Finite Group Databasebeta

← #10691591

Subgroups — #10691591

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×#4843C211yesmaximal, characteristic
2M16×D12C211yesmaximal, characteristic, complemented
3#70774C211yesmaximal, characteristic, complemented
4#70762C211yesmaximal, characteristic, complemented
5#69876C211yesmaximal, characteristic, complemented
6C3×#1283C211yesmaximal, characteristic, complemented
7#67705C211yesmaximal, characteristic, complemented
8#9253—31nomaximal, Sylow
9C2×D3×M16—21no—
10C22×D24C412yes—
11C2×#874C2211yescharacteristic
12C2×C4×D12C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C3×C4×M16C2211yescharacteristic
16C2×#823—21no—
17C2×#823—21no—
18#5126—21no—
19C2×#821C412yes—
20#4924C2211yescharacteristic
21#4924C2211yescharacteristic
22#4843—21no—
23#4843—21no—
24#4843—21no—
25#4843—21no—
26#4706—22no—
27#4688—21no—
28C8×D12—21no—
29#4485—22no—
30#4459—21no—
31#4459—21no—
32C2×#459—31no—
33D4×M16—31no—
34#1965—31no—
35#1545—31no—
36#1486—31no—
37#1292—31no—
38#1283D311yescharacteristic, complemented
39C22×D12—21no—
40C22×D12—21no—
41C22×D12C2×C411yescharacteristic
42C22×D12C2×C412yes—
43C22×C4×D3—21no—
44C2×C3×M16D411yescharacteristic
45C2×C3×M16D411yescharacteristic
46C22×C3×C8C2×C412yes—
47C2×C3×C42C2311yescharacteristic
48C2×(C2×D3)⋊C4—21no—
49C2×C4⋊Dic3C2×C411yescharacteristic
50C2×#208—21no—
51C2×#208—21no—
52D3×M16—42no—
53D3×M16—42no—
54C2×D24—22no—
55C2×D24—22no—
56C2×D24—44no—
57C2×ΓL(1,𝔽25)—22no—
58C2×D3×C8—22no—
59#874—21no—
60#874—21no—
61#874—21no—
62#874—21no—
63C4×D12—21no—
64C4×D12—21no—
65C4×D12D411yescharacteristic
66C4×D12D411yescharacteristic
67C4×D12D411yescharacteristic
68C4×D12D411yescharacteristic
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#97—21no—
74C3×C4×C8—21no—
75C3×C4×C8—21no—
76#823—42no—
77#823—42no—
78#821—22no—
79#821—22no—
80(C2×D3)⋊C8—22no—
81(C2×D3)⋊C8—22no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)D412yes—
86C22×D8—32no—
87C22×M16—61no—
88C2×#127—31no—
89C2×C4×D4—31no—
90#459—61no—
91#459—61no—
92#459—61no—
93#459—61no—
94#436—61no—
95#435—62no—
96C8×D4—61no—
97#431C2×D311yescharacteristic
98#423—31no—
99#423—31no—
100C2×C4⋊C8C2×D311yescharacteristic
101C2×#102—32no—
102C2×#100—61no—
103C2×#100—61no—
104#407—61no—
105C4×M16C2×D311yescharacteristic
106#331—62no—
107D4⋊C8—61no—
108D4⋊C8—61no—
109C23×D3—21no—
110C23×D3—22no—
111C22×C3×C4C22×C411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C22×Dic3—21no—
115C2×D12—22no—
116C2×D12—22no—
117C2×D12—21no—
118C2×D12—22no—
119C2×D12—41no—
120C2×D12—42no—
121C2×D12—42no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—41no—
128C2×D12—41no—
129C2×D12—41no—
130C2×D12—82no—
131C2×D12C22⋊C411yescharacteristic
132C2×D12C22⋊C411yescharacteristic
133C2×C4×D3—21no—
134C2×C4×D3—21no—
135C2×C4×D3—21no—
136C2×C4×D3—21no—
137C2×C4×D3—41no—
138C2×C4×D3—41no—
139C2×C4×D3—41no—
140C2×C4×D3—41no—
141C3×M16—21no—
142C3×M16—21no—
143C3×M16—21no—
144C3×M16D812yescomplemented
145C2×C3×C8—21no—
146C2×C3×C8—22no—
147C2×C3×C8—21no—
148C2×C3×C8—42no—
149C2×C3×C8—41no—
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42#4111yescharacteristic
154(C2×D3)⋊C4—21no—
155(C2×D3)⋊C4—21no—
156(C2×D3)⋊C4—21no—
157(C2×D3)⋊C4—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3C22⋊C411yescharacteristic
160C4⋊Dic3C22⋊C411yescharacteristic
161#208—41no—
162#208—41no—
163#208—41no—
164#208—41no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—41no—
167D24—44no—
168D24—44no—
169ΓL(1,𝔽25)—44no—
170D3×C8—44no—
171C22×D4—31no—
172C22×D4—32no—
173C22×D4—61no—
174C22×D4—61no—
175C23×C4—61no—
176C2×D8—62no—
177C2×D8—62no—
178C2×D8—124no—
179C2×M16—62no—
180C2×M16—61no—
181C2×M16—61no—
182C2×M16—122no—
183C2×M16—122no—
184C2×M16D1211yescharacteristic
185C2×M16D1211yescharacteristic
186C22×C8—62no—
187C22×C8C4×D312yes—
188#127—61no—
189#127—61no—
190#127—61no—
191#127—61no—
192C4×D4—31no—
193C4×D4—31no—
194C4×D4—31no—
195C4×D4—31no—
196C4×D4—61no—
197C4×D4—61no—
198C2×C4⋊C4—31no—
199C2×C22⋊C4—61no—
200C2×C42C22×D311yescharacteristic
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—32no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208#102—62no—
209#102—62no—
210#100—122no—
211#100—122no—
212C22⋊C8—21no—
213C22⋊C8—62no—
214C22⋊C8—62no—
215#97—21no—
216C4×C8—21no—
217C4×C8—21no—
218C23×C3C4×D411yescharacteristic
219C22×D3—21no—
220C22×D3—21no—
221C22×D3—42no—
222C22×D3—41no—
223C22×D3—41no—
224C22×D3—42no—
225C22×D3—42no—
226C22×D3—42no—
227C22×D3—81no—
228C22×D3—82no—
229C22×D3—82no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—41no—
233C2×C3×C4C2×C22⋊C411yescharacteristic
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12311yescharacteristic
239C2×C3×C4C2×D811yescharacteristic
240C2×C3×C4#13511yescharacteristic
241C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
242C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247D12—21no—
248D12—21no—
249D12—21no—
250D12—42no—
251D12—42no—
252D12—42no—
253D12—82no—
254D12—82no—
255D12—82no—
256D12—81no—
257D12—81no—
258D12—81no—
259D12—81no—
260D12#10012yescomplemented
261C4×D3—41no—
262C4×D3—41no—
263C4×D3—41no—
264C4×D3—41no—
265C4×D3—41no—
266C4×D3—41no—
267C4×D3—41no—
268C4×D3—41no—
269C3×C8—22no—
270C3×C8—42no—
271C3×C8—41no—
272C3×C8—81no—
273C3⋊C8—42no—
274C3⋊C8—81no—
275C24—61no—
276C24—62no—
277C2×D4—31no—
278C2×D4—31no—
279C2×D4—62no—
280C2×D4—62no—
281C2×D4—61no—
282C2×D4—62no—
283C2×D4—121no—
284C2×D4—122no—
285C2×D4—122no—
286C2×D4—121no—
287C2×D4—121no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4—121no—
291C2×D4—121no—
292C2×D4—121no—
293C2×D4—121no—
294C2×D4—242no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—121no—
301C22×C4—121no—
302C22×C4—121no—
303C22×C4—121no—
304C22×C4C2×C4×D311yescharacteristic
305C22×C4C2×D1211yescharacteristic
306C22×C4#23511yescharacteristic
307D8—124no—
308D8—124no—
309M16—21no—
310M16—21no—
311M16—21no—
312M16—124no—
313M16—121no—
314M16—121no—
315M16—121no—
316M16—121no—
317M16D2412yescomplemented
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—21no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—62no—
324C2×C8—124no—
325C2×C8—121no—
326C4⋊C4—31no—
327C4⋊C4—31no—
328C4⋊C4—61no—
329C22⋊C4—61no—
330C22⋊C4—61no—
331C22⋊C4—61no—
332C22⋊C4—61no—
333C42D3×D411yescharacteristic
334C42D3×D411yescharacteristic
335C42D3×D411yescharacteristic
336C42#23911yescharacteristic
337C22×C3—21no—
338C22×C3—21no—
339C22×C3#38611yescharacteristic
340C22×C3C4×D811yescharacteristic
341C22×C3#44211yescharacteristic
342C2×D3—22no—
343C2×D3—41no—
344C2×D3—41no—
345C2×D3—41no—
346C2×D3—41no—
347C2×D3—41no—
348C2×D3—82no—
349C2×D3—82no—
350C2×D3—82no—
351C2×D3—82no—
352C2×D3—82no—
353C2×D3—162no—
354C2×D3—162no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4C2×#10011yescharacteristic
359C3×C4#41311yescharacteristic
360C3×C4#44711yescharacteristic
361C3×C4#44911yescharacteristic
362C3×C4#45911yescharacteristic
363C3×C4#46311yescharacteristic
364Dic3—41no—
365Dic3—41no—
366C23—61no—
367C23—61no—
368C23—122no—
369C23—121no—
370C23—121no—
371C23—122no—
372C23—122no—
373C23—122no—
374C23—241no—
375C23—242no—
376C23—242no—
377C23C4×D1211yescharacteristic
378D4—32no—
379D4—61no—
380D4—61no—
381D4—61no—
382D4—122no—
383D4—122no—
384D4—122no—
385D4—242no—
386D4—242no—
387D4—242no—
388D4—241no—
389D4—241no—
390D4—241no—
391D4—241no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—41no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4D3×C22⋊C411yescharacteristic
408C2×C4#88211yescharacteristic
409C2×C4#88411yescharacteristic
410C2×C4#89311yescharacteristic
411C2×C4#89411yescharacteristic
412C2×C4#89511yescharacteristic
413C2×C4C2×D2411yescharacteristic
414C2×C4#90411yescharacteristic
415C2×C4#90811yescharacteristic
416C2×C4#90811yescharacteristic
417C8—22no—
418C8—41no—
419C8—42no—
420C8—81no—
421C8—122no—
422C8—241no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3#187911yescharacteristic
426C2×C3#190011yescharacteristic
427C2×C3#190311yescharacteristic
428D3—42no—
429D3—81no—
430D3—162no—
431C22—21no—
432C22—21no—
433C22—62no—
434C22—121no—
435C22—121no—
436C22—121no—
437C22—121no—
438C22—121no—
439C22—242no—
440C22—242no—
441C22—242no—
442C22—242no—
443C22—242no—
444C22—482no—
445C22—482no—
446C22#466511yescharacteristic
447C22C4×D2411yescharacteristic
448C22#471111yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—121no—
453C4—121no—
454C4#473211yescharacteristic
455C4#473411yescharacteristic
456C4D3×#10011yescharacteristic
457C4#475111yescharacteristic
458C4#484311yescharacteristic
459C4#484411yescharacteristic
460C3#925311yescharacteristic, complemented, Sylow
461C2—21no—
462C2—21no—
463C2—122no—
464C2—241no—
465C2—482no—
466C2#6867311yescharacteristic
467C2#6899211yescharacteristic
468C2#6903811yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1283C2
socleC22×C3#386
prime radicalC2×#4843C2
cosocleC2×C3×C42C23
hypercenter#1283D3
hypocenterC3#9253
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1283

Complements

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

NH ≅ G/Ntype
2. M16×D12C2semidirect
3. #70774C2semidirect
4. #70762C2semidirect
5. #69876C2semidirect
6. C3×#1283C2semidirect
7. #67705C2semidirect
38. #1283D3semidirect
144. C3×M16D8semidirect
260. D12#100semidirect
317. M16D24semidirect
460. C3#9253semidirect

Maximal subgroups

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

subgroupindex
C2×#48432
M16×D122
#707742
#707622
#698762
C3×#12832
#677052
#92533