Finite Group Databasebeta

← #10845425

Subgroups — #10845425

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
1#83240C211yesmaximal, characteristic
2C2×#5094C211yesmaximal, characteristic, complemented
3C2×#4688C211yesmaximal, characteristic
4#74369C211yesmaximal, characteristic
5C3×#1990C211yesmaximal, characteristic, complemented
6C4×#816C211yesmaximal, characteristic, complemented
7#69567C211yesmaximal, characteristic, complemented
8#68884C212yesmaximal, complemented
9#68870C212yesmaximal, complemented
10#68833C212yesmaximal, complemented
11#68825C212yesmaximal, complemented
12#25348—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#947—21no—
15C22×ΓL(1,𝔽25)C2212yes—
16C2×#897C2212yes—
17C2×C4×D12C2211yescharacteristic
18C2×C4×Dic6C2211yescharacteristic
19C2×C3×C4×C8C2211yescharacteristic
20C3×#384C2211yescharacteristic
21#5229—21no—
22C2×(C2×D3)⋊C8C2212yes—
23C2×#816C2211yescharacteristic
24C2×#816C2211yescharacteristic
25#5094—24no—
26#4994—22no—
27#4992—22no—
28#4988—21no—
29#4987C2212yes—
30#4987C2212yes—
31#4987C2212yes—
32#4984—21no—
33C2×#810C2212yes—
34C2×#808C2212yes—
35C2×#807C2212yes—
36C4×C4⋊Dic3C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38#4688—24no—
39C3×#340C2212yes—
40C3×#340C2212yes—
41#4551C2211yescharacteristic
42#4551C2211yescharacteristic
43#4529C2212yes—
44C2×#475—31no—
45C2×#436—31no—
46#2547—31no—
47#2538—31no—
48#2077—32no—
49#2036—32no—
50#1990D311yescharacteristic, complemented
51#1947—31no—
52#1894—32no—
53#1809—32no—
54C4×#103—31no—
55C22×C3×Q8—21no—
56C22×D12D411yescharacteristic
57C22×C4×D3C2312yes—
58C22×Dic6C2311yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8—22no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C42C2311yescharacteristic
69#947—44no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4D412yes—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3D411yescharacteristic
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2311yescharacteristic
79C2×C4⋊Dic3C2311yescharacteristic
80C2×Dic3⋊C4—21no—
81C2×C4×Dic3—21no—
82C2×C4×Dic3—21no—
83C22×C3⋊C8C2312yes—
84C2×ΓL(1,𝔽25)—24no—
85C2×ΓL(1,𝔽25)—24no—
86C2×ΓL(1,𝔽25)—42no—
87#897—24no—
88#897—42no—
89C4×D12—22no—
90C4×D12—22no—
91C4×Dic6—22no—
92C4×Dic6—22no—
93C3×C4×C8—22no—
94C3⋊#95—21no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98(C2×D3)⋊C8—24no—
99#816—22no—
100#816—22no—
101#810—24no—
102#808—24no—
103#807—42no—
104C4⋊(C3⋊C8)D414yes—
105C22×M16—32no—
106C2×C4⋊Q8C2×D311yescharacteristic
107C2×#124—61no—
108C2×C22⋊Q8—32no—
109C2×C4×Q8—31no—
110C2×C4×D4—31no—
111#475—64no—
112#436—64no—
113C2×#107—32no—
114C2×#106—32no—
115C2×C4⋊C8—31no—
116C2×#103—31no—
117C2×#103—31no—
118C2×#103—32no—
119C2×C22⋊C8—32no—
120C2×C4×C8C2×D311yescharacteristic
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⋊C4—31no—
131#340—31no—
132#340—31no—
133#340—32no—
134#340C2×D312yes—
135#340C2×D312yes—
136C23×D3C2×D412yes—
137C2×C3×Q8—42no—
138C2×C3×Q8—42no—
139C2×C3×Q8—42no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2411yescharacteristic
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C22×Dic3C2×D412yes—
150C2×D12—22no—
151C2×D12—22no—
152C2×D12—22no—
153C2×C4×D3—22no—
154C2×C4×D3—22no—
155C2×C4×D3—22no—
156C2×C4×D3—22no—
157C2×C4×D3—22no—
158C2×C4×D3—22no—
159C2×Dic6—21no—
160C2×Dic6—21no—
161C2×Dic6—21no—
162C2×Dic6—21no—
163C2×Dic6—21no—
164C2×Dic6#4112yes—
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—21no—
169C2×C3×C8—21no—
170C2×C3×C8#4112yes—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—42no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—81no—
178C3×C42C2×D412yes—
179C3×C42#4112yes—
180(C2×D3)⋊C4—24no—
181(C2×D3)⋊C4—44no—
182(C2×D3)⋊C4—44no—
183(C2×D3)⋊C4—44no—
184(C2×D3)⋊C4—82no—
185C4⋊Dic3—22no—
186C4⋊Dic3—21no—
187C4⋊Dic3—21no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3—21no—
191C4⋊Dic3—21no—
192C4⋊Dic3—44no—
193Dic3⋊C4—42no—
194C4×Dic3—42no—
195C4×Dic3—42no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8C2×D414yes—
199ΓL(1,𝔽25)—48no—
200C22×Q8—21no—
201C22×Q8—31no—
202C22×D4—31no—
203C23×C4—32no—
204C2×M16—64no—
205C2×M16—64no—
206C2×M16—122no—
207C22×C8—32no—
208C22×C8C22×D311yescharacteristic
209C22×C8C22×D311yescharacteristic
210C4⋊Q8—22no—
211C4⋊Q8—22no—
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—21no—
222C2×C4⋊C4—31no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4—31no—
225C2×C4⋊C4—31no—
226C2×C4⋊C4—62no—
227C2×C4⋊C4—61no—
228C2×C4⋊C4C22×D312yes—
229C2×C4⋊C4C22×D312yes—
230C2×C22⋊C4—32no—
231C2×C22⋊C4—62no—
232C2×C22⋊C4—62no—
233C2×C22⋊C4—62no—
234C2×C22⋊C4—62no—
235C2×C42—61no—
236C2×C42—61no—
237C2×C42C22×D311yescharacteristic
238#107—122no—
239#106—64no—
240C4⋊C8—34no—
241#103—64no—
242#103—62no—
243#103—62no—
244C22⋊C8—64no—
245C4×C8—22no—
246#95—62no—
247#95—62no—
248#95—62no—
249#95—61no—
250C23×C3C22×D411yescharacteristic
251C22×D3—22no—
252C22×D3—22no—
253C22×D3—22no—
254C22×D3—22no—
255C22×D3—22no—
256C22×D3—22no—
257C22×D3—22no—
258C3×Q8—84no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4#12112yes—
268C2×C3×C4#12112yes—
269C2×C3×C4AGL(1,ℤ/8)12yes—
270C2×C3×C4#13712yes—
271C2×C3×C4C22×D411yescharacteristic
272C2×C3×C4C2×#4111yescharacteristic
273C2×C3×C4C2×#4111yescharacteristic
274C2×C3×C4C2×#4111yescharacteristic
275C2×C3×C4H(5,𝔽2)11yescharacteristic
276C2×C3×C42−1+411yescharacteristic
277C2×Dic3—22no—
278C2×Dic3—22no—
279C2×Dic3—22no—
280C2×Dic3—22no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292D12—44no—
293C4×D3—42no—
294C4×D3—42no—
295C4×D3—42no—
296C4×D3—42no—
297Dic6—22no—
298Dic6—22no—
299Dic6—41no—
300Dic6—41no—
301C3×C8—22no—
302C3×C8—41no—
303C3⋊C8—24no—
304C24—32no—
305C2×Q8—32no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—61no—
310C2×Q8—61no—
311C2×Q8—61no—
312C2×Q8—61no—
313C2×Q8—61no—
314C2×D4—62no—
315C2×D4—62no—
316C2×D4—62no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—32no—
320C22×C4—61no—
321C22×C4—62no—
322C22×C4—62no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—62no—
326C22×C4—62no—
327C22×C4—61no—
328C22×C4—61no—
329C22×C4—61no—
330C22×C4D3×D411yescharacteristic
331C22×C4D3×D411yescharacteristic
332C22×C4C23×D311yescharacteristic
333M16—128no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—34no—
340C2×C8—62no—
341C2×C8—62no—
342C2×C8#23912yes—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—62no—
350C4⋊C4—61no—
351C4⋊C4—61no—
352C4⋊C4—61no—
353C4⋊C4—61no—
354C4⋊C4—61no—
355C4⋊C4—61no—
356C4⋊C4—81no—
357C4⋊C4—124no—
358C4⋊C4—122no—
359C22⋊C4—64no—
360C22⋊C4—124no—
361C22⋊C4—124no—
362C22⋊C4—124no—
363C22⋊C4—242no—
364C42—122no—
365C42—122no—
366C42#23512yes—
367C42D3×D412yes—
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—82no—
385C3×C4#46814yes—
386C3×C4#52911yescharacteristic
387C3×C4#54211yescharacteristic
388C3×C4#54611yescharacteristic
389C3×C4#54811yescharacteristic
390Dic3—42no—
391Dic3—42no—
392Dic3—81no—
393Dic3—81no—
394Dic3—81no—
395C23—62no—
396C23—62no—
397C23—62no—
398C23—62no—
399C23—62no—
400C23—62no—
401C23—62no—
402C23C2×D3×D411yescharacteristic
403Q8—62no—
404Q8—62no—
405Q8—84no—
406Q8—121no—
407Q8—121no—
408D4—124no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4#88412yes—
437C2×C4#88412yes—
438C2×C4#91112yes—
439C2×C4#91512yes—
440C2×C4C2×#23511yescharacteristic
441C2×C4C2×D3×D411yescharacteristic
442C2×C4C2×#23911yescharacteristic
443C2×C4#100711yescharacteristic
444C2×C4D3×#4111yescharacteristic
445C2×C4#100911yescharacteristic
446C8—22no—
447C8—41no—
448C8—64no—
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×#88411yescharacteristic
465C22#567611yescharacteristic
466C22#572611yescharacteristic
467C22C2×#91111yescharacteristic
468C22C2×#91511yescharacteristic
469C22#576211yescharacteristic
470C22#576611yescharacteristic
471C4—22no—
472C4—82no—
473C4—82no—
474C4—122no—
475C4—122no—
476C4—241no—
477C4—241no—
478C4—241no—
479C4#478014yes—
480C4#558011yescharacteristic
481C4#558211yescharacteristic
482C4#563011yescharacteristic
483C4#563311yescharacteristic
484C3#2534811yescharacteristic, complemented, Sylow
485C2—242no—
486C2#7662411yescharacteristic
487C2C2×#478011yescharacteristic
488C2#7880011yescharacteristic
489C2#8158211yescharacteristic
490C2#8161611yescharacteristic
491C2#8190211yescharacteristic
492C2#8195011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1990C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenter#1990D3
hypocenterC3#25348
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
2. C2×#5094C2semidirect
5. C3×#1990C2semidirect
6. C4×#816C2semidirect
7. #69567C2semidirect
8. #68884C2semidirect
9. #68870C2semidirect
10. #68833C2semidirect
11. #68825C2semidirect
50. #1990D3semidirect
484. C3#25348semidirect

Maximal subgroups

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

subgroupindex
#832402
C2×#50942
C2×#46882
#743692
C3×#19902
C4×#8162
#695672
#688842
#688702
#688332
#688252
#253483