#388—22no—
145#382C2×D311yescharacteristic
146#380C2×D312yes—
147#340—32no—
148#340—32no—
149#340—32no—
150C24×C3C2×D412yes—
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—24no—
158C2×C3×D4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—22no—
175C22×Dic3—28no—
176C22×Dic3—82no—
177C22×Dic3—82no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—24no—
184C2×Dic6—82no—
185C2×Dic6C2×D412yes—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—44no—
192C3×C4⋊C4—42no—
193C3×C22⋊C4—24no—
194C3×C22⋊C4—44no—
195C3×C22⋊C4—44no—
196C3×C42—42no—
197C3×C42C2×D412yes—
198C3×C42C2×D412yes—
199#217—28no—
200#216—28no—
201C4⋊Dic3—28no—
202C4⋊Dic3—24no—
203C4⋊Dic3—21no—
204C4⋊Dic3—22no—
205C4⋊Dic3C2×D412yes—
206C4⋊Dic3C2×D414yes—
207Dic3⋊C4—28no—
208Dic3⋊C4—24no—
209C4×Dic3—24no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8#4112yes—
219C22×Q8—31no—
220C22×Q8—61no—
221C22×Q8C22×D311yescharacteristic
222C22×D4#6711yescharacteristic
223C23×C4—62no—
224C23×C4C22×D312yes—
225C2×Q16—62no—
226C2×Q16—62no—
227C2×Q16—62no—
228C22×C8—31no—
229C22×C8—31no—
230C22×C8—32no—
231C4⋊Q8—34no—
232C4⋊Q8—62no—
233#125—68no—
234C22⋊Q8—24no—
235C22⋊Q8—42no—
236C22⋊Q8—616no—
237C22⋊Q8—64no—
238C22⋊Q8—64no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8—34no—
242C4×Q8—62no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—68no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—61no—
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—22no—
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—22no—
277#95—21no—
278#95—21no—
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×D4—28no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4#12012yes—
310C2×C3×C4#12012yes—
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4H(5,𝔽2)11yescharacteristic
316C2×C3×C42−1+411yescharacteristic
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—24no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—88no—
326Dic6—22no—
327Dic6—22no—
328Dic6—41no—
329Dic6—41no—
330Dic6—84no—
331C3⋊C8—22no—
332C3⋊C8—41no—
333C3⋊C8—42no—
334C24C2×#6712yes—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—32no—
341C2×Q8—64no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—61no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—242no—
348C2×Q8#23712yes—
349C2×D4—24no—
350C2×D4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—61no—
361C22×C4—61no—
362C22×C4—62no—
363C22×C4—68no—
364C22×C4—242no—
365C22×C4—242no—
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—61no—
375C2×C8—62no—
376C2×C8—61no—
377C2×C8—62no—
378C2×C8—62no—
379C2×C8—61no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—32no—
386C4⋊C4—34no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—68no—
390C4⋊C4—68no—
391C4⋊C4—64no—
392C4⋊C4—64no—
393C4⋊C4—61no—
394C4⋊C4—62no—
395C22⋊C4—24no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—68no—
399C42—42no—
400C42—64no—
401C42D3×D412yes—
402C42D3×D412yes—
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—81no—
420C3×C4—82no—
421C3×C4#52511yescharacteristic
422C3×C4#53711yescharacteristic
423C3×C4#54811yescharacteristic
424C3×C4#55011yescharacteristic
425Dic3—42no—
426Dic3—44no—
427Dic3—42no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23#95114yes—
435C23C22×#6711yescharacteristic
436Q8—22no—
437Q8—22no—
438Q8—41no—
439Q8—41no—
440Q8—62no—
441Q8—62no—
442Q8—121no—
443Q8—121no—
444Q8—244no—
445D4—28no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—62no—
464C2×C4—62no—
465C2×C4—64no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—248no—
472C2×C4#93712yes—
473C2×C4#93712yes—
474C2×C4C2×D3×D411yescharacteristic
475C2×C4C2×D3×D411yescharacteristic
476C2×C4C2×#23711yescharacteristic
477C2×C4#100411yescharacteristic
478C2×C4D3×#4111yescharacteristic
479C2×C4#101011yescharacteristic
480C8—62no—
481C8—122no—
482C8—121no—
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×#93711yescharacteristic
497C22#580611yescharacteristic
498C22C2×#95112yes—
499C22#582411yescharacteristic
500C22#583712yes—
501C4—22no—
502C4—22no—
503C4—42no—
504C4—42no—
505C4—82no—
506C4—81no—
507C4—122no—
508C4—122no—
509C4—124no—
510C4#559811yescharacteristic
511C4#559911yescharacteristic
512C4#560611yescharacteristic
513C4#560811yescharacteristic
514C3#2413411yescharacteristic, complemented, Sylow
515C2—44no—
516C2#8270111yescharacteristic
517C2#8293311yescharacteristic
518C2#8340411yescharacteristic
519C2#8353812yes—
520C2#8359312yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2536C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#2536D3
hypocenterC3#24134
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2536

Complements

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

NH ≅ G/Ntype
3. C2×#5020C2semidirect
4. C2×#5030C2semidirect
5. #71700C2semidirect
6. #71692C2semidirect
8. #69559C2semidirect
9. #69555C2semidirect
10. #69526C2semidirect
11. #69505C2semidirect
514. C3#24134semidirect

Maximal subgroups

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

subgroupindex
C2×#55432
C3×#25362
C2×#50202
C2×#50302
#717002
#716922
#716152
#695592
#695552
#695262
#695052
#241343