Finite Group Databasebeta

← #10845321

Subgroups — #10845321

Order 768.

874 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
1#83260C211yesmaximal, characteristic
2C2×#5091C211yesmaximal, characteristic, complemented
3C2×C8×D12C211yesmaximal, characteristic
4#74369C211yesmaximal, characteristic
5C3×#1990C211yesmaximal, characteristic, complemented
6C4×#816C211yesmaximal, characteristic, complemented
7#69567C211yesmaximal, characteristic, complemented
8#68890C212yesmaximal, complemented
9#68882C212yesmaximal, complemented
10#68869C212yesmaximal, complemented
11#68825C212yesmaximal, complemented
12#25340—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C22×D3×C8C2212yes—
15C2×#897C2212yes—
16C2×C4×D12C2211yescharacteristic
17C2×C4×Dic6C2211yescharacteristic
18C2×C3×C4×C8C2211yescharacteristic
19C3×#384C2211yescharacteristic
20#5231—22no—
21C2×(C2×D3)⋊C8C2212yes—
22C2×#816C2211yescharacteristic
23C2×#816C2211yescharacteristic
24#5091—24no—
25#4994—22no—
26#4988—21no—
27#4988—21no—
28#4987C2212yes—
29#4987C2212yes—
30#4987C2212yes—
31#4984—21no—
32#4977—21no—
33C2×#810C2212yes—
34C2×#807C2212yes—
35C2×#807C2212yes—
36C4×C4⋊Dic3C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38C8×D12—24no—
39C3×#340C2212yes—
40C3×#340C2212yes—
41#4551C2211yescharacteristic
42#4551C2211yescharacteristic
43#4529C2212yes—
44C2×#477—31no—
45C2×C8×D4—31no—
46#2563—31no—
47#2538—31no—
48#2076—32no—
49#2036—32no—
50#1990D311yescharacteristic, complemented
51#1947—31no—
52#1892—32no—
53#1808—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
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4D412yes—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3D411yescharacteristic
76C2×C4⋊Dic3C2311yescharacteristic
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2311yescharacteristic
79C2×Dic3⋊C4—21no—
80C2×C4×Dic3—21no—
81C2×C4×Dic3—21no—
82C22×C3⋊C8C2312yes—
83C2×D3×C8—24no—
84C2×D3×C8—24no—
85C2×D3×C8—42no—
86#897—24no—
87#897—42no—
88C4×D12—22no—
89C4×D12—22no—
90C4×Dic6—22no—
91C4×Dic6—22no—
92C3×C4×C8—22no—
93C3⋊#95—21no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97(C2×D3)⋊C8—24no—
98#816—22no—
99#816—22no—
100#810—24no—
101#807—24no—
102#807—42no—
103C4⋊(C3⋊C8)D414yes—
104C23×C8—32no—
105C2×C4⋊Q8C2×D311yescharacteristic
106C2×C22⋊Q8—32no—
107C2×C4×Q8—31no—
108C2×C4×D4—31no—
109#477—64no—
110C8×D4—64no—
111C2×#107—32no—
112C2×#107—32no—
113C2×C4⋊C8—31no—
114C2×#103—31no—
115C2×#103—31no—
116C2×#103—32no—
117C2×C22⋊C8—32no—
118C2×C4×C8C2×D311yescharacteristic
119#396—62no—
120#393—62no—
121#390—61no—
122#390—61no—
123#389—61no—
124#384C2×D311yescharacteristic
125#382—61no—
126#380—32no—
127#380—32no—
128#380—32no—
129C4×C4⋊C4—31no—
130#340—31no—
131#340—31no—
132#340—32no—
133#340C2×D312yes—
134#340C2×D312yes—
135C23×D3C2×D412yes—
136C2×C3×Q8—42no—
137C2×C3×Q8—81no—
138C2×C3×Q8—81no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2411yescharacteristic
144C22×Dic3—21no—
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—21no—
148C22×Dic3C2×D412yes—
149C2×D12—22no—
150C2×D12—22no—
151C2×D12Q1614yes—
152C2×C4×D3—22no—
153C2×C4×D3—22no—
154C2×C4×D3—22no—
155C2×C4×D3—22no—
156C2×C4×D3—22no—
157C2×C4×D3—22no—
158C2×Dic6—21no—
159C2×Dic6—21no—
160C2×Dic6—21no—
161C2×Dic6—21no—
162C2×Dic6—21no—
163C2×Dic6#4112yes—
164C2×C3×C8—21no—
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—21no—
169C2×C3×C8#4112yes—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4—42no—
176C3×C4⋊C4—81no—
177C3×C42C2×D412yes—
178C3×C42#4112yes—
179(C2×D3)⋊C4—24no—
180(C2×D3)⋊C4—44no—
181(C2×D3)⋊C4—82no—
182(C2×D3)⋊C4—82no—
183(C2×D3)⋊C4—82no—
184C4⋊Dic3—22no—
185C4⋊Dic3—21no—
186C4⋊Dic3—21no—
187C4⋊Dic3—21no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3—21no—
191C4⋊Dic3—44no—
192Dic3⋊C4—42no—
193C4×Dic3—42no—
194C4×Dic3—42no—
195C2×C3⋊C8—22no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8C2×D414yes—
198D3×C8—48no—
199C22×Q8—21no—
200C22×Q8—31no—
201C22×D4—31no—
202C23×C4—32no—
203C22×C8—32no—
204C22×C8—64no—
205C22×C8—64no—
206C22×C8—122no—
207C22×C8C22×D311yescharacteristic
208C22×C8C22×D311yescharacteristic
209C4⋊Q8—22no—
210C4⋊Q8—22no—
211C22⋊Q8—64no—
212C22⋊Q8—122no—
213C4×Q8—62no—
214C4×Q8—62no—
215C4×D4—62no—
216C4×D4—62no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—31no—
221C2×C4⋊C4—31no—
222C2×C4⋊C4—31no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4—62no—
225C2×C4⋊C4—61no—
226C2×C4⋊C4C22×D312yes—
227C2×C4⋊C4C22×D312yes—
228C2×C22⋊C4—32no—
229C2×C22⋊C4—62no—
230C2×C22⋊C4—62no—
231C2×C22⋊C4—62no—
232C2×C22⋊C4—62no—
233C2×C42—61no—
234C2×C42—61no—
235C2×C42C22×D311yescharacteristic
236#107—64no—
237#107—122no—
238C4⋊C8—34no—
239#103—64no—
240#103—62no—
241#103—62no—
242C22⋊C8—64no—
243C4×C8—22no—
244#95—62no—
245#95—62no—
246#95—62no—
247#95—61no—
248C23×C3C22×D411yescharacteristic
249C22×D3—22no—
250C22×D3—22no—
251C22×D3—22no—
252C22×D3—22no—
253C22×D3—22no—
254C22×D3—22no—
255C22×D3C2×Q1614yes—
256C3×Q8—84no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4#12112yes—
266C2×C3×C4#12112yes—
267C2×C3×C4C2×Q1612yes—
268C2×C3×C4#13512yes—
269C2×C3×C4C22×D411yescharacteristic
270C2×C3×C4C2×#4111yescharacteristic
271C2×C3×C4C2×#4111yescharacteristic
272C2×C3×C4C2×#4111yescharacteristic
273C2×C3×C4H(5,𝔽2)11yescharacteristic
274C2×C3×C42−1+411yescharacteristic
275C2×Dic3—22no—
276C2×Dic3—22no—
277C2×Dic3—22no—
278C2×Dic3—22no—
279C2×Dic3—41no—
280C2×Dic3—41no—
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—
290D12—28no—
291C4×D3—42no—
292C4×D3—42no—
293C4×D3—42no—
294C4×D3—42no—
295Dic6—22no—
296Dic6—22no—
297Dic6—41no—
298Dic6—41no—
299C3×C8—22no—
300C3×C8—41no—
301C3⋊C8—24no—
302C24—32no—
303C2×Q8—32no—
304C2×Q8—42no—
305C2×Q8—61no—
306C2×Q8—61no—
307C2×Q8—61no—
308C2×Q8—61no—
309C2×Q8—61no—
310C2×Q8—81no—
311C2×Q8—81no—
312C2×D4—34no—
313C2×D4—62no—
314C2×D4—62no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—32no—
318C22×C4—61no—
319C22×C4—62no—
320C22×C4—62no—
321C22×C4—62no—
322C22×C4—62no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—61no—
326C22×C4—61no—
327C22×C4—61no—
328C22×C4D3×D411yescharacteristic
329C22×C4D3×D411yescharacteristic
330C22×C4C23×D311yescharacteristic
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—34no—
337C2×C8—62no—
338C2×C8—62no—
339C2×C8—128no—
340C2×C8#23912yes—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—62no—
348C4⋊C4—61no—
349C4⋊C4—61no—
350C4⋊C4—61no—
351C4⋊C4—61no—
352C4⋊C4—61no—
353C4⋊C4—61no—
354C4⋊C4—81no—
355C4⋊C4—124no—
356C4⋊C4—122no—
357C22⋊C4—64no—
358C22⋊C4—124no—
359C22⋊C4—242no—
360C22⋊C4—242no—
361C22⋊C4—242no—
362C42—122no—
363C42—122no—
364C42#23512yes—
365C42D3×D412yes—
366C22×C3C2×#12111yescharacteristic
367C22×C3#54611yescharacteristic
368C22×C3#54711yescharacteristic
369C22×C3C22×Q1611yescharacteristic
370C22×C3C2×#13511yescharacteristic
371C22×C3#57611yescharacteristic
372C22×C3#57711yescharacteristic
373C2×D3—24no—
374C2×D3—24no—
375C2×D3—24no—
376C2×D3—42no—
377C2×D3—42no—
378C2×D3—42no—
379C2×D3—42no—
380C3×C4—22no—
381C3×C4—82no—
382C3×C4—82no—
383C3×C4#46714yes—
384C3×C4#52911yescharacteristic
385C3×C4#54211yescharacteristic
386C3×C4#54611yescharacteristic
387C3×C4#54811yescharacteristic
388Dic3—42no—
389Dic3—42no—
390Dic3—81no—
391Dic3—81no—
392Dic3—81no—
393C23—34no—
394C23—62no—
395C23—62no—
396C23—62no—
397C23—62no—
398C23—62no—
399C23—62no—
400C23C2×D3×D411yescharacteristic
401Q8—62no—
402Q8—62no—
403Q8—84no—
404Q8—121no—
405Q8—121no—
406D4—68no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—62no—
416C2×C4—62no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4#88412yes—
435C2×C4#88412yes—
436C2×C4#91612yes—
437C2×C4D3×Q1612yes—
438C2×C4C2×#23511yescharacteristic
439C2×C4C2×D3×D411yescharacteristic
440C2×C4C2×#23911yescharacteristic
441C2×C4#100711yescharacteristic
442C2×C4D3×#4111yescharacteristic
443C2×C4#100911yescharacteristic
444C8—22no—
445C8—41no—
446C8—64no—
447C2×C3#242711yescharacteristic
448C2×C3C2×#46711yescharacteristic
449C2×C3#306211yescharacteristic
450C2×C3#329911yescharacteristic
451C2×C3#330511yescharacteristic
452C2×C3#333811yescharacteristic
453C2×C3#334011yescharacteristic
454D3—44no—
455C22—64no—
456C22—64no—
457C22—64no—
458C22—122no—
459C22—122no—
460C22—122no—
461C22—122no—
462C22C2×#88411yescharacteristic
463C22#567611yescharacteristic
464C22#572611yescharacteristic
465C22#575711yescharacteristic
466C22C2×#91611yescharacteristic
467C22#576211yescharacteristic
468C22C2×D3×Q1611yescharacteristic
469C4—22no—
470C4—82no—
471C4—82no—
472C4—122no—
473C4—122no—
474C4—241no—
475C4—241no—
476C4—241no—
477C4#481314yes—
478C4#558011yescharacteristic
479C4#558211yescharacteristic
480C4#563011yescharacteristic
481C4#563311yescharacteristic
482C3#2534011yescharacteristic, complemented, Sylow
483C2—124no—
484C2#7662411yescharacteristic
485C2#7875211yescharacteristic
486C2C2×#481311yescharacteristic
487C2#8157911yescharacteristic
488C2#8161311yescharacteristic
489C2#8190011yescharacteristic
490C2#8194711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1990C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenter#1990D3
hypocenterC3#25340
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×#5091C2semidirect
5. C3×#1990C2semidirect
6. C4×#816C2semidirect
7. #69567C2semidirect
8. #68890C2semidirect
9. #68882C2semidirect
10. #68869C2semidirect
11. #68825C2semidirect
50. #1990D3semidirect
482. C3#25340semidirect

Maximal subgroups

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

subgroupindex
#832602
C2×#50912
C2×C8×D122
#743692
C3×#19902
C4×#8162
#695672
#688902
#688822
#688692
#688252
#253403