Finite Group Databasebeta

← #10822742

Subgroups — #10822742

Order 768.

1418 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic
2#84087C211yesmaximal, characteristic
3C2×#5234C211yesmaximal, characteristic
4C2×#5225C211yesmaximal, characteristic, complemented
5C22×#835C211yesmaximal, characteristic, complemented
6C2×#4998C211yesmaximal, characteristic, complemented
7#71695C214yesmaximal, complemented
8#71694C214yesmaximal, complemented
9#71581C211yesmaximal, characteristic, complemented
10#23487—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×Q8C2211yescharacteristic
13C3×#536—24no—
14C3×#536—24no—
15C2×C3×C4⋊Q8—21no—
16C2×C3×#124—21no—
17C2×C3×#123—22no—
18C2×C3×C22⋊Q8C2214yes—
19C2×C3×#117C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24#5234—24no—
25#5229—21no—
26#5225—24no—
27C2×#835—24no—
28C2×#835C2212yes—
29C2×#835C2214yes—
30C2×#835C2212yes—
31#5221—21no—
32#5220—22no—
33#5212C2214yes—
34C2×#830C2212yes—
35C2×#830C2212yes—
36#4998—24no—
37C2×#808C2212yes—
38C2×#807C2212yes—
39#4529C2214yes—
40C2×#536D311yescharacteristic
41C2×#427—31no—
42C2×#419—31no—
43C22×#103—31no—
44C2×#408—31no—
45#2311—31no—
46#2036—34no—
47#2035—34no—
48#1736—31no—
49C2×C3×#41—22no—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C22×C3×Q8—21no—
56C22×C3×Q8—22no—
57C22×C3×Q8—22no—
58C22×C3×Q8—22no—
59C22×C3×Q8—22no—
60C22×C3×Q8—22no—
61C22×C3×Q8D414yes—
62C22×C3×Q8D411yescharacteristic
63C22×C3×Q8C2312yes—
64C22×C3×D4D412yes—
65C22×C3×D4D411yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×Dic3—21no—
70C3×C4⋊Q8—28no—
71C3×#124—28no—
72C3×#123—216no—
73C3×C22⋊Q8—28no—
74C3×C22⋊Q8—28no—
75C3×#117—22no—
76C3×#117D414yes—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C42—21no—
82C2×#217—21no—
83C2×#217—24no—
84#926—22no—
85#926—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C2312yes—
90C2×C4⋊Dic3C2312yes—
91C2×C4×Dic3—21no—
92C2×#208—22no—
93C2×#208—22no—
94C2×#208—22no—
95C22×C3⋊C8—22no—
96C22×C3⋊C8—22no—
97C22×C3⋊C8D414yes—
98C22×C3⋊C8C2312yes—
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101#835—28no—
102#835—24no—
103#835—24no—
104C3⋊#95—24no—
105#830—24no—
106#830—24no—
107#808—24no—
108#807—24no—
109C22×#41C2×D311yescharacteristic
110C23×Q8C2×D311yescharacteristic
111C22×M16—31no—
112C23×C8—31no—
113#536—24no—
114#536—24no—
115C2×C4⋊Q8—21no—
116C2×#124—21no—
117C2×#123—22no—
118C2×C22⋊Q8C2×D314yes—
119C2×#117—31no—
120C2×#117C2×D311yescharacteristic
121C22×C4⋊C4—31no—
122#427—64no—
123C2×#107—32no—
124C2×#106—32no—
125#419—64no—
126C2×#103—32no—
127C2×#103—32no—
128C2×#103—34no—
129C2×#103—64no—
130#408—64no—
131C2×C22⋊C8—32no—
132C2×C22⋊C8—32no—
133#391—61no—
134#390—61no—
135#386—62no—
136#380—34no—
137#340—34no—
138C24×C3C2×D412yes—
139C24×C3C2×D411yescharacteristic
140C3×#41—44no—
141C3×#41—44no—
142C3×#41—44no—
143C3×#41—44no—
144C2×C3×Q8—24no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—28no—
150C2×C3×Q8—28no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—44no—
156C2×C3×Q8—42no—
157C2×C3×Q8—42no—
158C2×C3×Q8—44no—
159C2×C3×Q8C2×D414yes—
160C2×C3×D4—24no—
161C2×C3×D4—24no—
162C2×C3×D4—24no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—42no—
176C22×C3×C4—42no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4#4112yes—
183C22×C3×C4#4112yes—
184C22×C3×C4#4112yes—
185C22×C3×C4C2411yescharacteristic
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—42no—
189C22×Dic3—82no—
190C3×C4⋊C4—216no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4C2×D414yes—
193C3×C22⋊C4—216no—
194C3×C22⋊C4—24no—
195C3×C42—24no—
196#217—42no—
197#217—84no—
198C4⋊Dic3—22no—
199C4⋊Dic3—22no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
202C4⋊Dic3—42no—
203C4×Dic3—42no—
204#208—44no—
205C2×C3⋊C8—28no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8C2×D412yes—
214C2×C3⋊C8C2×D412yes—
215C2×#41—22no—
216C2×#41—22no—
217C2×#41—22no—
218C2×#41—22no—
219C2×#41—22no—
220C2×#41—22no—
221C22×Q8—21no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8#6714yes—
228C22×Q8#6711yescharacteristic
229C22×Q8C22×D312yes—
230C22×D4#6712yes—
231C22×D4#6711yescharacteristic
232C23×C4—21no—
233C23×C4—61no—
234C23×C4C22×D312yes—
235C23×C4C22×D311yescharacteristic
236C2×M16—62no—
237C2×M16—62no—
238C2×M16—62no—
239C22×C8—34no—
240C22×C8—31no—
241C22×C8—31no—
242C22×C8—32no—
243C22×C8—62no—
244C22×C8—62no—
245C4⋊Q8—28no—
246#124—28no—
247#123—216no—
248C22⋊Q8—28no—
249C22⋊Q8—28no—
250#117—22no—
251#117—62no—
252#117—62no—
253#117#6714yes—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—32no—
256C2×C4⋊C4—32no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4C22×D312yes—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—61no—
264C2×C22⋊C4—64no—
265C2×C42—21no—
266C2×C42—61no—
267#107—64no—
268#106—64no—
269#103—64no—
270#103—64no—
271#103—68no—
272C22⋊C8—64no—
273C22⋊C8—64no—
274#95—64no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—22no—
279C23×C3—22no—
280C23×C3—24no—
281C23×C3—24no—
282C23×C3#12012yes—
283C23×C3#12112yes—
284C23×C3#12312yes—
285C23×C3C22×D411yescharacteristic
286C3×Q8—24no—
287C3×Q8—24no—
288C3×Q8—42no—
289C3×Q8—42no—
290C3×Q8—48no—
291C3×Q8—44no—
292C3×D4—48no—
293C3×D4—44no—
294C2×C3×C4—24no—
295C2×C3×C4—24no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—48no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4#12012yes—
309C2×C3×C4#12114yes—
310C2×C3×C4#12114yes—
311C2×C3×C4#12112yes—
312C2×C3×C4#12314yes—
313C2×C3×C4#12312yes—
314C2×C3×C4#12414yes—
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—44no—
325C2×Dic3—42no—
326C2×Dic3—84no—
327C3⋊C8—22no—
328C3⋊C8—22no—
329C3⋊C8—42no—
330C24C2×#6712yes—
331C24C2×#6711yescharacteristic
332#41—44no—
333#41—44no—
334#41—44no—
335#41—44no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—24no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—28no—
342C2×Q8—28no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—42no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—42no—
351C2×Q8C2×#6714yes—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—122no—
372C22×C4—242no—
373C22×C4D3×D412yes—
374C22×C4#23712yes—
375C22×C4#23712yes—
376C22×C4#23712yes—
377C22×C4C2×#6712yes—
378C22×C4C2×#6711yescharacteristic
379C22×C4C2×#6712yes—
380C22×C4C2×#6712yes—
381C22×C4C23×D311yescharacteristic
382M16—124no—
383C2×C8—32no—
384C2×C8—32no—
385C2×C8—68no—
386C2×C8—61no—
387C2×C8—61no—
388C2×C8—61no—
389C2×C8—61no—
390C2×C8—62no—
391C2×C8—62no—
392C2×C8—62no—
393C4⋊C4—216no—
394C4⋊C4—22no—
395C4⋊C4—62no—
396C4⋊C4—62no—
397C4⋊C4—62no—
398C4⋊C4—62no—
399C4⋊C4—122no—
400C4⋊C4C2×#6714yes—
401C22⋊C4—216no—
402C22⋊C4—24no—
403C22⋊C4—122no—
404C22⋊C4—244no—
405C42—24no—
406C42—122no—
407C22×C3—24no—
408C22×C3—24no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—44no—
412C22×C3—42no—
413C22×C3—44no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3#39414yes—
417C22×C3C2×#12011yescharacteristic
418C22×C3C2×#12111yescharacteristic
419C22×C3C2×#12311yescharacteristic
420C22×C3#57712yes—
421C22×C3#57812yes—
422C3×C4—42no—
423C3×C4—44no—
424C3×C4—44no—
425C3×C4#39414yes—
426C3×C4C2×#12111yescharacteristic
427C3×C4C2×#12111yescharacteristic
428C3×C4C2×#12311yescharacteristic
429C3×C4C2×#12411yescharacteristic
430Dic3—82no—
431Dic3—82no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—22no—
439C23#93512yes—
440C23#93912yes—
441C23#95312yes—
442C23C22×#6711yescharacteristic
443Q8—24no—
444Q8—24no—
445Q8—48no—
446Q8—44no—
447Q8—42no—
448Q8—42no—
449D4—48no—
450D4—44no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—48no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—124no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—244no—
471C2×C4#93512yes—
472C2×C4#93514yes—
473C2×C4#93614yes—
474C2×C4#93814yes—
475C2×C4#93912yes—
476C2×C4#93914yes—
477C2×C4#95312yes—
478C2×C4C2×D3×D411yescharacteristic
479C2×C4C2×#23711yescharacteristic
480C2×C4C2×#23711yescharacteristic
481C2×C4C2×#23711yescharacteristic
482C2×C4C22×#6711yescharacteristic
483C2×C4C22×#6711yescharacteristic
484C8—62no—
485C8—62no—
486C8—122no—
487C2×C3—24no—
488C2×C3—82no—
489C2×C3C2×#39411yescharacteristic
490C2×C3#301312yes—
491C2×C3#305711yescharacteristic
492C2×C3#305911yescharacteristic
493C2×C3#309712yes—
494C22—24no—
495C22—24no—
496C22—22no—
497C22—22no—
498C22—42no—
499C22—44no—
500C22—44no—
501C22—42no—
502C22—42no—
503C22#522214yes—
504C22C2×#93511yescharacteristic
505C22C2×#93911yescharacteristic
506C22#583712yes—
507C22#583812yes—
508C22C2×#95311yescharacteristic
509C4—42no—
510C4—44no—
511C4—44no—
512C4—242no—
513C4—242no—
514C4#522214yes—
515C4C2×#93511yescharacteristic
516C4C2×#93611yescharacteristic
517C4C2×#93811yescharacteristic
518C4C2×#93911yescharacteristic
519C3#2348711yescharacteristic, complemented, Sylow
520C2—24no—
521C2—82no—
522C2#8339712yes—
523C2#8339811yescharacteristic
524C2#8339911yescharacteristic
525C2C2×#522211yescharacteristic
526C2#8411412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#536C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenterC2×#536D3
hypocenterC3#23487
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
4. C2×#5225C2semidirect
5. C22×#835C2semidirect
6. C2×#4998C2semidirect
7. #71695C2semidirect
8. #71694C2semidirect
9. #71581C2semidirect
519. C3#23487semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5362
#840872
C2×#52342
C2×#52252
C22×#8352
C2×#49982
#716952
#716942
#715812
#234873