Finite Group Databasebeta

← #10830083

Subgroups — #10830083

Order 768.

1116 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×#5542C211yesmaximal, characteristic, complemented
2C2×C3×#442C211yesmaximal, characteristic
3#82388C211yesmaximal, characteristic, complemented
4C2×#5019C211yesmaximal, characteristic, complemented
5#72045C212yesmaximal
6#71605C211yesmaximal, characteristic, complemented
7#70588C212yesmaximal, complemented
8#70216C212yesmaximal, complemented
9#70213C212yesmaximal, complemented
10#69838C211yesmaximal, characteristic, complemented
11#69822C211yesmaximal, characteristic, complemented
12#24197—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#936C2212yes—
17C22×#214C2212yes—
18C2×#926—21no—
19C2×#924—21no—
20#5542—24no—
21#5542—24no—
22C2×#887C2212yes—
23C2×#879—22no—
24C2×#869C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C3×#442—24no—
27C2×C3×#106C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31#5221C2212yes—
32C2×#832C2212yes—
33#5210—22no—
34C2×#816C2212yes—
35C2×#816C2212yes—
36#5019—24no—
37#4972C2212yes—
38C2×#810C2212yes—
39#4949—21no—
40#4947—21no—
41#4936C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#556—31no—
47C2×#461—31no—
48C2×#442D311yescharacteristic
49#2580—31no—
50#2042—32no—
51#2012—32no—
52#1985—31no—
53#1960—31no—
54#1880—32no—
55#1865—32no—
56#1804—31no—
57C22×C3×D4C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C23×C3×C4—21no—
61C22×Dic6D411yescharacteristic
62C22×Dic6C2312yes—
63C2×C3×D8—22no—
64C2×C3×D8—22no—
65C2×C3×D8—22no—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C3×C4×D4D414yes—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C42C2311yescharacteristic
77C2×#217—22no—
78C2×#217—22no—
79C2×#217—22no—
80C2×#217—22no—
81#936—24no—
82#936—24no—
83C2×#214—24no—
84C2×#214—24no—
85C2×#214—42no—
86#926—28no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3D411yescharacteristic
90C2×C4⋊Dic3C2312yes—
91#924—28no—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4D412yes—
95C2×C4×Dic3—22no—
96C2×C4×Dic3C2312yes—
97C22×C3⋊C8C2312yes—
98#887—24no—
99#887—24no—
100#879—216no—
101#869—22no—
102#869D414yes—
103C4×Dic6—22no—
104C4×Dic6—22no—
105C3×#106—22no—
106C3×#102—22no—
107C3×#102—22no—
108C3×#97—22no—
109#832—42no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—22no—
113#816—24no—
114#816—42no—
115#810—24no—
116C4⋊(C3⋊C8)—22no—
117C22×SD16—32no—
118C22×D8C2×D311yescharacteristic
119#556—64no—
120#556—64no—
121C2×C4⋊Q8—31no—
122C2×#126—62no—
123C2×#124—32no—
124C2×C22⋊Q8—61no—
125C2×C4×Q8—31no—
126C2×C4×Q8—32no—
127C2×C4×D4C2×D311yescharacteristic
128C2×C4×D4C2×D311yescharacteristic
129C2×#117—61no—
130#461—64no—
131#442—24no—
132C2×#106C2×D311yescharacteristic
133C2×C4⋊C8—31no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—32no—
137C2×#102—32no—
138C2×#102C2×D311yescharacteristic
139C2×#102C2×D311yescharacteristic
140C2×#97C2×D311yescharacteristic
141#399—61no—
142#393—61no—
143#390—32no—
144#389—32no—
145#385—62no—
146#382—31no—
147#340—32no—
148#336—32no—
149#336—32no—
150C24×C3—21no—
151C24×C3—21no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4C2×D412yes—
163C2×C3×D4#4112yes—
164C22×C3×C4—21no—
165C22×C3×C4—24no—
166C22×C3×C4—21no—
167C22×C3×C4—44no—
168C22×C3×C4—42no—
169C22×C3×C4—82no—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3C2×D412yes—
176C22×Dic3C2×D412yes—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—42no—
184C2×Dic6C2×D414yes—
185C3×D8—28no—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8#4112yes—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4C2×D412yes—
196C3×C22⋊C4—24no—
197C3×C22⋊C4—42no—
198C3×C42C2×D412yes—
199C3×C42C2×D412yes—
200#217—28no—
201#217—28no—
202#217—44no—
203#217—82no—
204#214—48no—
205C4⋊Dic3—22no—
206C4⋊Dic3—24no—
207C4⋊Dic3—24no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210Dic3⋊C4—28no—
211Dic3⋊C4—24no—
212Dic3⋊C4—82no—
213C4×Dic3—28no—
214C4×Dic3—22no—
215C4×Dic3—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C22×Q8—31no—
220C22×Q8—32no—
221C22×D4C22×D311yescharacteristic
222C22×D4C22×D311yescharacteristic
223C23×C4—21no—
224C23×C4—21no—
225C2×SD16—64no—
226C2×SD16—64no—
227C2×SD16—122no—
228C2×D8—22no—
229C2×D8—22no—
230C2×D8—22no—
231C22×C8—32no—
232C22×C8C22×D311yescharacteristic
233C22×C8C22×D311yescharacteristic
234C4⋊Q8—34no—
235C4⋊Q8—62no—
236#126—616no—
237#124—64no—
238#124—64no—
239C22⋊Q8—68no—
240C4×Q8—64no—
241C4×Q8—64no—
242C4×Q8—62no—
243C4×Q8—62no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4#6714yes—
248#117—68no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4C22×D311yescharacteristic
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C42—32no—
265C2×C42—62no—
266C2×C42C22×D311yescharacteristic
267#106—22no—
268C4⋊C8—62no—
269#103—64no—
270#103—64no—
271#103—122no—
272#102—22no—
273#102—22no—
274#102—122no—
275#97—22no—
276#95—62no—
277#95—61no—
278#95—61no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3C22×D411yescharacteristic
290C3×D4—22no—
291C3×D4—22no—
292C3×D4—22no—
293C3×D4—22no—
294C3×D4—41no—
295C3×D4—41no—
296C3×D4—41no—
297C3×D4—41no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—48no—
302C2×C3×C4—42no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—84no—
306C2×C3×C4#12012yes—
307C2×C3×C4#12012yes—
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C2×#4111yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312C2×C3×C4H(5,𝔽2)11yescharacteristic
313C2×C3×C42−1+411yescharacteristic
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—24no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326Dic6—24no—
327Dic6—24no—
328Dic6—44no—
329Dic6—42no—
330Dic6—42no—
331C3×C8—22no—
332C3×C8—41no—
333C3⋊C8—42no—
334C24—21no—
335C24—21no—
336C2×Q8—34no—
337C2×Q8—62no—
338C2×Q8—62no—
339C2×Q8—62no—
340C2×Q8—62no—
341C2×Q8—62no—
342C2×Q8—62no—
343C2×Q8—122no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4#23512yes—
355C2×D4C2×#6712yes—
356C22×C4—21no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—32no—
360C22×C4—32no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—82no—
366C22×C4D3×D411yescharacteristic
367C22×C4D3×D411yescharacteristic
368C22×C4C23×D311yescharacteristic
369SD16—128no—
370D8—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—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—68no—
384C4⋊C4—62no—
385C4⋊C4—64no—
386C4⋊C4—64no—
387C4⋊C4—64no—
388C4⋊C4—62no—
389C4⋊C4—62no—
390C4⋊C4—242no—
391C4⋊C4C2×#6712yes—
392C22⋊C4—24no—
393C22⋊C4—42no—
394C22⋊C4—68no—
395C22⋊C4—68no—
396C22⋊C4—124no—
397C22⋊C4—242no—
398C42—68no—
399C42—62no—
400C42—62no—
401C42D3×D412yes—
402C42C2×#6712yes—
403C22×C3—44no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—82no—
409C22×C3—82no—
410C22×C3—82no—
411C22×C3—81no—
412C22×C3—81no—
413C22×C3C2×#12011yescharacteristic
414C22×C3#54811yescharacteristic
415C22×C3#55011yescharacteristic
416C22×C3#57512yes—
417C22×C3#57712yes—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—42no—
421C3×C4—81no—
422C3×C4#52511yescharacteristic
423C3×C4#53711yescharacteristic
424C3×C4#54611yescharacteristic
425C3×C4#54711yescharacteristic
426Dic3—44no—
427Dic3—42no—
428Dic3—42no—
429Dic3—82no—
430C23—22no—
431C23—22no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23C2×D3×D411yescharacteristic
441Q8—64no—
442Q8—64no—
443Q8—124no—
444Q8—122no—
445Q8—122no—
446D4—22no—
447D4—22no—
448D4—22no—
449D4—22no—
450D4—41no—
451D4—41no—
452D4—41no—
453D4—41no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—48no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—64no—
468C2×C4—84no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4#93712yes—
475C2×C4#93712yes—
476C2×C4C2×#23511yescharacteristic
477C2×C4C2×#23511yescharacteristic
478C2×C4C2×D3×D411yescharacteristic
479C2×C4#100911yescharacteristic
480C2×C4#101011yescharacteristic
481C2×C4C22×#6711yescharacteristic
482C8—22no—
483C8—41no—
484C8—122no—
485C2×C3—82no—
486C2×C3—82no—
487C2×C3#240711yescharacteristic
488C2×C3#300511yescharacteristic
489C2×C3#301011yescharacteristic
490C2×C3#337612yes—
491C2×C3#340112yes—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—44no—
497C22—81no—
498C22—81no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22#555211yescharacteristic
503C22#555311yescharacteristic
504C22#577312yes—
505C22#577512yes—
506C22C2×#93711yescharacteristic
507C4—22no—
508C4—22no—
509C4—42no—
510C4—81no—
511C4—124no—
512C4—122no—
513C4—122no—
514C4—242no—
515C4#555011yescharacteristic
516C4#555111yescharacteristic
517C4#579111yescharacteristic
518C4#583311yescharacteristic
519C3#2419711yescharacteristic, complemented, Sylow
520C2—82no—
521C2—82no—
522C2#7985412yes—
523C2#7985812yes—
524C2#8240211yescharacteristic
525C2#8394511yescharacteristic
526C2#8394811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#442C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
hypercenterC2×#442D3
hypocenterC3#24197
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#442

Complements

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

NH ≅ G/Ntype
1. C2×#5542C2semidirect
3. #82388C2semidirect
4. C2×#5019C2semidirect
6. #71605C2semidirect
7. #70588C2semidirect
8. #70216C2semidirect
9. #70213C2semidirect
10. #69838C2semidirect
11. #69822C2semidirect
519. C3#24197semidirect

Maximal subgroups

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

subgroupindex
C2×#55422
C2×C3×#4422
#823882
C2×#50192
#720452
#716052
#705882
#702162
#702132
#698382
#698222
#241973