Finite Group Databasebeta

← #10690922

Subgroups — #10690922

Order 768.

1348 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#95C211yesmaximal, characteristic, complemented
2#84032C211yesmaximal, characteristic
3#72126C211yesmaximal, characteristic, complemented
4C2×#4451C212yesmaximal, complemented
5#67612C212yesmaximal, complemented
6#9147—31nomaximal, Sylow
7C24×C3×C4—21no—
8C24×C3×C4C2211yescharacteristic
9C2×#953—21no—
10C2×#930C412yescomplemented
11C2×#930C2211yescharacteristic
12C2×C3×#95—21no—
13C2×C3×#95—24no—
14C2×C3×#95C2212yes—
15#5213—21no—
16#5213—22no—
17#5212C412yescomplemented
18#5212C2211yescharacteristic
19C2×#830—22no—
20C2×#830C2212yes—
21#4451—24no—
22#2847—31no—
23C22×#95D311yescharacteristic, complemented
24#1779—31no—
25C2×#327—32no—
26#1294—32no—
27C25×C3D411yescharacteristic
28C23×C3×C4—28no—
29C23×C3×C4—24no—
30C23×C3×C4—22no—
31C23×C3×C4—22no—
32C23×C3×C4—22no—
33C23×C3×C4—44no—
34C23×C3×C4—44no—
35C23×C3×C4—42no—
36C23×C3×C4—42no—
37C23×C3×C4C2×C412yes—
38C23×C3×C4D412yes—
39C23×C3×C4D411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C22×#67—22no—
42C22×#67—21no—
43C22×D12C2×C411yescharacteristic, complemented
44#953—42no—
45#953—42no—
46C2×#217—22no—
47C2×#217—21no—
48#930—22no—
49#930—22no—
50#930—44no—
51C2×(C2×D3)⋊C4—22no—
52C2×(C2×D3)⋊C4—21no—
53C2×C4⋊Dic3C2×C411yescharacteristic, complemented
54C2×C4⋊Dic3C2×C412yescomplemented
55C22×C3⋊C8—22no—
56C3×#95—24no—
57C3×#95—22no—
58C3×#95D414yescomplemented
59C3⋊#95—22no—
60C3⋊#95—21no—
61#830—22no—
62#830—22no—
63#830—44no—
64C24×C4—21no—
65C24×C4C2×D311yescharacteristic
66C2×#121—31no—
67C2×#121—32no—
68C2×#120—61no—
69C2×C22⋊C8—32no—
70C2×C22⋊C8—62no—
71#386—62no—
72#386—61no—
73#380—32no—
74#380—31no—
75C2×#95—21no—
76C2×#95—24no—
77C2×#95C2×D312yes—
78#327—64no—
79C24×C3—24no—
80C24×C3—24no—
81C24×C3—21no—
82C24×C3—21no—
83C24×C3—21no—
84C24×C3—21no—
85C24×C3C22⋊C412yes—
86C24×C3D812yes—
87C24×C3SD1612yes—
88C24×C3C2×D411yescharacteristic
89C23×D3—21no—
90C22×C3×C4—216no—
91C22×C3×C4—28no—
92C22×C3×C4—24no—
93C22×C3×C4—24no—
94C22×C3×C4—21no—
95C22×C3×C4—24no—
96C22×C3×C4—21no—
97C22×C3×C4—24no—
98C22×C3×C4—24no—
99C22×C3×C4—42no—
100C22×C3×C4—42no—
101C22×C3×C4—42no—
102C22×C3×C4—42no—
103C22×C3×C4—48no—
104C22×C3×C4—48no—
105C22×C3×C4—48no—
106C22×C3×C4—416no—
107C22×C3×C4—416no—
108C22×C3×C4—44no—
109C22×C3×C4—48no—
110C22×C3×C4—42no—
111C22×C3×C4—42no—
112C22×C3×C4C22⋊C412yes—
113C22×C3×C4C22×C411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D412yes—
116C22×C3×C4#4112yes—
117C2×#67—44no—
118C2×#67—44no—
119C2×#67—44no—
120C2×#67—41no—
121C2×#67—41no—
122C2×#67—41no—
123C2×#67—41no—
124C2×#67—41no—
125C2×#67—41no—
126C22×Dic3—21no—
127C22×Dic3—22no—
128C2×D12—22no—
129C2×D12—42no—
130#217—44no—
131#217—41no—
132#217—41no—
133(C2×D3)⋊C4—44no—
134(C2×D3)⋊C4—42no—
135C4⋊Dic3—22no—
136C4⋊Dic3—42no—
137C2×C3⋊C8—44no—
138C2×C3⋊C8—42no—
139C25#6711yescharacteristic
140C22×D4—31no—
141C22×D4—62no—
142C22×D4—61no—
143C23×C4—24no—
144C23×C4—28no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4C4×D312yes—
153C23×C4D1212yes—
154C23×C4#6711yescharacteristic
155C23×C4C22×D311yescharacteristic
156C22×C8—62no—
157#121—62no—
158#121—62no—
159#121—124no—
160#120—122no—
161#120—122no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—32no—
164C2×C22⋊C4—62no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—62no—
167C2×C22⋊C4—61no—
168C22⋊C8—62no—
169C22⋊C8—62no—
170C22⋊C8—124no—
171#95—24no—
172#95—22no—
173#95—62no—
174#95—61no—
175#95#6714yescomplemented
176C23×C3—24no—
177C23×C3—24no—
178C23×C3—28no—
179C23×C3—24no—
180C23×C3—24no—
181C23×C3—28no—
182C23×C3—28no—
183C23×C3—28no—
184C23×C3—22no—
185C23×C3—22no—
186C23×C3—22no—
187C23×C3—22no—
188C23×C3—22no—
189C23×C3—22no—
190C23×C3—22no—
191C23×C3—22no—
192C23×C3—22no—
193C23×C3—22no—
194C23×C3#9914yes—
195C23×C3#10214yes—
196C23×C3#10414yes—
197C23×C3C2×C22⋊C411yescharacteristic
198C23×C3#12011yescharacteristic
199C23×C3#12311yescharacteristic
200C23×C3C2×D811yescharacteristic
201C23×C3C2×SD1611yescharacteristic
202C23×C3AGL(1,ℤ/8)11yescharacteristic
203C23×C3#13711yescharacteristic
204C22×D3—41no—
205C22×D3—42no—
206C22×D3—41no—
207C22×D3—41no—
208C22×D3—42no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—28no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—48no—
218C2×C3×C4—48no—
219C2×C3×C4—432no—
220C2×C3×C4—44no—
221C2×C3×C4—48no—
222C2×C3×C4—48no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—44no—
226C2×C3×C4C4×D412yes—
227C2×C3×C4#12112yes—
228#67—88no—
229#67—81no—
230#67—81no—
231#67—81no—
232#67—81no—
233C2×Dic3—41no—
234C2×Dic3—42no—
235C2×Dic3—41no—
236C2×Dic3—42no—
237C2×Dic3—41no—
238C2×Dic3—42no—
239D12—82no—
240C3⋊C8—82no—
241C24—24no—
242C24—24no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—61no—
248C24(C2×D3)⋊C412yes—
249C24#21312yes—
250C24#21512yes—
251C24C2×#6711yescharacteristic
252C2×D4—62no—
253C2×D4—122no—
254C2×D4—124no—
255C2×D4—124no—
256C2×D4—124no—
257C2×D4—121no—
258C2×D4—121no—
259C2×D4—121no—
260C2×D4—121no—
261C2×D4—121no—
262C2×D4—121no—
263C22×C4—216no—
264C22×C4—28no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—48no—
277C22×C4—48no—
278C22×C4—48no—
279C22×C4—416no—
280C22×C4—44no—
281C22×C4—416no—
282C22×C4—48no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—62no—
286C22×C4—61no—
287C22×C4(C2×D3)⋊C412yes—
288C22×C4C2×C4×D311yescharacteristic
289C22×C4C2×D1211yescharacteristic
290C22×C4#23512yes—
291C22×C4C2×#6712yes—
292C2×C8—124no—
293C2×C8—122no—
294C4⋊C4—62no—
295C4⋊C4—122no—
296C22⋊C4—124no—
297C22⋊C4—124no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C22⋊C4—122no—
301C22×C3—28no—
302C22×C3—28no—
303C22×C3—28no—
304C22×C3—28no—
305C22×C3—24no—
306C22×C3—21no—
307C22×C3—24no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—24no—
311C22×C3—21no—
312C22×C3—22no—
313C22×C3—24no—
314C22×C3—22no—
315C22×C3—22no—
316C22×C3—24no—
317C22×C3—24no—
318C22×C3—22no—
319C22×C3#32718yescomplemented
320C22×C3#38611yescharacteristic
321C22×C3C2×#9912yes—
322C22×C3C2×#10211yescharacteristic
323C22×C3#41711yescharacteristic
324C22×C3C2×#10412yes—
325C22×C3#44711yescharacteristic
326C22×C3#44811yescharacteristic
327C22×C3#48011yescharacteristic
328C22×C3#48111yescharacteristic
329C2×D3—81no—
330C2×D3—81no—
331C2×D3—81no—
332C2×D3—82no—
333C2×D3—82no—
334C3×C4—22no—
335C3×C4—22no—
336C3×C4—48no—
337C3×C4—42no—
338Dic3—81no—
339Dic3—82no—
340C23—24no—
341C23—24no—
342C23—28no—
343C23—24no—
344C23—28no—
345C23—24no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—28no—
353C23—28no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—122no—
359C23—122no—
360C23—121no—
361C23—121no—
362C23—121no—
363C23#80514yes—
364C23#80614yes—
365C23#80914yes—
366C23C2×(C2×D3)⋊C411yescharacteristic
367C23#92911yescharacteristic
368C23C2×#21311yescharacteristic
369C23#93211yescharacteristic
370C23C2×#21511yescharacteristic
371C23#94211yescharacteristic
372C23#95311yescharacteristic
373D4—242no—
374D4—241no—
375D4—248no—
376D4—241no—
377D4—241no—
378D4—241no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—28no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—432no—
390C2×C4—48no—
391C2×C4—48no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4—121no—
399C2×C4—122no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4C4×#6712yes—
403C2×C4#93012yes—
404C8—242no—
405C2×C3—21no—
406C2×C3—28no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3C2×#32712yes—
411C2×C3#150412yes—
412C2×C3#187411yescharacteristic
413C2×C3#189611yescharacteristic
414C2×C3#189811yescharacteristic
415D3—161no—
416C22—28no—
417C22—28no—
418C22—28no—
419C22—28no—
420C22—22no—
421C22—24no—
422C22—22no—
423C22—22no—
424C22—24no—
425C22—24no—
426C22—22no—
427C22—21no—
428C22—24no—
429C22—21no—
430C22—21no—
431C22—24no—
432C22—21no—
433C22—24no—
434C22—242no—
435C22—242no—
436C22—241no—
437C22—241no—
438C22—241no—
439C22#445118yescomplemented
440C22C2×#80512yes—
441C22C2×#80612yes—
442C22C2×#80911yescharacteristic
443C22#496611yescharacteristic
444C22#496711yescharacteristic
445C22#496811yescharacteristic
446C22#521311yescharacteristic
447C22#521711yescharacteristic
448C22#522711yescharacteristic
449C4—22no—
450C4—22no—
451C4—48no—
452C4—42no—
453C4—242no—
454C4—241no—
455C3#914711yescharacteristic, complemented, Sylow
456C2—21no—
457C2—28no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—481no—
462C2C2×#445112yes—
463C2#6939812yes—
464C2#7153211yescharacteristic
465C2#7154311yescharacteristic
466C2#7156011yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC22×C3×#95C2
socleC23×C3C2×C22⋊C4
prime radical#84032C2
cosocleC23×C3×C4C23
hypercenterC22×#95D3
hypocenterC3#9147
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
1. C22×C3×#95C2semidirect
3. #72126C2semidirect
4. C2×#4451C2semidirect
5. #67612C2semidirect
10. C2×#930C4semidirect
17. #5212C4semidirect
23. C22×#95D3semidirect
43. C22×D12C2×C4semidirect
53. C2×C4⋊Dic3C2×C4semidirect
54. C2×C4⋊Dic3C2×C4semidirect
58. C3×#95D4semidirect
175. #95#67semidirect
319. C22×C3#327semidirect
439. C22#4451semidirect
455. C3#9147semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#952
#840322
#721262
C2×#44512
#676122
#91473