Finite Group Databasebeta

← #10795100

Subgroups — #10795100

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×Dic3×#41C211yesmaximal, characteristic
2C2×#4720C212yesmaximal, complemented
3C3×#1780C211yesmaximal, characteristic
4#72136C211yesmaximal, characteristic
5#69910C212yesmaximal, complemented
6#69909C211yesmaximal, characteristic, complemented
7#69909C211yesmaximal, characteristic, complemented
8#69908C212yesmaximal, complemented
9#69907C212yesmaximal, complemented
10#69906C211yesmaximal, characteristic, complemented
11Dic3×C22⋊C8C211yesmaximal, characteristic, complemented
12#20720—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14Dic3×#41—28no—
15C2×Q8×Dic3C412yes—
16C2×D4×Dic3C412yes—
17C2×D4×Dic3C412yes—
18C2×D4×Dic3C412yes—
19C2×#926C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C22×C4×Dic3C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2212yes—
28C2×C3×C22⋊C8C2212yes—
29C2×#830C2211yescharacteristic
30C2×#830C2211yescharacteristic
31C2×#830C2212yes—
32C2×#830C2212yes—
33C2×#815C2212yes—
34C2×Dic3⋊C8C2212yes—
35C2×Dic3⋊C8C2212yes—
36C2×C8×Dic3C2212yes—
37#4720—28no—
38#4550C2212yes—
39#4550C2212yes—
40#4550C2212yes—
41#4550C2212yes—
42C2×C4×#41—31no—
43C2×#433—32no—
44#1817—32no—
45#1798—31no—
46#1798—31no—
47#1797—32no—
48#1780—31no—
49#1780D311yescharacteristic
50#1750—32no—
51#1749—31no—
52C4×C22⋊C8—31no—
53C2×C3×#41—24no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C2×C3×#41—24no—
57C22×C3×Q8C2×C411yescharacteristic
58C22×C3×D4C2×C411yescharacteristic
59C22×C3×D4C2×C411yescharacteristic
60C22×C3×D4C2×C411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×Dic3C2×C412yes—
65C23×Dic3C2×C412yes—
66C23×Dic3C2×C412yes—
67C22×C3×C8C2312yes—
68C22×C3×C8C2312yes—
69C2×#217C2×C412yes—
70C2×#217C2×C412yes—
71C2×#217C2×C412yes—
72Q8×Dic3—28no—
73D4×Dic3—28no—
74D4×Dic3—28no—
75D4×Dic3—28no—
76#926—24no—
77#926—24no—
78#926D418yes—
79C2×C4⋊Dic3C2×C412yes—
80C2×C4⋊Dic3C2×C412yes—
81C2×C4⋊Dic3C2×C412yes—
82C2×C4×Dic3—24no—
83C2×C4×Dic3—24no—
84C2×C4×Dic3—24no—
85C2×C4×Dic3C2×C412yes—
86C2×C4×Dic3C2×C412yes—
87C2×C4×Dic3C2×C412yes—
88C2×C4×Dic3C2311yescharacteristic
89C2×C4×Dic3C2311yescharacteristic
90C2×C4×Dic3C2311yescharacteristic
91C2×C4×Dic3C2311yescharacteristic
92C22×C3⋊C8C2312yes—
93C22×C3⋊C8C2312yes—
94C3×C22⋊C8—24no—
95C3×C22⋊C8—22no—
96C3×C22⋊C8—22no—
97C3×C22⋊C8—24no—
98#830—24no—
99#830—22no—
100#830—22no—
101#830—24no—
102#815—24no—
103Dic3⋊C8—24no—
104Dic3⋊C8—24no—
105C8×Dic3—24no—
106C22×#41C2×D311yescharacteristic
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×#117—31no—
115C22×C42—31no—
116C22×C42—31no—
117C22×C42—31no—
118#433—68no—
119C2×C4⋊C8—32no—
120C2×C4⋊C8—32no—
121C2×C22⋊C8—32no—
122C2×C22⋊C8—31no—
123C2×C22⋊C8—31no—
124C2×C22⋊C8—32no—
125C2×C22⋊C8C2×D312yes—
126C2×C22⋊C8C2×D311yescharacteristic
127C2×C22⋊C8C2×D311yescharacteristic
128C2×C22⋊C8C2×D312yes—
129C2×#97—32no—
130C2×C4×C8—32no—
131#336—32no—
132#336—32no—
133#336—32no—
134#336—32no—
135C24×C3C22×C411yescharacteristic
136C24×C3C22×C411yescharacteristic
137C24×C3C22×C411yescharacteristic
138C3×#41—216no—
139C3×#41—216no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4C22×C411yescharacteristic
161C22×C3×C4C22×C411yescharacteristic
162C22×C3×C4C22×C411yescharacteristic
163C22×C3×C4C2×D414yes—
164C22×C3×C4#4112yes—
165C22×C3×C4#4112yes—
166C22×C3×C4#4114yes—
167C22×C3×C4C2411yescharacteristic
168C22×Dic3—24no—
169C22×Dic3—24no—
170C22×Dic3—24no—
171C22×Dic3—24no—
172C22×Dic3—24no—
173C22×Dic3—24no—
174C22×Dic3—42no—
175C22×Dic3—42no—
176C22×Dic3—42no—
177C22×Dic3C22×C412yes—
178C22×Dic3C22×C412yes—
179C22×Dic3C22×C412yes—
180C22×Dic3C22×C412yes—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187#217—24no—
188#217—24no—
189#217C22⋊C418yes—
190C4⋊Dic3—24no—
191C4⋊Dic3—24no—
192C4⋊Dic3C22⋊C418yes—
193C4×Dic3—24no—
194C4×Dic3—22no—
195C4×Dic3—22no—
196C4×Dic3—24no—
197C4×Dic3—24no—
198C4×Dic3C2×D414yes—
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—24no—
207C2×#41—22no—
208C2×#41—22no—
209C2×#41—24no—
210C22×Q8C4×D311yescharacteristic
211C22×D4C4×D311yescharacteristic
212C22×D4C4×D311yescharacteristic
213C22×D4C4×D311yescharacteristic
214C23×C4—32no—
215C23×C4—32no—
216C23×C4—32no—
217C23×C4C22×D311yescharacteristic
218C23×C4C22×D311yescharacteristic
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—38no—
229#117—64no—
230#117—64no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—32no—
233C2×C4⋊C4—32no—
234C2×C22⋊C4—32no—
235C2×C22⋊C4—32no—
236C2×C22⋊C4—32no—
237C2×C42—31no—
238C2×C42—31no—
239C2×C42—32no—
240C2×C42—32no—
241C2×C42—32no—
242C2×C42—31no—
243C2×C42—31no—
244C2×C42—64no—
245C2×C42—64no—
246C2×C42—64no—
247C4⋊C8—64no—
248C4⋊C8—64no—
249C22⋊C8—24no—
250C22⋊C8—24no—
251C22⋊C8—22no—
252C22⋊C8—22no—
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—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3C2×C22⋊C412yes—
272C23×C3#11712yes—
273C23×C3#11712yes—
274C23×C3C23×C411yescharacteristic
275C3×Q8—28no—
276C3×D4—28no—
277C3×D4—28no—
278C3×D4—28no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4C2×C22⋊C412yes—
292C2×C3×C4#11712yes—
293C2×C3×C4#11712yes—
294C2×C3×C4C22×D412yes—
295C2×C3×C4C2×#4111yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4C2×#4112yes—
298C2×Dic3—22no—
299C2×Dic3—22no—
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—48no—
308C2×Dic3—48no—
309C2×Dic3—48no—
310C2×Dic3C2×C22⋊C414yes—
311C2×Dic3C2×C22⋊C414yes—
312C2×Dic3#13114yes—
313C3×C8—42no—
314C3×C8—42no—
315C3⋊C8—42no—
316C3⋊C8—42no—
317C24C2×C4×D311yescharacteristic
318C24C2×C4×D311yescharacteristic
319C24C2×C4×D311yescharacteristic
320#41—216no—
321#41—216no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—32no—
343C22×C4—32no—
344C22×C4—32no—
345C22×C4—32no—
346C22×C4—64no—
347C22×C4—64no—
348C22×C4—64no—
349C22×C4—64no—
350C22×C4—64no—
351C22×C4—64no—
352C22×C4—122no—
353C22×C4—122no—
354C22×C4—122no—
355C22×C4C2×C4×D311yescharacteristic
356C22×C4C2×C4×D311yescharacteristic
357C22×C4C2×C4×D311yescharacteristic
358C22×C4D3×D414yes—
359C22×C4#23712yes—
360C22×C4#23712yes—
361C22×C4#23714yes—
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—38no—
376C4⋊C4—64no—
377C4⋊C4—64no—
378C22⋊C4—38no—
379C22⋊C4—64no—
380C22⋊C4—64no—
381C42—34no—
382C42—34no—
383C42—64no—
384C42—64no—
385C42—64no—
386C42—62no—
387C42—62no—
388C22×C3—24no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—24no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3C22×C22⋊C411yescharacteristic
404C22×C3C2×#11711yescharacteristic
405C22×C3C2×#11711yescharacteristic
406C22×C3C2×#13112yes—
407C22×C3#56812yes—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4#52512yes—
412C3×C4#52512yes—
413Dic3—24no—
414Dic3—24no—
415Dic3—24no—
416Dic3—42no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23#87712yes—
430C23#87712yes—
431C23D3×C22⋊C412yes—
432C23C22×C4×D311yescharacteristic
433Q8—28no—
434D4—28no—
435D4—28no—
436D4—28no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—34no—
450C2×C4—34no—
451C2×C4—34no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—128no—
462C2×C4—128no—
463C2×C4—128no—
464C2×C4#87712yes—
465C2×C4#87712yes—
466C2×C4D3×C22⋊C412yes—
467C2×C4C2×D3×D412yes—
468C2×C4C2×#23711yescharacteristic
469C2×C4C2×#23711yescharacteristic
470C2×C4C2×#23712yes—
471C8—42no—
472C8—42no—
473C8—122no—
474C8—122no—
475C2×C3—42no—
476C2×C3—42no—
477C2×C3—42no—
478C2×C3#228011yescharacteristic
479C2×C3#288012yes—
480C2×C3#292211yescharacteristic
481C2×C3#292311yescharacteristic
482C2×C3#292512yes—
483C22—22no—
484C22—22no—
485C22—24no—
486C22—24no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22C2×#87711yescharacteristic
499C22C2×#87711yescharacteristic
500C22C2×D3×C22⋊C411yescharacteristic
501C22D3×#13112yes—
502C22#575012yes—
503C4—22no—
504C4—22no—
505C4—22no—
506C4—64no—
507C4—64no—
508C4—64no—
509C4—122no—
510C4#558612yes—
511C4#558612yes—
512C3#2072011yescharacteristic, complemented, Sylow
513C2—42no—
514C2—42no—
515C2—42no—
516C2#7381711yescharacteristic
517C2#7864811yescharacteristic
518C2#7864911yescharacteristic
519C2#7865012yes—
520C2#7866312yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1780C2
prime radicalC22×C3×#41C22
cosocleC22×C3×C4C24
hypercenter#1780D3
hypocenterC3#20720
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1780

Complements

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

NH ≅ G/Ntype
2. C2×#4720C2semidirect
5. #69910C2semidirect
6. #69909C2semidirect
7. #69909C2semidirect
8. #69908C2semidirect
9. #69907C2semidirect
10. #69906C2semidirect
11. Dic3×C22⋊C8C2semidirect
512. C3#20720semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×#412
C2×#47202
C3×#17802
#721362
#699102
#699092
#699082
#699072
#699062
Dic3×C22⋊C82
#207203