Finite Group Databasebeta

← #10702270

Subgroups — #10702270

Order 768.

1202 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#381C211yesmaximal, characteristic
2C2×#5211C212yesmaximal
3#82094C211yesmaximal, characteristic
4C2×#4527C212yesmaximal
5C2×#4526C211yesmaximal, characteristic
6#11070—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C22×C3×C22⋊C4C2211yescharacteristic
9C22×C3×C22⋊C4C2211yescharacteristic
10C22×#217C2212yes—
11C22×#217C2212yes—
12C3×#381—22no—
13C3×#381C414yescomplemented
14C2×C3×#95—22no—
15#5212—22no—
16#5211—24no—
17#5211—24no—
18C2×C3⋊#95—24no—
19#4950—24no—
20#4527—24no—
21#4526—22no—
22#2315—31no—
23C2×#381—32no—
24C2×#381D311yescharacteristic
25C2×#342—31no—
26C2×#342—32no—
27C25×C3C2311yescharacteristic
28C23×C3×C4—22no—
29C23×C3×C4—22no—
30C23×C3×C4—21no—
31C23×C3×C4—21no—
32C23×C3×C4—21no—
33C23×C3×C4—21no—
34C23×C3×C4—22no—
35C23×C3×C4—22no—
36C23×C3×C4—21no—
37C23×C3×C4C2×C412yes—
38C23×C3×C4D412yes—
39C23×C3×C4D412yes—
40C23×C3×C4Q812yes—
41C23×Dic3—22no—
42C23×Dic3D414yes—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4C2×C412yes—
49C2×C3×C22⋊C4C2×C412yes—
50C2×C3×C22⋊C4D412yes—
51C2×#217—22no—
52C2×#217—22no—
53C2×#217—22no—
54C2×#217—22no—
55C2×#217—22no—
56C2×#217—22no—
57C2×#217—22no—
58C2×#217—22no—
59C2×#217—22no—
60C2×#217—22no—
61C2×#217—22no—
62C2×#217D414yes—
63C2×C4⋊Dic3—24no—
64C3×#95—28no—
65C3⋊#95—24no—
66C3⋊#95—216no—
67C24×C4C2×D311yescharacteristic
68C22×C22⋊C4—32no—
69C22×C22⋊C4—32no—
70C22×C22⋊C4C2×D311yescharacteristic
71C22×C22⋊C4C2×D311yescharacteristic
72#385—64no—
73#381—22no—
74#381—64no—
75#381—64no—
76#381Dic314yescomplemented
77#380—62no—
78C2×#95—22no—
79C2×#95—64no—
80#342—64no—
81#342—62no—
82C24×C3—44no—
83C24×C3C22×C411yescharacteristic
84C24×C3C2×D411yescharacteristic
85C24×C3C2×D412yes—
86C24×C3C2×D412yes—
87C24×C3C2×D411yescharacteristic
88C24×C3C2×D411yescharacteristic
89C24×C3C2×Q811yescharacteristic
90C24×C3#4111yescharacteristic
91C24×C3#4111yescharacteristic
92C24×C3#4112yes—
93C24×C3#4112yes—
94C22×C3×C4—22no—
95C22×C3×C4—22no—
96C22×C3×C4—24no—
97C22×C3×C4—24no—
98C22×C3×C4—24no—
99C22×C3×C4—22no—
100C22×C3×C4—22no—
101C22×C3×C4—22no—
102C22×C3×C4—24no—
103C22×C3×C4—24no—
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4—24no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—41no—
110C22×C3×C4—42no—
111C22×C3×C4—42no—
112C22×C3×C4—42no—
113C22×C3×C4—42no—
114C22×C3×C4—41no—
115C22×C3×C4—42no—
116C22×C3×C4—41no—
117C22×C3×C4—42no—
118C22×C3×C4—41no—
119C22×C3×C4—44no—
120C22×C3×C4—44no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4C22⋊C414yes—
124C22×C3×C4C4⋊C414yescomplemented
125C22×Dic3—24no—
126C22×Dic3—24no—
127C22×Dic3—24no—
128C22×Dic3—24no—
129C22×Dic3—48no—
130C22×Dic3—48no—
131C22×Dic3—42no—
132C22×Dic3—42no—
133C3×C22⋊C4—24no—
134C3×C22⋊C4—24no—
135C3×C22⋊C4—24no—
136C3×C22⋊C4—42no—
137#217—24no—
138#217—24no—
139#217—42no—
140#217—42no—
141#217—42no—
142#217—42no—
143#217—42no—
144#217—42no—
145C4⋊Dic3—84no—
146C25C22×D311yescharacteristic
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—34no—
157C23×C4—62no—
158C23×C4Dic612yes—
159C23×C4D1212yes—
160C23×C4C2×Dic312yes—
161C23×C4#6712yes—
162C2×C4⋊C4—64no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—34no—
169C2×C22⋊C4—62no—
170C2×C22⋊C4—62no—
171C2×C22⋊C4—62no—
172C2×C22⋊C4—62no—
173C2×C22⋊C4—62no—
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⋊C4C2×Dic312yes—
181C2×C22⋊C4C2×Dic312yes—
182C2×C22⋊C4#6712yes—
183#95—28no—
184#95—616no—
185#95—64no—
186C23×C3—28no—
187C23×C3—24no—
188C23×C3—42no—
189C23×C3—42no—
190C23×C3—44no—
191C23×C3—42no—
192C23×C3—42no—
193C23×C3—42no—
194C23×C3—42no—
195C23×C3—44no—
196C23×C3—44no—
197C23×C3C2×C22⋊C411yescharacteristic
198C23×C3C2×C4⋊C411yescharacteristic
199C23×C3#11711yescharacteristic
200C23×C3C4×D412yes—
201C23×C3C4×D412yes—
202C23×C3#12012yes—
203C23×C3#12011yescharacteristic
204C23×C3#12112yes—
205C23×C3#12111yescharacteristic
206C23×C3#12111yescharacteristic
207C23×C3#12112yes—
208C23×C3#12112yes—
209C23×C3C22⋊Q812yes—
210C23×C3C22⋊Q812yes—
211C23×C3C22⋊Q811yescharacteristic
212C23×C3C22⋊Q811yescharacteristic
213C23×C3#12312yes—
214C23×C3#12312yes—
215C23×C3#12311yescharacteristic
216C23×C3#12412yes—
217C23×C3#12411yescharacteristic
218C23×C3#12612yes—
219C23×C3#12711yescharacteristic
220C2×C3×C4—24no—
221C2×C3×C4—28no—
222C2×C3×C4—24no—
223C2×C3×C4—42no—
224C2×C3×C4—44no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—44no—
229C2×C3×C4—44no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—44no—
233C2×C3×C4—44no—
234C2×C3×C4—44no—
235C2×C3×C4—42no—
236C2×C3×C4—82no—
237C2×C3×C4—82no—
238C2×C3×C4—82no—
239C2×Dic3—48no—
240C2×Dic3—48no—
241C2×Dic3—44no—
242C2×Dic3—44no—
243C2×Dic3—48no—
244C2×Dic3—84no—
245C2×Dic3—84no—
246C2×Dic3—84no—
247C24—44no—
248C24C2×Dic611yescharacteristic
249C24C2×D1211yescharacteristic
250C24#23511yescharacteristic
251C24#23511yescharacteristic
252C24D3×D412yes—
253C24D3×D412yes—
254C24#23712yes—
255C24#23712yes—
256C24C22×Dic311yescharacteristic
257C24C2×#6711yescharacteristic
258C24C2×#6711yescharacteristic
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—64no—
289C22×C4—64no—
290C22×C4—64no—
291C22×C4—64no—
292C22×C4—128no—
293C22×C4—128no—
294C22×C4—122no—
295C22×C4—122no—
296C22×C4C4⋊Dic314yescomplemented
297C22×C4#21714yes—
298C4⋊C4—244no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—24no—
302C22⋊C4—42no—
303C22⋊C4—64no—
304C22⋊C4—64no—
305C22⋊C4—122no—
306C22⋊C4—122no—
307C22⋊C4—122no—
308C22⋊C4—122no—
309C22⋊C4—122no—
310C22⋊C4—122no—
311C22×C3—24no—
312C22×C3—22no—
313C22×C3—22no—
314C22×C3—24no—
315C22×C3—24no—
316C22×C3—24no—
317C22×C3—22no—
318C22×C3—22no—
319C22×C3—24no—
320C22×C3—22no—
321C22×C3—24no—
322C22×C3—22no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—42no—
326C22×C3—41no—
327C22×C3—42no—
328C22×C3—42no—
329C22×C3—41no—
330C22×C3—42no—
331C22×C3—42no—
332C22×C3—42no—
333C22×C3#38011yescharacteristic
334C22×C3#38512yes—
335C22×C3#38611yescharacteristic
336C22×C3#38812yes—
337C22×C3#39011yescharacteristic
338C22×C3#39211yescharacteristic
339C22×C3#39311yescharacteristic
340C22×C3#39412yes—
341C22×C3#39612yes—
342C22×C3#39711yescharacteristic
343C22×C3#39911yescharacteristic
344C22×C3#45814yes—
345C3×C4—44no—
346C3×C4—44no—
347C3×C4—82no—
348Dic3—84no—
349Dic3—84no—
350C23—24no—
351C23—28no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—44no—
357C23—44no—
358C23—44no—
359C23—42no—
360C23—42no—
361C23#87812yes—
362C23#87812yes—
363C23#87912yes—
364C23#88212yes—
365C23#88312yes—
366C23#88412yes—
367C23#88512yes—
368C23#88612yes—
369C23#92411yescharacteristic
370C23#92411yescharacteristic
371C23C2×C4⋊Dic311yescharacteristic
372C23#92611yescharacteristic
373C23#93011yescharacteristic
374C23#93011yescharacteristic
375C23D4×Dic312yes—
376C23D4×Dic312yes—
377C23#93511yescharacteristic
378C23#93611yescharacteristic
379C23#93711yescharacteristic
380C23#93812yes—
381C23#93912yes—
382C23#94011yescharacteristic
383C23C2×#21711yescharacteristic
384C2×C4—28no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—128no—
404C2×C4—128no—
405C2×C4—124no—
406C2×C4—124no—
407C2×C4—128no—
408C2×C4—244no—
409C2×C4—244no—
410C2×C4—244no—
411C2×C3—22no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—22no—
415C2×C3—22no—
416C2×C3—22no—
417C2×C3—21no—
418C2×C3—21no—
419C2×C3#143911yescharacteristic
420C2×C3#189012yes—
421C2×C3#202212yes—
422C2×C3#203012yes—
423C22—22no—
424C22—22no—
425C22—24no—
426C22—24no—
427C22—22no—
428C22—24no—
429C22—24no—
430C22—22no—
431C22—22no—
432C22—24no—
433C22—24no—
434C22—22no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—42no—
444C22—42no—
445C22#494711yescharacteristic
446C22#494812yes—
447C22#494911yescharacteristic
448C22#495012yes—
449C22#495112yes—
450C22#495311yescharacteristic
451C22#496011yescharacteristic
452C22#496112yes—
453C22#505314yes—
454C22#521211yescharacteristic
455C22#522011yescharacteristic
456C22#522111yescharacteristic
457C4—44no—
458C4—44no—
459C4—82no—
460C4—244no—
461C4—244no—
462C3#1107011yescharacteristic, complemented, Sylow
463C2—21no—
464C2—22no—
465C2—22no—
466C2—21no—
467C2—22no—
468C2—22no—
469C2—21no—
470C2—21no—
471C2#6953612yes—
472C2#6953912yes—
473C2#7154611yescharacteristic
474C2#7170412yes—
subgroupG/H
centerC23#937
commutatorC24×C3C22×C4
FrattiniC25C22×D3
FittingC2×C3×#381C2
socleC23×C3#120
prime radical, cosocleC25×C3C23
hypercenterC2×#381D3
hypocenterC3#11070
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C25 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
13. C3×#381C4semidirect
76. #381Dic3semidirect
124. C22×C3×C4C4⋊C4semidirect
296. C22×C4C4⋊Dic3semidirect
462. C3#11070semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3812
C2×#52112
#820942
C2×#45272
C2×#45262
#110703