Finite Group Databasebeta

← #10691594

Subgroups — #10691594

Order 768.

612 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#459C211yesmaximal, characteristic
2D4×#208C211yesmaximal, characteristic, complemented
3#71766C211yesmaximal, characteristic, complemented
4#71759C211yesmaximal, characteristic, complemented
5#71598C211yesmaximal, characteristic, complemented
6#67706C211yesmaximal, characteristic, complemented
7#67705C211yesmaximal, characteristic, complemented
8#9253—31nomaximal, Sylow
9C22×C3×D8C412yes—
10C2×C3×#127C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C22×#208—21no—
13C3×#459—21no—
14C3×#459—21no—
15C3×#459—21no—
16C3×#459—21no—
17C2×C3×C4⋊C8C2211yescharacteristic
18C2×C3×#102C412yes—
19C2×#833—21no—
20C2×#833—21no—
21#5207—21no—
22#5012—21no—
23#5011—22no—
24D4×C3⋊C8—21no—
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27#4924C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4544—22no—
30D4⋊(C3⋊C8)—21no—
31D4⋊(C3⋊C8)—21no—
32C2×#459D311yescharacteristic
33D4×M16—31no—
34#1965—31no—
35#1545—31no—
36#1486—31no—
37#1292—31no—
38#1283—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4C2×C411yescharacteristic
42C22×C3×D4C2×C412yes—
43C23×C3×C4—21no—
44C2×C3×D8—22no—
45C2×C3×D8—22no—
46C2×C3×D8—44no—
47C22×C3×C8C2×C412yes—
48C3×#127—21no—
49C3×#127—21no—
50C3×#127—21no—
51C3×#127—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4D411yescharacteristic
55C3×C4×D4D411yescharacteristic
56C3×C4×D4D411yescharacteristic
57C3×C4×D4D411yescharacteristic
58C2×C3×C4⋊C4C2×C411yescharacteristic
59C2×C3×C22⋊C4—21no—
60C2×C3×C42C2311yescharacteristic
61C2×#208—22no—
62C2×#208—21no—
63C2×#208—21no—
64C2×#208—42no—
65C2×#208—42no—
66C2×#208D411yescharacteristic
67C2×#208D411yescharacteristic
68C22×C3⋊C8—22no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×#102—22no—
72C3×#102—22no—
73#833—42no—
74#833—42no—
75#830—22no—
76#830—22no—
77#830—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83#803—21no—
84C4×C3⋊C8—21no—
85C4×C3⋊C8—21no—
86C22×D8Dic312yes—
87C22×M16—61no—
88C2×#127C2×D311yescharacteristic
89C2×C4×D4C2×D311yescharacteristic
90#459—21no—
91#459—21no—
92#459—21no—
93#459—21no—
94#436—61no—
95#435—62no—
96C8×D4—61no—
97#431—31no—
98#423—31no—
99#423—31no—
100C2×C4⋊C8C2×D311yescharacteristic
101C2×#102Dic312yes—
102C2×#100—61no—
103C2×#100—61no—
104#407—61no—
105C4×M16—31no—
106#331—62no—
107D4⋊C8—61no—
108D4⋊C8—61no—
109C24×C3—21no—
110C24×C3—22no—
111C2×C3×D4—22no—
112C2×C3×D4—22no—
113C2×C3×D4—21no—
114C2×C3×D4—22no—
115C2×C3×D4—41no—
116C2×C3×D4—42no—
117C2×C3×D4—42no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—82no—
127C2×C3×D4C22⋊C411yescharacteristic
128C2×C3×D4C22⋊C411yescharacteristic
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4C22×C411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C3×D8—44no—
142C3×D8—44no—
143C2×C3×C8—22no—
144C2×C3×C8—42no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4C22⋊C411yescharacteristic
147C3×C4⋊C4C22⋊C411yescharacteristic
148C3×C22⋊C4—21no—
149C3×C22⋊C4—21no—
150C3×C22⋊C4—21no—
151C3×C22⋊C4—21no—
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42#4111yescharacteristic
156#208—21no—
157#208—21no—
158#208—21no—
159#208—44no—
160#208—41no—
161#208—41no—
162#208—41no—
163#208—41no—
164#208D812yescomplemented
165C2×C3⋊C8—21no—
166C2×C3⋊C8—21no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—44no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4C2×Dic311yescharacteristic
174C22×D4C2×Dic312yes—
175C23×C4—21no—
176C2×D8—22no—
177C2×D8—22no—
178C2×D8—44no—
179C2×M16—31no—
180C2×M16—31no—
181C2×M16—62no—
182C2×M16—61no—
183C2×M16—61no—
184C2×M16—122no—
185C2×M16—122no—
186C22×C8—62no—
187C22×C8C2×Dic312yes—
188#127—21no—
189#127—21no—
190#127—21no—
191#127—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4#6711yescharacteristic
195C4×D4#6711yescharacteristic
196C4×D4#6711yescharacteristic
197C4×D4#6711yescharacteristic
198C2×C4⋊C4C2×Dic311yescharacteristic
199C2×C22⋊C4—21no—
200C2×C42C22×D311yescharacteristic
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—32no—
204C4⋊C8—61no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208#102—22no—
209#102—22no—
210#100—122no—
211#100—122no—
212C22⋊C8—62no—
213C22⋊C8—62no—
214C22⋊C8—61no—
215#97—61no—
216C4×C8—61no—
217C4×C8—61no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—42no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3—42no—
226C23×C3—81no—
227C23×C3—82no—
228C23×C3—82no—
229C23×C3C4×D411yescharacteristic
230C3×D4—21no—
231C3×D4—21no—
232C3×D4—21no—
233C3×D4—42no—
234C3×D4—42no—
235C3×D4—42no—
236C3×D4—82no—
237C3×D4—82no—
238C3×D4—82no—
239C3×D4—81no—
240C3×D4—81no—
241C3×D4—81no—
242C3×D4—81no—
243C3×D4#10012yescomplemented
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4C2×C22⋊C411yescharacteristic
260C2×C3×C4C4×D411yescharacteristic
261C2×C3×C4#12011yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12311yescharacteristic
265C2×C3×C4C2×D811yescharacteristic
266C2×C3×C4#13511yescharacteristic
267C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
268C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
269C3×C8—42no—
270C3⋊C8—22no—
271C3⋊C8—42no—
272C3⋊C8—41no—
273C3⋊C8—81no—
274C3⋊C8—81no—
275C24—21no—
276C24—22no—
277C2×D4—22no—
278C2×D4—21no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—82no—
293C2×D4#21711yescharacteristic
294C2×D4#21711yescharacteristic
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4D3×D411yescharacteristic
305C22×C4#23711yescharacteristic
306C22×C4C22×Dic311yescharacteristic
307D8—44no—
308D8—44no—
309M16—32no—
310M16—61no—
311M16—61no—
312M16—61no—
313M16—124no—
314M16—121no—
315M16—121no—
316M16—121no—
317M16—121no—
318C2×C8—22no—
319C2×C8—42no—
320C2×C8—61no—
321C2×C8—61no—
322C2×C8—62no—
323C2×C8—124no—
324C2×C8—121no—
325C2×C8—121no—
326C4⋊C4—21no—
327C4⋊C4#21711yescharacteristic
328C4⋊C4#21711yescharacteristic
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—21no—
333C42D3×D411yescharacteristic
334C42#23711yescharacteristic
335C42C2×#6711yescharacteristic
336C42C2×#6711yescharacteristic
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—22no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—82no—
346C22×C3—82no—
347C22×C3—82no—
348C22×C3—82no—
349C22×C3—82no—
350C22×C3—162no—
351C22×C3—162no—
352C22×C3#38611yescharacteristic
353C22×C3C4×D811yescharacteristic
354C22×C3#44211yescharacteristic
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4C2×#10011yescharacteristic
361C3×C4#41311yescharacteristic
362C3×C4#44711yescharacteristic
363C3×C4#44911yescharacteristic
364C3×C4#45911yescharacteristic
365C3×C4#46311yescharacteristic
366C23—21no—
367C23—21no—
368C23—42no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—81no—
375C23—82no—
376C23—82no—
377C23D4×Dic311yescharacteristic
378D4—21no—
379D4—21no—
380D4—21no—
381D4—42no—
382D4—42no—
383D4—42no—
384D4—82no—
385D4—82no—
386D4—82no—
387D4—81no—
388D4—81no—
389D4—81no—
390D4—81no—
391D4#83312yescomplemented
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4D3×D811yescharacteristic
408C2×C4#91111yescharacteristic
409C2×C4#91111yescharacteristic
410C2×C4#91211yescharacteristic
411C2×C4D4×Dic311yescharacteristic
412C2×C4#93511yescharacteristic
413C2×C4#93711yescharacteristic
414C2×C4#93911yescharacteristic
415C2×C4#93911yescharacteristic
416C2×C4C2×#21711yescharacteristic
417C8—42no—
418C8—62no—
419C8—122no—
420C8—121no—
421C8—241no—
422C8—241no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—42no—
426C2×C3—81no—
427C2×C3—162no—
428C2×C3#187911yescharacteristic
429C2×C3#190011yescharacteristic
430C2×C3#190311yescharacteristic
431C22—21no—
432C22—21no—
433C22—22no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—82no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22—162no—
445C22—162no—
446C22Dic3×D811yescharacteristic
447C22#515211yescharacteristic
448C22#522011yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—41no—
453C4—41no—
454C4#515011yescharacteristic
455C4#515311yescharacteristic
456C4#515611yescharacteristic
457C4#515811yescharacteristic
458C4C2×#83311yescharacteristic
459C4#523711yescharacteristic
460C3#925311yescharacteristic, complemented, Sylow
461C2—21no—
462C2—21no—
463C2—42no—
464C2—81no—
465C2—162no—
466C2#7168311yescharacteristic
467C2#7203711yescharacteristic
468C2#7209011yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC2×C3×#459C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenterC2×#459D3
hypocenterC3#9253
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
2. D4×#208C2semidirect
3. #71766C2semidirect
4. #71759C2semidirect
5. #71598C2semidirect
6. #67706C2semidirect
7. #67705C2semidirect
164. #208D8semidirect
243. C3×D4#100semidirect
391. D4#833semidirect
460. C3#9253semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4592
D4×#2082
#717662
#717592
#715982
#677062
#677052
#92533