Finite Group Databasebeta

← #10829979

Subgroups — #10829979

Order 768.

1082 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5542C211yesmaximal, characteristic
2C2×C3×C4×SD16C211yesmaximal, characteristic
3#82482C211yesmaximal, characteristic, complemented
4C2×#5031C211yesmaximal, characteristic, complemented
5#72058C212yesmaximal
6#71604C211yesmaximal, characteristic
7#70588C212yesmaximal, complemented
8#70281C212yesmaximal, complemented
9#70216C212yesmaximal
10#69788C211yesmaximal, characteristic, complemented
11#69771C211yesmaximal, characteristic, complemented
12#24189—31nomaximal, Sylow
13C22×C3×SD16C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C22×#216C2212yes—
17C2×#936C2212yes—
18C2×#926—21no—
19C2×#924—21no—
20#5542—24no—
21#5542—24no—
22C2×#887C2212yes—
23C2×#879—22no—
24C2×#869C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C3×C4×SD16—24no—
27C2×C3×#106C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×C4×C8C2211yescharacteristic
31#5229C2212yes—
32C2×#832C2212yes—
33C2×#816C2212yes—
34C2×#816C2212yes—
35#5031—24no—
36#4982—22no—
37#4972C2212yes—
38C2×#810C2212yes—
39#4935C2211yescharacteristic
40#4929—21no—
41#4928—21no—
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#556—31no—
47C2×#462—31no—
48C2×C4×SD16D311yescharacteristic
49#2582—31no—
50#2042—32no—
51#2015—32no—
52#1984—31no—
53#1956—31no—
54#1878—32no—
55#1865—32no—
56#1802—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×Dic6D411yescharacteristic
61C22×Dic6C2312yes—
62C2×C3×SD16—22no—
63C2×C3×SD16—22no—
64C2×C3×SD16—22no—
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C3×C4×D4—22no—
70C3×C4×D4D414yes—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×C3×C42—21no—
76C2×C3×C42C2311yescharacteristic
77C2×#217—22no—
78C2×#217—22no—
79C2×#216—24no—
80C2×#216—24no—
81C2×#216—42no—
82#936—24no—
83#936—24no—
84#926—28no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
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×#103—22no—
107C3×#102—22no—
108C3×C4×C8—22no—
109#832—42no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113#816—24no—
114#816—42no—
115#810—24no—
116C4⋊(C3⋊C8)—22no—
117C22×Q16—32no—
118C22×SD16C2×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×Q8C2×D311yescharacteristic
128C2×C4×D4C2×D311yescharacteristic
129C2×#117—61no—
130#462—64no—
131C4×SD16—24no—
132C2×#106C2×D311yescharacteristic
133C2×C4⋊C8—31no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—32no—
137C2×#103C2×D311yescharacteristic
138C2×#102—32no—
139C2×#102C2×D311yescharacteristic
140C2×C4×C8C2×D311yescharacteristic
141#398—62no—
142#391—32no—
143#391—61no—
144#389—32no—
145#389—61no—
146#384—31no—
147#340—32no—
148#336—32no—
149#336—32no—
150C24×C3—21no—
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—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4C2×D412yes—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—24no—
166C22×C3×C4—21no—
167C22×C3×C4—42no—
168C22×C3×C4—42no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3C2×D412yes—
175C22×Dic3C2×D412yes—
176C2×Dic6—22no—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—22no—
184C2×Dic6C2×D414yes—
185C3×SD16—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⋊C4—42no—
196C3×C4⋊C4C2×D412yes—
197C3×C22⋊C4—24no—
198C3×C42—42no—
199C3×C42C2×D412yes—
200C3×C42C2×D412yes—
201#217—28no—
202#217—28no—
203#216—48no—
204C4⋊Dic3—22no—
205C4⋊Dic3—24no—
206C4⋊Dic3—24no—
207C4⋊Dic3—22no—
208C4⋊Dic3—22no—
209C4⋊Dic3—42no—
210C4⋊Dic3—81no—
211Dic3⋊C4—28no—
212Dic3⋊C4—24no—
213Dic3⋊C4—82no—
214C4×Dic3—28no—
215C4×Dic3—22no—
216C4×Dic3—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C22×Q8—31no—
221C22×Q8—32no—
222C22×Q8C22×D311yescharacteristic
223C22×D4C22×D311yescharacteristic
224C23×C4—21no—
225C2×Q16—64no—
226C2×Q16—64no—
227C2×Q16—122no—
228C2×SD16—22no—
229C2×SD16—22no—
230C2×SD16—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—22no—
241C4×Q8—22no—
242C4×Q8—64no—
243C4×Q8—64no—
244C4×Q8—62no—
245C4×Q8—62no—
246C4×D4—22no—
247C4×D4#6714yes—
248#117—68no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4C22×D311yescharacteristic
260C2×C4⋊C4C22×D311yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C42—21no—
265C2×C42—32no—
266C2×C42—62no—
267C2×C42C22×D311yescharacteristic
268#106—22no—
269C4⋊C8—62no—
270#103—22no—
271#103—64no—
272#103—64no—
273#103—122no—
274#102—22no—
275#102—122no—
276C4×C8—22no—
277#95—62no—
278#95—62no—
279#95—62no—
280C23×C3—22no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3C22×D411yescharacteristic
288C3×Q8—22no—
289C3×Q8—22no—
290C3×Q8—41no—
291C3×Q8—41no—
292C3×D4—22no—
293C3×D4—22no—
294C3×D4—41no—
295C3×D4—41no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
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—48no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4#12012yes—
309C2×C3×C4#12012yes—
310C2×C3×C4#13512yes—
311C2×C3×C4#13712yes—
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4H(5,𝔽2)11yescharacteristic
317C2×C3×C42−1+411yescharacteristic
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—24no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330Dic6—24no—
331Dic6—24no—
332Dic6—42no—
333Dic6—42no—
334Dic6—44no—
335C3×C8—22no—
336C3×C8—41no—
337C3⋊C8—42no—
338C24—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—34no—
345C2×Q8—62no—
346C2×Q8—62no—
347C2×Q8—62no—
348C2×Q8—62no—
349C2×Q8—62no—
350C2×Q8—62no—
351C2×Q8—62no—
352C2×Q8—62no—
353C2×Q8#23512yes—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4C2×#6712yes—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—32no—
365C22×C4—32no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4D3×D411yescharacteristic
371C22×C4D3×D411yescharacteristic
372C22×C4C23×D311yescharacteristic
373Q16—128no—
374SD16—28no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—62no—
381C2×C8—62no—
382C2×C8—62no—
383C2×C8#23512yes—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—42no—
388C4⋊C4—68no—
389C4⋊C4—62no—
390C4⋊C4—64no—
391C4⋊C4—64no—
392C4⋊C4—64no—
393C4⋊C4—62no—
394C4⋊C4—62no—
395C4⋊C4—122no—
396C4⋊C4—241no—
397C4⋊C4—242no—
398C4⋊C4C2×#6712yes—
399C22⋊C4—24no—
400C22⋊C4—68no—
401C22⋊C4—68no—
402C42—42no—
403C42—68no—
404C42—62no—
405C42—62no—
406C42D3×D412yes—
407C42C2×#6712yes—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—81no—
412C22×C3—81no—
413C22×C3—81no—
414C22×C3—81no—
415C22×C3C2×#12011yescharacteristic
416C22×C3#54811yescharacteristic
417C22×C3#55011yescharacteristic
418C22×C3C2×#13511yescharacteristic
419C22×C3C2×#13711yescharacteristic
420C22×C3#57711yescharacteristic
421C22×C3#57811yescharacteristic
422C3×C4—22no—
423C3×C4—42no—
424C3×C4—42no—
425C3×C4—81no—
426C3×C4#45214yes—
427C3×C4#52511yescharacteristic
428C3×C4#53711yescharacteristic
429C3×C4#54611yescharacteristic
430C3×C4#54711yescharacteristic
431Dic3—44no—
432Dic3—42no—
433Dic3—42no—
434Dic3—82no—
435C23—22no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23C2×D3×D411yescharacteristic
443Q8—22no—
444Q8—22no—
445Q8—41no—
446Q8—41no—
447Q8—64no—
448Q8—64no—
449Q8—124no—
450Q8—122no—
451Q8—122no—
452D4—22no—
453D4—22no—
454D4—41no—
455D4—41no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—48no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—64no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4#91512yes—
481C2×C4#91612yes—
482C2×C4#93712yes—
483C2×C4#93712yes—
484C2×C4C2×#23511yescharacteristic
485C2×C4C2×#23511yescharacteristic
486C2×C4C2×D3×D411yescharacteristic
487C2×C4#100911yescharacteristic
488C2×C4#101011yescharacteristic
489C2×C4C22×#6711yescharacteristic
490C8—22no—
491C8—41no—
492C8—122no—
493C2×C3—82no—
494C2×C3#240711yescharacteristic
495C2×C3C2×#45211yescharacteristic
496C2×C3#301311yescharacteristic
497C2×C3#336411yescharacteristic
498C2×C3#336911yescharacteristic
499C2×C3#339411yescharacteristic
500C2×C3#339811yescharacteristic
501C22—42no—
502C22—42no—
503C22—42no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22#555211yescharacteristic
509C22#555311yescharacteristic
510C22C2×#91511yescharacteristic
511C22C2×#91611yescharacteristic
512C22#577011yescharacteristic
513C22#577111yescharacteristic
514C22C2×#93711yescharacteristic
515C4—22no—
516C4—42no—
517C4—42no—
518C4—81no—
519C4—122no—
520C4—122no—
521C4—124no—
522C4—242no—
523C4#517314yes—
524C4#555011yescharacteristic
525C4#555111yescharacteristic
526C4#579111yescharacteristic
527C4#583311yescharacteristic
528C3#2418911yescharacteristic, complemented, Sylow
529C2—82no—
530C2#7962311yescharacteristic
531C2#7962611yescharacteristic
532C2#7963811yescharacteristic
533C2#7964011yescharacteristic
534C2#8240211yescharacteristic
535C2C2×#517311yescharacteristic
536C2#8391311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×SD16C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×C4×SD16D3
hypocenterC3#24189
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
3. #82482C2semidirect
4. C2×#5031C2semidirect
7. #70588C2semidirect
8. #70281C2semidirect
10. #69788C2semidirect
11. #69771C2semidirect
528. C3#24189semidirect

Maximal subgroups

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

subgroupindex
C2×#55422
C2×C3×C4×SD162
#824822
C2×#50312
#720582
#716042
#705882
#702812
#702162
#697882
#697712
#241893