Finite Group Databasebeta

← #10823955

Subgroups — #10823955

Order 768.

1010 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4802C212yesmaximal
2D3×#391C211yesmaximal, characteristic
3#74318C211yesmaximal, characteristic
4C3×#2072C211yesmaximal, characteristic, complemented
5#70234C212yesmaximal, complemented
6#69670C211yesmaximal, characteristic, complemented
7#69589C211yesmaximal, characteristic, complemented
8#68915C212yesmaximal, complemented
9#68867C212yesmaximal
10#68660C211yesmaximal, characteristic
11#68636C211yesmaximal, characteristic, complemented
12#23629—31nomaximal, Sylow
13C22×D3×Q8C2211yescharacteristic
14C22×ΓL(1,𝔽25)C2212yes—
15C2×#892C2212yes—
16C2×D3×C4⋊C4C2211yescharacteristic
17C2×C42×D3C2211yescharacteristic
18C2×C3×#103C2212yes—
19C3×#391C2211yescharacteristic
20C3×#384C2211yescharacteristic
21#5229C2211yescharacteristic
22C2×#835C2212yes—
23C2×#816C2212yes—
24#4987—21no—
25#4987—21no—
26#4977C2212yes—
27Dic3×C4⋊C4—21no—
28Dic3×C4⋊C4—21no—
29#4973C2211yescharacteristic
30C2×#810C2212yes—
31#4935C2211yescharacteristic
32#4929C2211yescharacteristic
33#4802—28no—
34D3×#95—21no—
35D3×#95—21no—
36#4646—21no—
37#4646—21no—
38C3×#340C2212yes—
39C3×#336C2211yescharacteristic
40#4551C2212yes—
41#4550C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2211yescharacteristic
45C2×#419—32no—
46C2×#391—31no—
47#2285—31no—
48#2074—32no—
49#2074—32no—
50#2072—31no—
51#2072—31no—
52#2072—31no—
53#2072D311yescharacteristic, complemented
54#1736—32no—
55#1728—31no—
56C22×C3×Q8C2311yescharacteristic
57C2×D3×Q8—22no—
58C2×D3×Q8—24no—
59C2×D3×Q8—24no—
60C2×D3×Q8D414yes—
61C22×C4×D3—21no—
62C22×C4×D3—21no—
63C22×C4×D3D411yescharacteristic
64C22×C4×D3D411yescharacteristic
65C22×C4×D3C2311yescharacteristic
66C22×Dic6—21no—
67C22×Dic6C2311yescharacteristic
68C22×C3×C8C2312yes—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×(C2×D3)⋊C4—22no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2312yes—
79C2×Dic3⋊C4—21no—
80C2×C4×Dic3—22no—
81C2×C4×Dic3D411yescharacteristic
82C2×C4×Dic3D411yescharacteristic
83C2×C4×Dic3C2311yescharacteristic
84C22×C3⋊C8C2312yes—
85C2×ΓL(1,𝔽25)—24no—
86C2×ΓL(1,𝔽25)—24no—
87C2×ΓL(1,𝔽25)—24no—
88#892—24no—
89#892—24no—
90D3×C4⋊C4—24no—
91C42×D3—24no—
92C3×#103—24no—
93C3×#95—21no—
94C3×#95—21no—
95#835—24no—
96C3⋊#95—21no—
97C3⋊#95—21no—
98C3⋊#95—21no—
99C3⋊#95—21no—
100C3⋊#95—21no—
101C3⋊#95—21no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104#816—24no—
105#810—24no—
106C23×Q8—31no—
107C22×M16—32no—
108C2×#117—32no—
109C22×C4⋊C4—31no—
110C22×C42—31no—
111#419—68no—
112C2×#103—32no—
113C2×#103—32no—
114C2×#103—32no—
115C2×#103C2×D312yes—
116#391—31no—
117#391—31no—
118#391—31no—
119#391—61no—
120#391—61no—
121#391C2×D311yescharacteristic
122#384—31no—
123#384C2×D311yescharacteristic
124#382—32no—
125#380—61no—
126#380—61no—
127C4×C4⋊C4—61no—
128C4×C4⋊C4—61no—
129C2×#95—61no—
130C2×#95—61no—
131#340—32no—
132#340—32no—
133#340—32no—
134#340C2×D312yes—
135#336—32no—
136#336—31no—
137#336C2×D311yescharacteristic
138C23×D3C2×D411yescharacteristic
139C2×C3×Q8—21no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8C2×D412yes—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2411yescharacteristic
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3—22no—
152C22×Dic3C2×D411yescharacteristic
153C22×Dic3C2×D411yescharacteristic
154C22×Dic3C2×D411yescharacteristic
155D3×Q8—216no—
156D3×Q8—48no—
157C2×C4×D3—24no—
158C2×C4×D3—22no—
159C2×C4×D3—22no—
160C2×C4×D3—22no—
161C2×C4×D3—22no—
162C2×C4×D3—41no—
163C2×C4×D3—41no—
164C2×C4×D3—42no—
165C2×C4×D3—42no—
166C2×C4×D3—42no—
167C2×C4×D3—42no—
168C2×C4×D3—42no—
169C2×C4×D3C2×D412yes—
170C2×C4×D3#4114yes—
171C2×C4×D3#4114yes—
172C2×C4×D3#4112yes—
173C2×Dic6—22no—
174C2×Dic6—22no—
175C2×Dic6—21no—
176C2×Dic6—24no—
177C2×Dic6—42no—
178C2×Dic6—42no—
179C2×Dic6C2×D412yes—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—42no—
187C3×C4⋊C4—42no—
188C3×C42—22no—
189(C2×D3)⋊C4—44no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192C4⋊Dic3—22no—
193C4⋊Dic3—42no—
194C4⋊Dic3—42no—
195Dic3⋊C4—42no—
196C4×Dic3—22no—
197C4×Dic3—22no—
198C4×Dic3—22no—
199C4×Dic3—44no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203ΓL(1,𝔽25)—48no—
204C22×Q8—34no—
205C22×Q8—31no—
206C22×Q8—61no—
207C22×Q8—62no—
208C22×Q8—64no—
209C22×Q8—64no—
210C22×Q8C22×D311yescharacteristic
211C23×C4—31no—
212C23×C4—31no—
213C23×C4—31no—
214C23×C4—61no—
215C23×C4—61no—
216C2×M16—64no—
217C2×M16—64no—
218C2×M16—64no—
219C22×C8—32no—
220C22×C8C22×D312yes—
221#117—64no—
222#117—64no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—31no—
226C2×C4⋊C4—32no—
227C2×C4⋊C4—64no—
228C2×C4⋊C4—61no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4—61no—
231C2×C4⋊C4C22×D311yescharacteristic
232C2×C4⋊C4C22×D312yes—
233C2×C22⋊C4—62no—
234C2×C42—31no—
235C2×C42—31no—
236C2×C42—31no—
237C2×C42—64no—
238C2×C42—62no—
239C2×C42C22×D311yescharacteristic
240#103—24no—
241#103—64no—
242#103—64no—
243#103—64no—
244#95—21no—
245#95—21no—
246#95—61no—
247#95—61no—
248#95—61no—
249#95—61no—
250#95—62no—
251#95—62no—
252#95—61no—
253#95—61no—
254C23×C3C22×D411yescharacteristic
255C22×D3—21no—
256C22×D3—21no—
257C22×D3—21no—
258C22×D3—21no—
259C22×D3#12112yes—
260C22×D3#12312yes—
261C22×D3#12412yes—
262C3×Q8—24no—
263C3×Q8—42no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—22no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4#13712yes—
278C2×C3×C4#13712yes—
279C2×C3×C4C22×D411yescharacteristic
280C2×C3×C4C2×#4112yes—
281C2×C3×C4C2×#4112yes—
282C2×C3×C4C2×#4111yescharacteristic
283C2×Dic3—22no—
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3—21no—
287C2×Dic3—21no—
288C2×Dic3—21no—
289C2×Dic3—21no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—42no—
293C2×Dic3—41no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3#12112yes—
299C2×Dic3#12312yes—
300C2×Dic3#12412yes—
301C4×D3—44no—
302C4×D3—44no—
303C4×D3—48no—
304C4×D3—84no—
305C4×D3#12318yes—
306C4×D3#12618yes—
307Dic6—24no—
308Dic6—42no—
309Dic6—48no—
310C3×C8—42no—
311C3⋊C8—42no—
312C24—31no—
313C2×Q8—21no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—32no—
317C2×Q8—64no—
318C2×Q8—616no—
319C2×Q8—61no—
320C2×Q8—62no—
321C2×Q8—62no—
322C2×Q8—122no—
323C2×Q8—122no—
324C2×Q8—128no—
325C2×Q8D3×D412yes—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—31no—
330C22×C4—32no—
331C22×C4—34no—
332C22×C4—34no—
333C22×C4—32no—
334C22×C4—31no—
335C22×C4—31no—
336C22×C4—62no—
337C22×C4—62no—
338C22×C4—64no—
339C22×C4—61no—
340C22×C4—62no—
341C22×C4—62no—
342C22×C4—61no—
343C22×C4—62no—
344C22×C4—121no—
345C22×C4—121no—
346C22×C4—122no—
347C22×C4—122no—
348C22×C4—122no—
349C22×C4—122no—
350C22×C4—122no—
351C22×C4D3×D411yescharacteristic
352C22×C4D3×D411yescharacteristic
353C22×C4C23×D311yescharacteristic
354M16—128no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—62no—
359C2×C8—62no—
360C2×C8—62no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—62no—
367C4⋊C4—62no—
368C4⋊C4—62no—
369C4⋊C4—122no—
370C4⋊C4—122no—
371C4⋊C4—122no—
372C22⋊C4—124no—
373C42—22no—
374C42—62no—
375C42—62no—
376C42—62no—
377C42—124no—
378C22×C3C2×#12111yescharacteristic
379C22×C3C2×#12311yescharacteristic
380C22×C3C2×#12411yescharacteristic
381C22×C3C2×#13711yescharacteristic
382C22×C3C2×#13711yescharacteristic
383C22×C3#57511yescharacteristic
384C22×C3#57511yescharacteristic
385C2×D3—22no—
386C2×D3—22no—
387C2×D3—22no—
388C2×D3—22no—
389C2×D3—22no—
390C2×D3—22no—
391C2×D3#39714yes—
392C3×C4—22no—
393C3×C4—22no—
394C3×C4—42no—
395C3×C4—81no—
396C3×C4—82no—
397C3×C4C2×#12312yes—
398C3×C4C2×#12612yes—
399Dic3—41no—
400Dic3—41no—
401Dic3—42no—
402Dic3—81no—
403Dic3—82no—
404Dic3#39714yes—
405C23—32no—
406C23—32no—
407C23—32no—
408C23—61no—
409C23—61no—
410C23—61no—
411C23—61no—
412C23C2×D3×D411yescharacteristic
413Q8—24no—
414Q8—42no—
415Q8—64no—
416Q8—128no—
417Q8—122no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—32no—
424C2×C4—32no—
425C2×C4—32no—
426C2×C4—38no—
427C2×C4—38no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—62no—
443C2×C4—124no—
444C2×C4—124no—
445C2×C4—128no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—121no—
449C2×C4—122no—
450C2×C4—121no—
451C2×C4—122no—
452C2×C4—121no—
453C2×C4—122no—
454C2×C4—244no—
455C2×C4#91512yes—
456C2×C4#91812yes—
457C2×C4C2×D3×D411yescharacteristic
458C2×C4D3×#4112yes—
459C2×C4D3×#4112yes—
460C2×C4D3×#4111yescharacteristic
461C8—42no—
462C8—122no—
463C2×C3C2×#39711yescharacteristic
464C2×C3#303711yescharacteristic
465C2×C3#303711yescharacteristic
466C2×C3#309511yescharacteristic
467C2×C3#309511yescharacteristic
468C2×C3#313911yescharacteristic
469C2×C3#314011yescharacteristic
470D3—24no—
471C22—34no—
472C22—62no—
473C22—62no—
474C22—62no—
475C22—62no—
476C22—62no—
477C22—62no—
478C22D3×#12111yescharacteristic
479C22D3×#12311yescharacteristic
480C22D3×#12411yescharacteristic
481C22C2×#91511yescharacteristic
482C22C2×#91811yescharacteristic
483C22#577311yescharacteristic
484C22#577711yescharacteristic
485C4—22no—
486C4—22no—
487C4—34no—
488C4—42no—
489C4—81no—
490C4—82no—
491C4—121no—
492C4—121no—
493C4—122no—
494C4—241no—
495C4—242no—
496C4D3×#12312yes—
497C4D3×#12612yes—
498C3#2362911yescharacteristic, complemented, Sylow
499C2—64no—
500C2D3×#39711yescharacteristic
501C2#8038311yescharacteristic
502C2#8047611yescharacteristic
503C2#8142811yescharacteristic
504C2#8146911yescharacteristic
505C2#8160011yescharacteristic
506C2#8168211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2072C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2072D3
hypocenterC3#23629
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2072

Complements

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

NH ≅ G/Ntype
4. C3×#2072C2semidirect
5. #70234C2semidirect
6. #69670C2semidirect
7. #69589C2semidirect
8. #68915C2semidirect
11. #68636C2semidirect
53. #2072D3semidirect
498. C3#23629semidirect

Maximal subgroups

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

subgroupindex
C2×#48022
D3×#3912
#743182
C3×#20722
#702342
#696702
#695892
#689152
#688672
#686602
#686362
#236293