Finite Group Databasebeta

← #10801323

Subgroups — #10801323

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2392C211yesmaximal, characteristic
2C2×#5012C212yesmaximal, complemented
3C2×#5012C211yesmaximal, characteristic, complemented
4C2×#4997C211yesmaximal, characteristic, complemented
5#71615C211yesmaximal, characteristic, complemented
6#71610C212yesmaximal, complemented
7#71583C211yesmaximal, characteristic, complemented
8#69485C211yesmaximal, characteristic, complemented
9C4⋊C4×C3⋊C8C211yesmaximal, characteristic, complemented
10#69451C212yesmaximal, complemented
11#69448C212yesmaximal, complemented
12#21494—31nomaximal, Sylow
13C2×C3×C4×D4C2212yes—
14C2×C3×C4×D4C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C22×#208C2212yes—
17C22×#208C2211yescharacteristic
18C23×C3⋊C8C2211yescharacteristic
19C3×#398C411yescharacteristic
20C3×#397C411yescharacteristic
21C3×#396C412yes—
22C3×#394C412yes—
23C3×#394C411yescharacteristic
24C3×#392C411yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C3×C4×C22⋊C4C2212yes—
27C2×#830C2212yes—
28C2×#830C2212yes—
29C2×#830C2212yes—
30C2×#830C2212yes—
31#5012—28no—
32#5012—24no—
33#4997—24no—
34C2×C4⋊(C3⋊C8)C2212yes—
35C2×C4⋊(C3⋊C8)C2211yescharacteristic
36C2×C4⋊(C3⋊C8)C2211yescharacteristic
37C2×#803C2211yescharacteristic
38C2×#803C2211yescharacteristic
39C2×C4×C3⋊C8C2211yescharacteristic
40C2×C4×C3⋊C8C2212yes—
41C2×C4×C3⋊C8C2211yescharacteristic
42#4524C2212yes—
43#4524C2212yes—
44#4524C2212yes—
45C2×#436—32no—
46C2×#436—31no—
47C2×#405—31no—
48#2392D311yescharacteristic
49#1955—31no—
50#1940—31no—
51#1923—32no—
52#1869—32no—
53#1798—31no—
54C8×C4⋊C4—31no—
55#1753—32no—
56C22×C3×D4C2×C412yes—
57C22×C3×D4C2×C411yescharacteristic
58C23×C3×C4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C3×C4×D4—24no—
61C3×C4×D4—24no—
62C3×C4×D4—22no—
63C3×C4×D4—22no—
64C3×#117—22no—
65C3×#117—22no—
66C2×C3×C4⋊C4C2×C412yes—
67C2×C3×C4⋊C4C2×C411yescharacteristic
68C2×C3×C4⋊C4C2×C411yescharacteristic
69C2×C3×C4⋊C4C2×C412yes—
70C2×C3×C4⋊C4C2×C411yescharacteristic
71C2×C3×C22⋊C4C2×C412yes—
72C2×C3×C22⋊C4C2×C411yescharacteristic
73C2×C3×C22⋊C4C2×C411yescharacteristic
74C2×C3×C22⋊C4C2×C412yes—
75C2×C3×C22⋊C4C2×C412yes—
76C2×C3×C22⋊C4C2×C412yes—
77C2×C3×C22⋊C4C2×C412yes—
78C2×C3×C42C2311yescharacteristic
79C2×C3×C42C2312yes—
80C2×C3×C42C2311yescharacteristic
81C2×#208—22no—
82C2×#208—24no—
83C2×#208—24no—
84C2×#208—22no—
85C2×#208—24no—
86C2×#208D414yescomplemented
87C22×C3⋊C8—22no—
88C22×C3⋊C8—22no—
89C22×C3⋊C8—22no—
90C22×C3⋊C8C2312yes—
91C22×C3⋊C8C2312yes—
92C22×C3⋊C8C2311yescharacteristic
93C22×C3⋊C8C2311yescharacteristic
94C22×C3⋊C8C2311yescharacteristic
95C22×C3⋊C8C2311yescharacteristic
96C3×#95C2×C412yes—
97C3×#95C2×C412yes—
98C3×#95C2×C411yescharacteristic
99C3×#95C2×C411yescharacteristic
100#830—24no—
101#830—24no—
102#830—24no—
103#830—24no—
104C4⋊(C3⋊C8)—22no—
105C4⋊(C3⋊C8)—22no—
106C4⋊(C3⋊C8)—24no—
107#803—22no—
108#803—22no—
109C4×C3⋊C8—22no—
110C4×C3⋊C8—24no—
111C4×C3⋊C8D414yescomplemented
112C22×M16—31no—
113C22×M16—32no—
114C23×C8—31no—
115C2×C4×D4C2×D312yes—
116C2×C4×D4C2×D311yescharacteristic
117C2×#117C2×D311yescharacteristic
118#436—64no—
119#436—68no—
120C2×C4⋊C8—31no—
121C2×C4⋊C8—31no—
122C2×C4⋊C8—32no—
123C2×C22⋊C8—32no—
124C2×C22⋊C8—32no—
125C2×C22⋊C8—32no—
126C2×C22⋊C8—32no—
127#405—64no—
128C2×#97—31no—
129C2×#97—31no—
130C2×C4×C8—31no—
131C2×C4×C8—31no—
132C2×C4×C8—32no—
133#398Dic311yescharacteristic
134#397Dic311yescharacteristic
135#396Dic312yes—
136#394Dic312yes—
137#394Dic311yescharacteristic
138#392Dic311yescharacteristic
139C4×C4⋊C4C2×D311yescharacteristic
140C4×C22⋊C4C2×D312yes—
141#336—32no—
142#336—32no—
143#336—32no—
144C24×C3C22×C412yes—
145C24×C3C22×C411yescharacteristic
146C2×C3×D4—24no—
147C2×C3×D4—22no—
148C2×C3×D4—24no—
149C2×C3×D4—42no—
150C2×C3×D4—42no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4C22×C412yes—
162C22×C3×C4C22×C411yescharacteristic
163C22×C3×C4C22×C411yescharacteristic
164C22×C3×C4C22×C412yes—
165C22×C3×C4C22×C411yescharacteristic
166C22×C3×C4C22×C411yescharacteristic
167C22×C3×C4C22×C411yescharacteristic
168C22×C3×C4C22×C412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4#4112yes—
171C22×C3×C4C2411yescharacteristic
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—24no—
177C3×C22⋊C4—24no—
178C3×C22⋊C4—22no—
179C3×C22⋊C4—22no—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—24no—
182C3×C22⋊C4—24no—
183C3×C22⋊C4—42no—
184C3×C42—21no—
185C3×C42—22no—
186C3×C42—21no—
187C3×C42—22no—
188C3×C42C2×D412yes—
189C3×C42#4112yes—
190#208—28no—
191#208—48no—
192C2×C3⋊C8—28no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—21no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8C2×D412yes—
206C2×C3⋊C8C2×D412yes—
207C2×C3⋊C8C2×D412yes—
208C2×C3⋊C8C2×D412yes—
209C2×C3⋊C8#4112yes—
210C2×C3⋊C8#4112yes—
211C22×D4C2×Dic312yes—
212C22×D4C2×Dic311yescharacteristic
213C23×C4C22×D312yes—
214C23×C4C22×D311yescharacteristic
215C2×M16—34no—
216C2×M16—62no—
217C2×M16—62no—
218C2×M16—64no—
219C2×M16—64no—
220C2×M16—64no—
221C22×C8—31no—
222C22×C8—31no—
223C22×C8—31no—
224C22×C8—31no—
225C22×C8—32no—
226C22×C8—32no—
227C22×C8—62no—
228C22×C8—62no—
229C22×C8—62no—
230C4×D4—24no—
231C4×D4—24no—
232C4×D4—22no—
233C4×D4—22no—
234#117—22no—
235#117—22no—
236C2×C4⋊C4C2×Dic311yescharacteristic
237C2×C4⋊C4C2×Dic311yescharacteristic
238C2×C4⋊C4C2×Dic312yes—
239C2×C4⋊C4C2×Dic311yescharacteristic
240C2×C4⋊C4C2×Dic312yes—
241C2×C22⋊C4C2×Dic311yescharacteristic
242C2×C22⋊C4C2×Dic311yescharacteristic
243C2×C22⋊C4C2×Dic312yes—
244C2×C22⋊C4C2×Dic312yes—
245C2×C22⋊C4C2×Dic312yes—
246C2×C22⋊C4C2×Dic312yes—
247C2×C22⋊C4C2×Dic312yes—
248C2×C42C22×D311yescharacteristic
249C2×C42C22×D312yes—
250C2×C42C22×D311yescharacteristic
251C4⋊C8—64no—
252C4⋊C8—62no—
253C4⋊C8—62no—
254C22⋊C8—64no—
255C22⋊C8—64no—
256C22⋊C8—64no—
257C22⋊C8—64no—
258#97—62no—
259#97—62no—
260C4×C8—34no—
261C4×C8—62no—
262C4×C8—64no—
263#95C2×Dic312yes—
264#95C2×Dic312yes—
265#95C2×Dic311yescharacteristic
266#95C2×Dic311yescharacteristic
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3C4×D412yes—
279C23×C3C23×C411yescharacteristic
280C3×D4—48no—
281C3×D4—44no—
282C2×C3×C4—22no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4C4×D412yes—
307C2×C3×C4C4×D412yes—
308C2×C3×C4C4×D412yes—
309C2×C3×C4#13114yes—
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C3⋊C8—22no—
317C3⋊C8—22no—
318C3⋊C8—22no—
319C3⋊C8—42no—
320C3⋊C8—42no—
321C3⋊C8#12114yes—
322C24C22×Dic312yes—
323C24C22×Dic311yescharacteristic
324C2×D4—24no—
325C2×D4—22no—
326C2×D4—24no—
327C2×D4—42no—
328C2×D4—42no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4D3×D412yes—
340C22×C4#23712yes—
341C22×C4C22×Dic312yes—
342C22×C4C22×Dic311yescharacteristic
343C22×C4C22×Dic311yescharacteristic
344C22×C4C22×Dic312yes—
345C22×C4C22×Dic311yescharacteristic
346C22×C4C22×Dic311yescharacteristic
347C22×C4C22×Dic311yescharacteristic
348C22×C4C22×Dic312yes—
349C22×C4C23×D311yescharacteristic
350M16—68no—
351M16—128no—
352C2×C8—32no—
353C2×C8—32no—
354C2×C8—32no—
355C2×C8—32no—
356C2×C8—32no—
357C2×C8—32no—
358C2×C8—68no—
359C2×C8—61no—
360C2×C8—61no—
361C2×C8—61no—
362C2×C8—61no—
363C2×C8—61no—
364C2×C8—61no—
365C2×C8—62no—
366C2×C8—62no—
367C2×C8—62no—
368C2×C8—62no—
369C2×C8—62no—
370C2×C8—62no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—42no—
383C42—21no—
384C42—22no—
385C42—21no—
386C42—22no—
387C42D3×D412yes—
388C42#23712yes—
389C22×C3—24no—
390C22×C3—22no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—42no—
400C22×C3—42no—
401C22×C3C2×C4×D411yescharacteristic
402C22×C3C2×C4×D411yescharacteristic
403C22×C3C4×#4111yescharacteristic
404C22×C3C2×#13112yes—
405C22×C3#56812yes—
406C3×C4—22no—
407C3×C4—22no—
408C3×C4—24no—
409C3×C4—22no—
410C3×C4—41no—
411C3×C4—41no—
412C3×C4—42no—
413C3×C4—42no—
414C3×C4C2×#12111yescharacteristic
415C3×C4#52511yescharacteristic
416C3×C4#52911yescharacteristic
417C3×C4#53211yescharacteristic
418C23—21no—
419C23—22no—
420C23—22no—
421C23—21no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23D4×Dic312yes—
430C23C23×Dic311yescharacteristic
431D4—48no—
432D4—44no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4D4×Dic312yes—
458C2×C4D4×Dic312yes—
459C2×C4D4×Dic312yes—
460C2×C4#94814yes—
461C2×C4C2×D3×D411yescharacteristic
462C2×C4C2×D3×D411yescharacteristic
463C2×C4C2×#23711yescharacteristic
464C2×C4C2×#23711yescharacteristic
465C2×C4D3×#4111yescharacteristic
466C2×C4D3×#4111yescharacteristic
467C8—34no—
468C8—62no—
469C8—62no—
470C8—62no—
471C8—122no—
472C8—122no—
473C2×C3—42no—
474C2×C3—82no—
475C2×C3C4×#12111yescharacteristic
476C2×C3#293011yescharacteristic
477C2×C3#293111yescharacteristic
478C2×C3#293312yes—
479C2×C3#296912yes—
480C22—22no—
481C22—24no—
482C22—41no—
483C22—42no—
484C22—41no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22C2×D4×Dic311yescharacteristic
493C22C2×D4×Dic311yescharacteristic
494C22C2×#94812yes—
495C22#581912yes—
496C22Dic3×#4111yescharacteristic
497C4—22no—
498C4—22no—
499C4—22no—
500C4—24no—
501C4—41no—
502C4—42no—
503C4—41no—
504C4—42no—
505C4#559611yescharacteristic
506C4#560011yescharacteristic
507C4D3×#12111yescharacteristic
508C4#560611yescharacteristic
509C3#2149411yescharacteristic, complemented, Sylow
510C2—42no—
511C2—82no—
512C2Dic3×#12111yescharacteristic
513C2#8291711yescharacteristic
514C2#8291812yes—
515C2#8312912yes—
516C2#8327511yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#2392C2
prime radicalC3×#392C4
cosocleC22×C3×C4C24
hypercenter#2392D3
hypocenterC3#21494
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #2392

Complements

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

NH ≅ G/Ntype
2. C2×#5012C2semidirect
3. C2×#5012C2semidirect
4. C2×#4997C2semidirect
5. #71615C2semidirect
6. #71610C2semidirect
7. #71583C2semidirect
8. #69485C2semidirect
9. C4⋊C4×C3⋊C8C2semidirect
10. #69451C2semidirect
11. #69448C2semidirect
86. C2×#208D4semidirect
111. C4×C3⋊C8D4semidirect
509. C3#21494semidirect

Maximal subgroups

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

subgroupindex
C3×#23922
C2×#50122
C2×#49972
#716152
#716102
#715832
#694852
C4⋊C4×C3⋊C82
#694512
#694482
#214943