Finite Group Databasebeta

← #10845422

Subgroups — #10845422

Order 768.

862 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4851C211yesmaximal, characteristic, complemented
2C2×#4838C211yesmaximal, characteristic
3#74369C211yesmaximal, characteristic
4#73802C211yesmaximal, characteristic
5C3×#1947C211yesmaximal, characteristic, complemented
6#70635C212yesmaximal, complemented
7#70631C212yesmaximal, complemented
8#69582C211yesmaximal, characteristic, complemented
9C4×#810C211yesmaximal, characteristic, complemented
10#68884C212yesmaximal, complemented
11#68832C212yesmaximal, complemented
12#25348—31nomaximal, Sylow
13C22×ΓL(1,𝔽25)C2212yes—
14C2×#897C2212yes—
15C2×#875—21no—
16C2×C4×D12C2211yescharacteristic
17C2×#869C2211yescharacteristic
18C2×C4×Dic6C2211yescharacteristic
19C2×C3×#107C2212yes—
20C2×C3×C4⋊C8C2211yescharacteristic
21C3×#384C2211yescharacteristic
22C3×C4×C4⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C8C2212yes—
24C2×#817C2212yes—
25C2×#816C2212yes—
26#4994—22no—
27#4988—21no—
28#4987C2212yes—
29#4987C2212yes—
30#4987C2212yes—
31#4984—21no—
32C2×#810C2211yescharacteristic
33C2×#810C2211yescharacteristic
34#4929—21no—
35C2×C4×C3⋊C8C2211yescharacteristic
36#4851—24no—
37#4838—24no—
38#4674—22no—
39C3×#340C2212yes—
40#4529C2212yes—
41#4529C2211yescharacteristic
42#4529C2211yescharacteristic
43#4529C2212yes—
44C2×#475—31no—
45C2×#436—31no—
46#2547—31no—
47#2538—31no—
48#2077—32no—
49#2036—32no—
50#1990—31no—
51#1947D311yescharacteristic, complemented
52#1894—32no—
53#1809—32no—
54C4×#103—31no—
55C22×D12D411yescharacteristic
56C22×C4×D3C2312yes—
57C22×Dic6—21no—
58C22×Dic6C2311yescharacteristic
59C22×C3×C8C2312yes—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4C2311yescharacteristic
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C42—21no—
66C2×C3×C42C2311yescharacteristic
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4D412yes—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3D411yescharacteristic
75C2×C4⋊Dic3C2311yescharacteristic
76C2×C4⋊Dic3C2312yes—
77C2×Dic3⋊C4—21no—
78C2×C4×Dic3—21no—
79C22×C3⋊C8C2311yescharacteristic
80C22×C3⋊C8C2311yescharacteristic
81C2×ΓL(1,𝔽25)—24no—
82C2×ΓL(1,𝔽25)—24no—
83C2×ΓL(1,𝔽25)—42no—
84#897—24no—
85#897—42no—
86#875—44no—
87C4×D12—22no—
88C4×D12—22no—
89#869—22no—
90#869—22no—
91C4×Dic6—22no—
92C4×Dic6—22no—
93C3×#107—42no—
94C3×C4⋊C8D414yes—
95C3×#95—21no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99(C2×D3)⋊C8—24no—
100#817—24no—
101#816—24no—
102#810—22no—
103#810—22no—
104C4×C3⋊C8—22no—
105C22×M16—32no—
106C2×C4⋊Q8—31no—
107C2×#124—61no—
108C2×C22⋊Q8—32no—
109C2×C4×Q8—31no—
110C2×C4×D4—31no—
111#475—64no—
112#436—64no—
113C2×#107C2×D312yes—
114C2×#106—32no—
115C2×C4⋊C8C2×D311yescharacteristic
116C2×#103—32no—
117C2×#103—31no—
118C2×#103—31no—
119C2×C22⋊C8—32no—
120C2×C4×C8—31no—
121#397—62no—
122#396—62no—
123#391—61no—
124#390—61no—
125#389—61no—
126#384C2×D311yescharacteristic
127#380—32no—
128#380—32no—
129#380—32no—
130C4×C4⋊C4C2×D311yescharacteristic
131#340—31no—
132#340—32no—
133#340—31no—
134#340—32no—
135#340C2×D312yes—
136C23×D3C2×D412yes—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2411yescharacteristic
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3—22no—
146C22×Dic3C2×D412yes—
147C2×D12—22no—
148C2×D12—22no—
149C2×D12—22no—
150C2×C4×D3—22no—
151C2×C4×D3—22no—
152C2×C4×D3—22no—
153C2×C4×D3—22no—
154C2×C4×D3—22no—
155C2×C4×D3—22no—
156C2×Dic6—21no—
157C2×Dic6—21no—
158C2×Dic6—21no—
159C2×Dic6—21no—
160C2×Dic6—21no—
161C2×Dic6—42no—
162C2×Dic6—42no—
163C2×Dic6—42no—
164C2×Dic6#4112yes—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8C2×D414yes—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4—81no—
176C3×C42—42no—
177C3×C42C2×D412yes—
178C3×C42#4112yes—
179(C2×D3)⋊C4—24no—
180(C2×D3)⋊C4—44no—
181(C2×D3)⋊C4—44no—
182(C2×D3)⋊C4—44no—
183(C2×D3)⋊C4—82no—
184C4⋊Dic3—22no—
185C4⋊Dic3—22no—
186C4⋊Dic3—21no—
187C4⋊Dic3—21no—
188C4⋊Dic3—22no—
189C4⋊Dic3—44no—
190C4⋊Dic3—42no—
191Dic3⋊C4—42no—
192C4×Dic3—42no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—21no—
198C2×C3⋊C8#4112yes—
199ΓL(1,𝔽25)—48no—
200C22×Q8—31no—
201C22×Q8—61no—
202C22×D4—31no—
203C23×C4—32no—
204C2×M16—64no—
205C2×M16—64no—
206C2×M16—122no—
207C22×C8—31no—
208C22×C8—31no—
209C22×C8C22×D312yes—
210C4⋊Q8—62no—
211C4⋊Q8—62no—
212#124—124no—
213C22⋊Q8—64no—
214C22⋊Q8—122no—
215C4×Q8—62no—
216C4×Q8—62no—
217C4×D4—62no—
218C4×D4—62no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—31no—
222C2×C4⋊C4—31no—
223C2×C4⋊C4—32no—
224C2×C4⋊C4—62no—
225C2×C4⋊C4—61no—
226C2×C4⋊C4—61no—
227C2×C4⋊C4C22×D312yes—
228C2×C4⋊C4C22×D311yescharacteristic
229C2×C4⋊C4C22×D311yescharacteristic
230C2×C22⋊C4—32no—
231C2×C22⋊C4—62no—
232C2×C22⋊C4—62no—
233C2×C22⋊C4—62no—
234C2×C22⋊C4—62no—
235C2×C42—21no—
236C2×C42—61no—
237C2×C42C22×D311yescharacteristic
238#107—42no—
239#106—64no—
240C4⋊C8D1214yes—
241#103—62no—
242#103—62no—
243#103—64no—
244C22⋊C8—64no—
245C4×C8—62no—
246#95—21no—
247#95—62no—
248#95—62no—
249#95—62no—
250C23×C3C22×D411yescharacteristic
251C22×D3—22no—
252C22×D3—22no—
253C22×D3—22no—
254C22×D3—22no—
255C22×D3—22no—
256C22×D3—22no—
257C22×D3—22no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—42no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4#12112yes—
270C2×C3×C4#12112yes—
271C2×C3×C4AGL(1,ℤ/8)12yes—
272C2×C3×C4#13712yes—
273C2×C3×C4C22×D411yescharacteristic
274C2×C3×C4C2×#4111yescharacteristic
275C2×C3×C4C2×#4111yescharacteristic
276C2×C3×C4C2×#4111yescharacteristic
277C2×C3×C4H(5,𝔽2)11yescharacteristic
278C2×C3×C42−1+411yescharacteristic
279C2×Dic3—22no—
280C2×Dic3—22no—
281C2×Dic3—22no—
282C2×Dic3—22no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—42no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291D12—44no—
292C4×D3—42no—
293C4×D3—42no—
294C4×D3—42no—
295C4×D3—42no—
296Dic6—22no—
297Dic6—22no—
298Dic6—41no—
299Dic6—41no—
300Dic6—84no—
301C3×C8—24no—
302C3⋊C8—22no—
303C3⋊C8—41no—
304C24—32no—
305C2×Q8—32no—
306C2×Q8—61no—
307C2×Q8—61no—
308C2×Q8—61no—
309C2×Q8—61no—
310C2×Q8—61no—
311C2×Q8—122no—
312C2×Q8—122no—
313C2×Q8—122no—
314C2×D4—62no—
315C2×D4—62no—
316C2×D4—62no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—32no—
321C22×C4—61no—
322C22×C4—62no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—62no—
326C22×C4—62no—
327C22×C4—62no—
328C22×C4—61no—
329C22×C4—62no—
330C22×C4D3×D411yescharacteristic
331C22×C4D3×D411yescharacteristic
332C22×C4C23×D311yescharacteristic
333M16—128no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—32no—
337C2×C8—61no—
338C2×C8—61no—
339C2×C8—61no—
340C2×C8—61no—
341C2×C8—61no—
342C2×C8C2×D1214yes—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—22no—
349C4⋊C4—42no—
350C4⋊C4—62no—
351C4⋊C4—62no—
352C4⋊C4—61no—
353C4⋊C4—61no—
354C4⋊C4—62no—
355C4⋊C4—81no—
356C4⋊C4—124no—
357C4⋊C4—122no—
358C4⋊C4—122no—
359C22⋊C4—64no—
360C22⋊C4—124no—
361C22⋊C4—124no—
362C22⋊C4—124no—
363C22⋊C4—242no—
364C42—42no—
365C42—122no—
366C42C2×D1212yes—
367C42#23912yes—
368C22×C3C2×#12111yescharacteristic
369C22×C3#54611yescharacteristic
370C22×C3#54711yescharacteristic
371C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
372C22×C3C2×#13711yescharacteristic
373C22×C3#57711yescharacteristic
374C22×C3#57811yescharacteristic
375C2×D3—42no—
376C2×D3—42no—
377C2×D3—42no—
378C2×D3—42no—
379C2×D3—42no—
380C2×D3—42no—
381C2×D3—42no—
382C3×C4—22no—
383C3×C4—82no—
384C3×C4—81no—
385C3×C4—81no—
386C3×C4#46814yes—
387C3×C4#52911yescharacteristic
388C3×C4#54211yescharacteristic
389C3×C4#54611yescharacteristic
390C3×C4#54811yescharacteristic
391Dic3—42no—
392Dic3—42no—
393Dic3—82no—
394Dic3—81no—
395C23—62no—
396C23—62no—
397C23—62no—
398C23—62no—
399C23—62no—
400C23—62no—
401C23—62no—
402C23C2×D3×D411yescharacteristic
403Q8—62no—
404Q8—62no—
405Q8—121no—
406Q8—121no—
407Q8—244no—
408D4—124no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—62no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—121no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4#89512yes—
437C2×C4#89512yes—
438C2×C4#91112yes—
439C2×C4#91812yes—
440C2×C4C22×D1211yescharacteristic
441C2×C4#100411yescharacteristic
442C2×C4C2×#23911yescharacteristic
443C2×C4#100711yescharacteristic
444C2×C4D3×#4111yescharacteristic
445C2×C4D3×#4111yescharacteristic
446C8—24no—
447C8—62no—
448C8—121no—
449C2×C3#242711yescharacteristic
450C2×C3C2×#46811yescharacteristic
451C2×C3#306311yescharacteristic
452C2×C3#332811yescharacteristic
453C2×C3#333211yescharacteristic
454C2×C3#335311yescharacteristic
455C2×C3#335611yescharacteristic
456D3—82no—
457C22—122no—
458C22—122no—
459C22—122no—
460C22—122no—
461C22—122no—
462C22—122no—
463C22—122no—
464C22C2×#89511yescharacteristic
465C22#561211yescharacteristic
466C22#563111yescharacteristic
467C22C2×#91111yescharacteristic
468C22C2×#91811yescharacteristic
469C22#577511yescharacteristic
470C22#577911yescharacteristic
471C4—22no—
472C4—81no—
473C4—81no—
474C4—82no—
475C4—122no—
476C4—122no—
477C4—242no—
478C4—241no—
479C4#488614yes—
480C4#555411yescharacteristic
481C4#557611yescharacteristic
482C4#569111yescharacteristic
483C4#573311yescharacteristic
484C3#2534811yescharacteristic, complemented, Sylow
485C2—242no—
486C2#7621911yescharacteristic
487C2C2×#488611yescharacteristic
488C2#7921411yescharacteristic
489C2#8035711yescharacteristic
490C2#8049511yescharacteristic
491C2#8099011yescharacteristic
492C2#8112911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1947C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenter#1947D3
hypocenterC3#25348
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
1. C2×#4851C2semidirect
5. C3×#1947C2semidirect
6. #70635C2semidirect
7. #70631C2semidirect
8. #69582C2semidirect
9. C4×#810C2semidirect
10. #68884C2semidirect
11. #68832C2semidirect
51. #1947D3semidirect
484. C3#25348semidirect

Maximal subgroups

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

subgroupindex
C2×#48512
C2×#48382
#743692
#738022
C3×#19472
#706352
#706312
#695822
C4×#8102
#688842
#688322
#253483