Finite Group Databasebeta

← #10702889

Subgroups — #10702889

Order 768.

1134 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2308C211yesmaximal, characteristic
2#84043C211yesmaximal, characteristic
3C22×#834C211yesmaximal, characteristic
4C2×#4527C212yesmaximal, complemented
5C2×#4527C212yesmaximal, complemented
6#11172—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C22×C3×C22⋊C4C2212yes—
9C22×#217C2211yescharacteristic
10C22×#217C2211yescharacteristic
11C22×#217C2212yes—
12C3×#386—24no—
13C3×#381—21no—
14C3×#379—21no—
15#5247—21no—
16#5220—22no—
17#5220C414yescomplemented
18C2×#834—22no—
19C2×#834—24no—
20#5211—21no—
21#4527—24no—
22#4527—24no—
23C22×#99—31no—
24#2308—31no—
25#2308D311yescharacteristic
26C2×#342—32no—
27C2×#342—32no—
28C25×C3D411yescharacteristic
29C25×C3C2311yescharacteristic
30C22×C3×D4—22no—
31C22×C3×D4—22no—
32C22×C3×D4—22no—
33C22×C3×D4—22no—
34C22×C3×D4—22no—
35C22×C3×D4—22no—
36C22×C3×D4C2×C412yes—
37C22×C3×D4D412yes—
38C23×C3×C4—22no—
39C23×C3×C4D411yescharacteristic
40C23×Dic3—21no—
41C23×Dic3—22no—
42C23×Dic3C2×C412yes—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—24no—
49C2×C3×C22⋊C4D414yes—
50C2×#217—22no—
51C2×#217—22no—
52C2×#217—22no—
53C2×#217—22no—
54C2×#217—22no—
55C2×#217—22no—
56C2×#217—21no—
57C2×#217—21no—
58C2×#217—21no—
59C2×#217—22no—
60C2×#217—22no—
61C2×#217—22no—
62C2×#217—24no—
63C2×#217C2×C412yes—
64C2×#217D414yes—
65C3×#95—24no—
66#834—28no—
67C3⋊#95—24no—
68C23×D4C2×D311yescharacteristic
69C22×C22⋊C4—31no—
70C22×C22⋊C4—32no—
71C22×C22⋊C4—31no—
72C22×C22⋊C4C2×D312yes—
73C2×#99—64no—
74C2×#99—62no—
75#386—24no—
76#386—34no—
77#386—62no—
78#381—21no—
79#381—61no—
80#379—21no—
81#379—61no—
82#342—64no—
83#342—64no—
84C24×C3—21no—
85C24×C3—22no—
86C24×C3—21no—
87C24×C3—22no—
88C24×C3—82no—
89C24×C3—82no—
90C24×C3C22⋊C412yes—
91C24×C3C22×C411yescharacteristic
92C24×C3C2×D411yescharacteristic
93C24×C3C2×D412yes—
94C24×C3C2×D411yescharacteristic
95C24×C3C2×D412yes—
96C24×C3#4112yes—
97C24×C3#4112yes—
98C24×C3#4112yes—
99C24×C3#4112yes—
100C2×C3×D4—22no—
101C2×C3×D4—22no—
102C2×C3×D4—24no—
103C2×C3×D4—24no—
104C2×C3×D4—22no—
105C2×C3×D4—24no—
106C2×C3×D4—24no—
107C2×C3×D4—22no—
108C2×C3×D4—42no—
109C2×C3×D4—42no—
110C2×C3×D4—42no—
111C2×C3×D4—42no—
112C2×C3×D4—42no—
113C2×C3×D4—42no—
114C2×C3×D4—42no—
115C2×C3×D4—42no—
116C22×C3×C4—21no—
117C22×C3×C4—22no—
118C22×C3×C4—21no—
119C22×C3×C4—22no—
120C22×C3×C4—24no—
121C22×C3×C4—42no—
122C22×C3×C4—42no—
123C22×C3×C4—84no—
124C22×C3×C4C22⋊C412yes—
125C22×Dic3—22no—
126C22×Dic3—22no—
127C22×Dic3—24no—
128C22×Dic3—24no—
129C22×Dic3—41no—
130C22×Dic3—42no—
131C22×Dic3—41no—
132C22×Dic3—42no—
133C22×Dic3—82no—
134C22×Dic3—82no—
135C22×Dic3—84no—
136C3×C22⋊C4—24no—
137C3×C22⋊C4—24no—
138C3×C22⋊C4—42no—
139C3×C22⋊C4—42no—
140C3×C22⋊C4—84no—
141#217—24no—
142#217—24no—
143#217—24no—
144#217—41no—
145#217—42no—
146#217—42no—
147#217—42no—
148#217—42no—
149#217—41no—
150#217—84no—
151C25#6711yescharacteristic
152C25C22×D311yescharacteristic
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4C4×D312yes—
160C22×D4D1212yes—
161C23×C4—22no—
162C23×C4—32no—
163C23×C4—61no—
164C23×C4—62no—
165C23×C4#6711yescharacteristic
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—32no—
173C2×C22⋊C4—34no—
174C2×C22⋊C4—62no—
175C2×C22⋊C4—62no—
176C2×C22⋊C4—62no—
177C2×C22⋊C4—62no—
178C2×C22⋊C4—62no—
179C2×C22⋊C4—62no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—62no—
182C2×C22⋊C4—62no—
183C2×C22⋊C4—62no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—64no—
187C2×C22⋊C4#6714yes—
188#99—68no—
189#95—24no—
190#95—64no—
191C23×C3—22no—
192C23×C3—22no—
193C23×C3—22no—
194C23×C3—42no—
195C23×C3—42no—
196C23×C3—44no—
197C23×C3—44no—
198C23×C3—42no—
199C23×C3—44no—
200C23×C3—44no—
201C23×C3—42no—
202C23×C3—42no—
203C23×C3—82no—
204C23×C3—82no—
205C23×C3—82no—
206C23×C3—82no—
207C23×C3—82no—
208C23×C3—84no—
209C23×C3—84no—
210C23×C3C2×C22⋊C411yescharacteristic
211C23×C3#11711yescharacteristic
212C23×C3#11711yescharacteristic
213C23×C3C4×D412yes—
214C23×C3C4×D412yes—
215C23×C3#12011yescharacteristic
216C23×C3#12011yescharacteristic
217C23×C3#12112yes—
218C23×C3#12112yes—
219C23×C3#12112yes—
220C23×C3#12112yes—
221C23×C3#12312yes—
222C23×C3#12312yes—
223C23×C3#12311yescharacteristic
224C23×C3#12311yescharacteristic
225C23×C3#12312yes—
226C23×C3#12312yes—
227C23×C3#12412yes—
228C23×C3#12412yes—
229C23×C3#12612yes—
230C23×C3#12612yes—
231C3×D4—44no—
232C3×D4—44no—
233C3×D4—44no—
234C3×D4—44no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—48no—
241C2×C3×C4—44no—
242C2×C3×C4—84no—
243C2×C3×C4—84no—
244C2×Dic3—44no—
245C2×Dic3—44no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—44no—
249C2×Dic3—84no—
250C2×Dic3—82no—
251C2×Dic3—82no—
252C2×Dic3—84no—
253C2×Dic3—84no—
254C2×Dic3—84no—
255C24—21no—
256C24—21no—
257C24—22no—
258C24—22no—
259C24—82no—
260C24—82no—
261C24(C2×D3)⋊C412yes—
262C24C2×C4×D311yescharacteristic
263C24C2×D1211yescharacteristic
264C24#23512yes—
265C24D3×D412yes—
266C24#23712yes—
267C24#23712yes—
268C24#23712yes—
269C24C2×#6711yescharacteristic
270C24C2×#6712yes—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—22no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—24no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—62no—
295C22×C4—62no—
296C22×C4—64no—
297C22×C4—64no—
298C22×C4—84no—
299C22×C4—121no—
300C22×C4—122no—
301C22×C4—121no—
302C22×C4—122no—
303C22×C4—242no—
304C22×C4—242no—
305C22×C4—244no—
306C22×C4(C2×D3)⋊C412yes—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—64no—
312C22⋊C4—64no—
313C22⋊C4—64no—
314C22⋊C4—84no—
315C22⋊C4—122no—
316C22⋊C4—121no—
317C22⋊C4—122no—
318C22⋊C4—122no—
319C22⋊C4—122no—
320C22⋊C4—121no—
321C22⋊C4—244no—
322C22×C3—22no—
323C22×C3—24no—
324C22×C3—24no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—41no—
328C22×C3—42no—
329C22×C3—41no—
330C22×C3—42no—
331C22×C3—42no—
332C22×C3—42no—
333C22×C3—42no—
334C22×C3—42no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—42no—
341C22×C3—42no—
342C22×C3—42no—
343C22×C3—44no—
344C22×C3—44no—
345C22×C3—42no—
346C22×C3—82no—
347C22×C3—84no—
348C22×C3—84no—
349C22×C3#38111yescharacteristic
350C22×C3#38611yescharacteristic
351C22×C3#38611yescharacteristic
352C22×C3#38812yes—
353C22×C3#38812yes—
354C22×C3#39412yes—
355C22×C3#39412yes—
356C22×C3#39712yes—
357C22×C3#39712yes—
358C3×C4—42no—
359C3×C4—84no—
360Dic3—82no—
361Dic3—84no—
362Dic3—82no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—44no—
367C23—42no—
368C23—42no—
369C23—44no—
370C23—42no—
371C23—44no—
372C23—44no—
373C23—42no—
374C23—42no—
375C23—82no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—84no—
381C23—84no—
382C23#87711yescharacteristic
383C23#87711yescharacteristic
384C23#87912yes—
385C23#87912yes—
386C23#67⋊C412yes—
387C23#88211yescharacteristic
388C23#88312yes—
389C23#88512yes—
390C23#88612yes—
391C23#88611yescharacteristic
392C23C2×(C2×D3)⋊C411yescharacteristic
393C23C4×#6712yes—
394C23#92911yescharacteristic
395C23#93012yes—
396C23#93512yes—
397C23#93512yes—
398C23#93612yes—
399C23#93812yes—
400C23#93912yes—
401C23#93912yes—
402C23#95311yescharacteristic
403D4—44no—
404D4—44no—
405D4—44no—
406D4—44no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—48no—
413C2×C4—44no—
414C2×C4—84no—
415C2×C4—84no—
416C2×C4—124no—
417C2×C4—124no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4—124no—
421C2×C4—244no—
422C2×C4—242no—
423C2×C4—242no—
424C2×C4—244no—
425C2×C4—244no—
426C2×C4—244no—
427C2×C3—21no—
428C2×C3—22no—
429C2×C3—22no—
430C2×C3—21no—
431C2×C3—22no—
432C2×C3—22no—
433C2×C3—22no—
434C2×C3—22no—
435C2×C3—82no—
436C2×C3#144011yescharacteristic
437C2×C3#191312yes—
438C22—22no—
439C22—22no—
440C22—24no—
441C22—24no—
442C22—22no—
443C22—42no—
444C22—42no—
445C22—41no—
446C22—42no—
447C22—42no—
448C22—41no—
449C22—42no—
450C22—42no—
451C22—41no—
452C22—41no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—44no—
460C22—42no—
461C22—44no—
462C22—82no—
463C22—84no—
464C22—84no—
465C22#494311yescharacteristic
466C22#494412yes—
467C22#494512yes—
468C22#495212yes—
469C22#495312yes—
470C22#495811yescharacteristic
471C22#496112yes—
472C22#521311yescharacteristic
473C22#522212yes—
474C4—42no—
475C4—84no—
476C4—242no—
477C4—244no—
478C4—242no—
479C3#1117211yescharacteristic, complemented, Sylow
480C2—22no—
481C2—21no—
482C2—22no—
483C2—21no—
484C2—22no—
485C2—22no—
486C2—22no—
487C2—22no—
488C2—82no—
489C2#7154911yescharacteristic
490C2#7165612yes—
subgroupG/H
centerC22#5213
commutatorC24×C3C22×C4
FrattiniC25C22×D3
FittingC3×#2308C2
socleC22×C3#386
prime radicalC23×C3×D4C22
cosocleC25×C3C23
hypercenter#2308D3
hypocenterC3#11172
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #2308

Complements

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

NH ≅ G/Ntype
4. C2×#4527C2semidirect
5. C2×#4527C2semidirect
17. #5220C4semidirect
479. C3#11172semidirect

Maximal subgroups

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

subgroupindex
C3×#23082
#840432
C22×#8342
C2×#45272
#111723