Finite Group Databasebeta

← #10795104

Subgroups — #10795104

Order 768.

1342 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×#5522C211yesmaximal, characteristic
2C2×#4840C212yesmaximal, complemented
3C3×#1798C211yesmaximal, characteristic, complemented
4#71583C211yesmaximal, characteristic, complemented
5#70475C212yesmaximal, complemented
6#69956C211yesmaximal, characteristic
7#69956C211yesmaximal, characteristic
8#69906C211yesmaximal, characteristic, complemented
9Dic3×C22⋊C8C211yesmaximal, characteristic, complemented
10#68664C212yesmaximal, complemented
11#68601C212yesmaximal, complemented
12#20720—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C4×Dic3C2211yescharacteristic
16#5522—28no—
17C2×#893C412yes—
18C2×#887C412yes—
19C2×#67⋊C4C412yes—
20C2×#67⋊C4C412yes—
21C2×#872C2211yescharacteristic
22C2×#872C2211yescharacteristic
23C2×C42×D3C2211yescharacteristic
24C2×C42×D3C2211yescharacteristic
25C2×C3×C4⋊C8C2212yes—
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×#830C2211yescharacteristic
29C2×#830C2211yescharacteristic
30C2×(C2×D3)⋊C8C2212yes—
31C2×(C2×D3)⋊C8C2212yes—
32C2×(C2×D3)⋊C8C2212yes—
33C2×(C2×D3)⋊C8C2212yes—
34C2×#815C2212yes—
35C2×C8×Dic3C2212yes—
36C2×C4⋊(C3⋊C8)C2212yes—
37#4840—28no—
38C3×#336C2212yes—
39#4550C2212yes—
40#4550C2212yes—
41#4524C2212yes—
42C2×C4×#41—31no—
43C2×#433—32no—
44#1817—32no—
45#1798—31no—
46#1798D311yescharacteristic, complemented
47#1797—32no—
48#1780—31no—
49#1780—31no—
50#1750—32no—
51#1749—31no—
52C4×C22⋊C8—31no—
53C23×C3×C4C2311yescharacteristic
54C22×#67C2×C411yescharacteristic
55C22×#67C2×C411yescharacteristic
56C23×Dic3C2×C412yes—
57C2×#235—22no—
58C2×#235—22no—
59C2×#235—24no—
60C2×#235—24no—
61C22×D12C2×C411yescharacteristic
62C22×C4×D3C2×C412yes—
63C22×C4×D3C2×C412yes—
64C22×C4×D3C2311yescharacteristic
65C22×C4×D3C2311yescharacteristic
66C22×Dic6C2×C411yescharacteristic
67C22×C3×C8C2312yes—
68C22×C3×C8C2312yes—
69C3×#117—24no—
70C2×C3×C4⋊C4C2×C412yes—
71C2×C3×C22⋊C4C2×C412yes—
72C2×C3×C42C2311yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×(C2×D3)⋊C4C2×C412yes—
75C2×(C2×D3)⋊C4C2×C412yes—
76C2×Dic3⋊C4C2×C412yes—
77C2×Dic3⋊C4C2×C412yes—
78C2×C4×Dic3—24no—
79C2×C4×Dic3C2×C412yes—
80C2×C4×Dic3C2×C412yes—
81C2×C4×Dic3C2×C412yes—
82C2×C4×Dic3C2311yescharacteristic
83C2×C4×Dic3C2311yescharacteristic
84C22×C3⋊C8C2312yes—
85C22×C3⋊C8C2312yes—
86#893—28no—
87#887—28no—
88#67⋊C4—28no—
89#67⋊C4—28no—
90#872—24no—
91#872D418yes—
92C42×D3—24no—
93C42×D3—24no—
94C3×C4⋊C8—24no—
95C3×C22⋊C8—22no—
96C3×C22⋊C8—22no—
97#830—22no—
98#830—22no—
99(C2×D3)⋊C8—24no—
100(C2×D3)⋊C8—24no—
101(C2×D3)⋊C8—24no—
102(C2×D3)⋊C8—24no—
103#815—24no—
104C8×Dic3—24no—
105C4⋊(C3⋊C8)—24no—
106C22×#41—31no—
107C4×#41—68no—
108C2×C4×Q8—32no—
109C2×C4×D4—32no—
110C2×C4×D4—32no—
111C2×C4×D4—32no—
112C2×#117—31no—
113C2×#117—31no—
114C2×#117C2×D311yescharacteristic
115C22×C42—31no—
116C22×C42—31no—
117C22×C42—31no—
118#433—68no—
119C2×C4⋊C8—32no—
120C2×C4⋊C8C2×D312yes—
121C2×C22⋊C8—31no—
122C2×C22⋊C8—31no—
123C2×C22⋊C8—32no—
124C2×C22⋊C8—32no—
125C2×C22⋊C8—32no—
126C2×C22⋊C8—32no—
127C2×C22⋊C8C2×D311yescharacteristic
128C2×C22⋊C8C2×D311yescharacteristic
129C2×#97—32no—
130C2×C4×C8—32no—
131#336—32no—
132#336—32no—
133#336—32no—
134#336C2×D312yes—
135C24×C3C22×C411yescharacteristic
136C23×D3C22×C411yescharacteristic
137C23×D3C22×C411yescharacteristic
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4C22×C412yes—
141C22×C3×C4C22×C412yes—
142C22×C3×C4C22×C411yescharacteristic
143C22×C3×C4#4112yes—
144C22×C3×C4#4112yes—
145C22×C3×C4C2411yescharacteristic
146C2×#67—22no—
147C2×#67—22no—
148C2×#67—22no—
149C2×#67—22no—
150C2×#67—22no—
151C2×#67—22no—
152C22×Dic3—24no—
153C22×Dic3—24no—
154C22×Dic3—42no—
155C22×Dic3C22×C411yescharacteristic
156C22×Dic3C22×C412yes—
157C22×Dic3C22×C411yescharacteristic
158C22×Dic3C22×C412yes—
159#235—216no—
160#235—216no—
161C2×D12—22no—
162C2×D12—22no—
163C2×D12—22no—
164C2×C4×D3—22no—
165C2×C4×D3—21no—
166C2×C4×D3—21no—
167C2×C4×D3—21no—
168C2×C4×D3—21no—
169C2×C4×D3—22no—
170C2×C4×D3—24no—
171C2×C4×D3—24no—
172C2×C4×D3—24no—
173C2×C4×D3—24no—
174C2×C4×D3—42no—
175C2×C4×D3—42no—
176C2×C4×D3C2×D414yes—
177C2×C4×D3#4114yes—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C3×C4⋊C4—24no—
188C3×C22⋊C4—24no—
189C3×C42—22no—
190C3×C42C2×D414yes—
191(C2×D3)⋊C4—24no—
192(C2×D3)⋊C4C22⋊C418yes—
193Dic3⋊C4—24no—
194Dic3⋊C4C22⋊C418yes—
195C4×Dic3—24no—
196C4×Dic3—22no—
197C4×Dic3—24no—
198C4×Dic3—24no—
199C4×Dic3C2×D414yes—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—22no—
206C2×#41—64no—
207C2×#41—64no—
208C2×#41—62no—
209C2×#41—62no—
210C22×Q8—31no—
211C22×D4—31no—
212C22×D4—31no—
213C22×D4—31no—
214C23×C4—31no—
215C23×C4—31no—
216C23×C4—32no—
217C23×C4—32no—
218C23×C4—32no—
219C23×C4C22×D311yescharacteristic
220C22×C8—32no—
221C22×C8—32no—
222C22×C8C22×D312yes—
223C22×C8C22×D312yes—
224C4×Q8—68no—
225C4×D4—68no—
226C4×D4—68no—
227C4×D4—68no—
228#117—24no—
229#117—38no—
230#117—64no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—32no—
233C2×C4⋊C4C4×D312yes—
234C2×C22⋊C4—32no—
235C2×C22⋊C4—32no—
236C2×C22⋊C4C4×D312yes—
237C2×C42—31no—
238C2×C42—32no—
239C2×C42—32no—
240C2×C42—32no—
241C2×C42—31no—
242C2×C42—64no—
243C2×C42—64no—
244C2×C42—64no—
245C2×C42C22×D311yescharacteristic
246C2×C42C22×D311yescharacteristic
247C4⋊C8—24no—
248C4⋊C8—64no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—64no—
252C22⋊C8—64no—
253C22⋊C8—64no—
254C22⋊C8—64no—
255C22⋊C8—62no—
256C22⋊C8—62no—
257#97—64no—
258C4×C8—64no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3#11712yes—
264C23×C3C23×C411yescharacteristic
265C22×D3—21no—
266C22×D3—21no—
267C22×D3—21no—
268C22×D3—21no—
269C22×D3—22no—
270C22×D3—22no—
271C22×D3—22no—
272C22×D3—22no—
273C22×D3C2×C22⋊C412yes—
274C22×D3#11712yes—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4C2×C22⋊C414yes—
285C2×C3×C4#11712yes—
286C2×C3×C4C22×D412yes—
287C2×C3×C4C2×#4111yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C4C2×#4112yes—
290#67—28no—
291#67—28no—
292C2×Dic3—22no—
293C2×Dic3—21no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—22no—
298C2×Dic3—22no—
299C2×Dic3—22no—
300C2×Dic3—48no—
301C2×Dic3C2×C22⋊C412yes—
302C2×Dic3C2×C22⋊C414yes—
303C2×Dic3#11712yes—
304C2×Dic3#13114yes—
305D12—28no—
306C4×D3—24no—
307C4×D3—24no—
308C4×D3—24no—
309C4×D3—24no—
310C4×D3—48no—
311C4×D3—48no—
312Dic6—28no—
313C3×C8—42no—
314C3×C8—42no—
315C3⋊C8—42no—
316C3⋊C8—42no—
317C24—31no—
318C24—31no—
319C24C2×C4×D311yescharacteristic
320#41—616no—
321#41—616no—
322C2×Q8—62no—
323C2×Q8—62no—
324C2×Q8—62no—
325C2×D4—62no—
326C2×D4—62no—
327C2×D4—62no—
328C2×D4—62no—
329C2×D4—62no—
330C2×D4—62no—
331C2×D4—62no—
332C2×D4—62no—
333C2×D4—62no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—32no—
337C22×C4—32no—
338C22×C4—31no—
339C22×C4—31no—
340C22×C4—34no—
341C22×C4—34no—
342C22×C4—64no—
343C22×C4—64no—
344C22×C4—64no—
345C22×C4—64no—
346C22×C4—64no—
347C22×C4—62no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—62no—
353C22×C4—64no—
354C22×C4—122no—
355C22×C4—122no—
356C22×C4—122no—
357C22×C4C2×C4×D312yes—
358C22×C4C2×C4×D312yes—
359C22×C4C2×C4×D311yescharacteristic
360C22×C4#23712yes—
361C22×C4#23912yes—
362C22×C4C23×D311yescharacteristic
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—62no—
370C2×C8—62no—
371C2×C8—62no—
372C2×C8—62no—
373C2×C8—62no—
374C2×C8—62no—
375C4⋊C4—24no—
376C4⋊C4—38no—
377C4⋊C4—64no—
378C22⋊C4—24no—
379C22⋊C4—38no—
380C22⋊C4—64no—
381C42—22no—
382C42—34no—
383C42—64no—
384C42—64no—
385C42—64no—
386C42—62no—
387C42D3×D414yes—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3C22×C22⋊C411yescharacteristic
394C22×C3C2×#11711yescharacteristic
395C22×C3C2×#11711yescharacteristic
396C22×C3C2×#13112yes—
397C22×C3#56812yes—
398C2×D3—24no—
399C2×D3—24no—
400C2×D3—22no—
401C2×D3—22no—
402C2×D3—42no—
403C2×D3—41no—
404C2×D3—41no—
405C2×D3—41no—
406C2×D3—41no—
407C2×D3—42no—
408C3×C4—24no—
409C3×C4—22no—
410C3×C4—42no—
411C3×C4#52512yes—
412C3×C4#52512yes—
413Dic3—24no—
414Dic3—24no—
415Dic3—22no—
416Dic3—22no—
417C23—21no—
418C23—22no—
419C23—22no—
420C23—21no—
421C23—32no—
422C23—32no—
423C23—61no—
424C23—62no—
425C23—62no—
426C23—62no—
427C23—62no—
428C23—61no—
429C23—61no—
430C23—61no—
431C23#89212yes—
432C23C22×C4×D311yescharacteristic
433Q8—68no—
434D4—68no—
435D4—68no—
436D4—68no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—34no—
447C2×C4—32no—
448C2×C4—34no—
449C2×C4—32no—
450C2×C4—62no—
451C2×C4—61no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—64no—
459C2×C4—64no—
460C2×C4—64no—
461C2×C4—64no—
462C2×C4—128no—
463C2×C4—128no—
464C2×C4—128no—
465C2×C4D3×C22⋊C414yes—
466C2×C4#89212yes—
467C2×C4C2×D3×D412yes—
468C2×C4C2×#23711yescharacteristic
469C2×C4C2×#23911yescharacteristic
470C2×C4D3×#4112yes—
471C8—42no—
472C8—42no—
473C8—122no—
474C8—122no—
475C2×C3—42no—
476C2×C3#228011yescharacteristic
477C2×C3#288012yes—
478C2×C3#292211yescharacteristic
479C2×C3#292311yescharacteristic
480C2×C3#292512yes—
481D3—42no—
482D3—42no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—42no—
487C22—42no—
488C22—64no—
489C22—64no—
490C22—62no—
491C22—62no—
492C22—122no—
493C22—122no—
494C22—121no—
495C22—121no—
496C22—121no—
497C22—121no—
498C22C2×D3×C22⋊C411yescharacteristic
499C22C2×#89211yescharacteristic
500C22D3×#11711yescharacteristic
501C22D3×#13112yes—
502C22#575012yes—
503C4—24no—
504C4—22no—
505C4—42no—
506C4—64no—
507C4—64no—
508C4—62no—
509C4—62no—
510C4#560612yes—
511C4#562712yes—
512C3#2072011yescharacteristic, complemented, Sylow
513C2—42no—
514C2—122no—
515C2—122no—
516C2#7386211yescharacteristic
517C2#7866312yes—
518C2#7907011yescharacteristic
519C2#7907111yescharacteristic
520C2#7937512yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1798C2
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenter#1798D3
hypocenterC3#20720
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1798

Complements

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

NH ≅ G/Ntype
2. C2×#4840C2semidirect
3. C3×#1798C2semidirect
4. #71583C2semidirect
5. #70475C2semidirect
8. #69906C2semidirect
9. Dic3×C22⋊C8C2semidirect
10. #68664C2semidirect
11. #68601C2semidirect
46. #1798D3semidirect
512. C3#20720semidirect

Maximal subgroups

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

subgroupindex
C2×#55222
C2×#48402
C3×#17982
#715832
#704752
#699562
#699062
Dic3×C22⋊C82
#686642
#686012
#207203