Finite Group Databasebeta

← #10896359

Subgroups — #10896359

Order 768.

800 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76480C211yesmaximal, characteristic
2#74320C211yesmaximal, characteristic, complemented
3C3×#2096C211yesmaximal, characteristic, complemented
4#70584C212yesmaximal
5#70564C212yesmaximal, complemented
6#70235C212yesmaximal, complemented
7#70230C212yesmaximal
8#69668C211yesmaximal, characteristic, complemented
9#69586C211yesmaximal, characteristic
10#68678C211yesmaximal, characteristic
11#68605C211yesmaximal, characteristic, complemented
12#29262—31nomaximal, Sylow
13C2×#947C2211yescharacteristic
14C2×Q8×Dic3C2211yescharacteristic
15C2×#893C2211yescharacteristic
16C2×#887—21no—
17C2×#869C2211yescharacteristic
18C2×C3×#106C2212yes—
19C2×C3×#103C2212yes—
20C3×#391C2211yescharacteristic
21C3×#384C2211yescharacteristic
22C2×#835C2212yes—
23C2×#821C2212yes—
24C2×Dic3⋊C8C2212yes—
25#4977C2211yescharacteristic
26Dic3×C4⋊C4—21no—
27Dic3×C4⋊C4C2212yes—
28#4973—21no—
29#4972—21no—
30#4972C2212yes—
31C2×#809C2212yes—
32C2×#807C2212yes—
33#4939C2211yescharacteristic
34C42×Dic3C2211yescharacteristic
35#4670—21no—
36#4670—21no—
37#4666—21no—
38#4666—21no—
39C3×#336C2211yescharacteristic
40#4551C2212yes—
41#4550C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44C4×C4⋊Q8—31no—
45C4×#124—31no—
46#2096D311yescharacteristic, complemented
47#2093—31no—
48#2024—31no—
49#2023—31no—
50#1983—32no—
51#1949—32no—
52#1864—32no—
53#1863—32no—
54#1831—31no—
55C22×C3×Q8C2311yescharacteristic
56C22×D12C2311yescharacteristic
57C22×C4×D3—21no—
58C22×Dic6—21no—
59C22×C3×C8C2312yes—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C42C2311yescharacteristic
65#947—22no—
66#947Q814yescomplemented
67Q8×Dic3—42no—
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—21no—
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—21no—
72C2×(C2×D3)⋊C4—21no—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3C2311yescharacteristic
76C2×C4⋊Dic3C2312yes—
77C2×Dic3⋊C4—22no—
78C2×C4×Dic3—21no—
79C2×C4×Dic3—22no—
80C2×C4×Dic3D411yescharacteristic
81C2×C4×Dic3D411yescharacteristic
82C2×C4×Dic3D411yescharacteristic
83C2×C4×Dic3D411yescharacteristic
84C2×C4×Dic3C2311yescharacteristic
85C2×C4×Dic3C2311yescharacteristic
86C22×C3⋊C8C2312yes—
87#893—42no—
88#887—44no—
89#869—24no—
90C3×#106—42no—
91C3×#103—42no—
92C3×#95—21no—
93C3×#95—21no—
94#835—24no—
95C3⋊#95—21no—
96C3⋊#95—21no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99#821—24no—
100Dic3⋊C8—24no—
101#809—42no—
102#807—42no—
103C2×C4⋊Q8—31no—
104C2×#124—31no—
105C2×C4×Q8—31no—
106C2×C4×Q8—61no—
107C2×C4×D4—31no—
108C2×#107—32no—
109C2×#106C2×D312yes—
110C2×C4⋊C8—32no—
111C2×#103—32no—
112C2×#103C2×D312yes—
113C2×#102—32no—
114C2×#102—32no—
115#391—61no—
116#391C2×D311yescharacteristic
117#390—31no—
118#389—32no—
119#389—61no—
120#388—61no—
121#388—61no—
122#384C2×D311yescharacteristic
123#382—31no—
124C4×C4⋊C4—32no—
125C4×C4⋊C4—61no—
126C4×C22⋊C4—61no—
127C4×C22⋊C4—61no—
128C43—31no—
129#340—32no—
130#340—32no—
131#336—32no—
132#336—31no—
133#336C2×D311yescharacteristic
134C23×D3—21no—
135C2×C3×Q8—21no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8C2×Q812yes—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2411yescharacteristic
145C22×Dic3—22no—
146C22×Dic3—21no—
147C22×Dic3C2×D411yescharacteristic
148C22×Dic3C2×D411yescharacteristic
149C22×Dic3C2×D411yescharacteristic
150C22×Dic3C2×D411yescharacteristic
151C2×D12—22no—
152C2×D12—22no—
153C2×D12—21no—
154C2×D12C2×Q812yes—
155C2×C4×D3—42no—
156C2×C4×D3—42no—
157C2×C4×D3—42no—
158C2×Dic6—24no—
159C2×Dic6—42no—
160C2×Dic6—42no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—42no—
168C3×C4⋊C4—42no—
169C3×C42—22no—
170(C2×D3)⋊C4—24no—
171(C2×D3)⋊C4—24no—
172(C2×D3)⋊C4—42no—
173(C2×D3)⋊C4—42no—
174(C2×D3)⋊C4—42no—
175C4⋊Dic3—24no—
176C4⋊Dic3—22no—
177C4⋊Dic3—22no—
178C4⋊Dic3—42no—
179C4⋊Dic3#4114yes—
180Dic3⋊C4—44no—
181C4×Dic3—22no—
182C4×Dic3—22no—
183C4×Dic3—22no—
184C4×Dic3—22no—
185C4×Dic3—41no—
186C4×Dic3—44no—
187C4×Dic3—42no—
188C4×Dic3C2×Q812yes—
189C4×Dic3#4112yes—
190C2×C3⋊C8—22no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—22no—
193C22×Q8—61no—
194C22×Q8C22×D311yescharacteristic
195C22×D4—31no—
196C23×C4—61no—
197C22×C8—32no—
198C22×C8C22×D312yes—
199C4⋊Q8—64no—
200#124—34no—
201#124—62no—
202C4×Q8—124no—
203C4×Q8—122no—
204C4×D4—122no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—32no—
208C2×C4⋊C4—31no—
209C2×C4⋊C4—61no—
210C2×C4⋊C4—61no—
211C2×C4⋊C4—62no—
212C2×C4⋊C4C22×D311yescharacteristic
213C2×C4⋊C4C22×D312yes—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—61no—
217C2×C22⋊C4—61no—
218C2×C22⋊C4—61no—
219C2×C42—31no—
220C2×C42—31no—
221C2×C42—31no—
222C2×C42—31no—
223C2×C42—31no—
224C2×C42—31no—
225C2×C42—61no—
226C2×C42—62no—
227C2×C42C22×D311yescharacteristic
228#107—122no—
229#106—42no—
230C4⋊C8—64no—
231#103—42no—
232#103—64no—
233#102—64no—
234#102—122no—
235#95—21no—
236#95—21no—
237#95—61no—
238#95—61no—
239#95—62no—
240#95—62no—
241C23×C3C22×D411yescharacteristic
242C22×D3—22no—
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—41no—
248C22×D3—41no—
249C3×Q8—24no—
250C3×Q8—42no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—22no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4#13512yes—
265C2×C3×C4AGL(1,ℤ/8)12yes—
266C2×C3×C4C22×Q811yescharacteristic
267C2×C3×C4C2×#4112yes—
268C2×C3×C4C2×#4111yescharacteristic
269C2×C3×C4H(5,𝔽2)12yes—
270C2×Dic3—21no—
271C2×Dic3—21no—
272C2×Dic3—21no—
273C2×Dic3—21no—
274C2×Dic3—21no—
275C2×Dic3—21no—
276C2×Dic3—24no—
277C2×Dic3—21no—
278C2×Dic3—21no—
279C2×Dic3—21no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3C22⋊Q812yes—
286C2×Dic3C22⋊Q812yes—
287C2×Dic3#13712yes—
288D12—24no—
289D12—42no—
290C4×D3—84no—
291Dic6—48no—
292C3×C8—42no—
293C3⋊C8—42no—
294C24—61no—
295C2×Q8—21no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—64no—
299C2×Q8—122no—
300C2×Q8—122no—
301C2×Q8D3×Q812yes—
302C2×D4—32no—
303C2×D4—62no—
304C2×D4—62no—
305C2×D4—61no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—31no—
310C22×C4—31no—
311C22×C4—31no—
312C22×C4—31no—
313C22×C4—62no—
314C22×C4—61no—
315C22×C4—122no—
316C22×C4—122no—
317C22×C4—122no—
318C22×C4D3×D411yescharacteristic
319C22×C4D3×D411yescharacteristic
320C22×C4C23×D311yescharacteristic
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—62no—
325C2×C8—62no—
326C2×C8—62no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—34no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—64no—
334C4⋊C4—62no—
335C4⋊C4—62no—
336C4⋊C4—124no—
337C4⋊C4—122no—
338C22⋊C4—64no—
339C22⋊C4—64no—
340C22⋊C4—122no—
341C22⋊C4—122no—
342C22⋊C4—122no—
343C42—22no—
344C42—32no—
345C42—32no—
346C42—62no—
347C42—62no—
348C42—62no—
349C42—62no—
350C42—121no—
351C42—122no—
352C42—124no—
353C22×C3C2×C22⋊Q811yescharacteristic
354C22×C3#52511yescharacteristic
355C22×C3#53511yescharacteristic
356C22×C3C2×#13511yescharacteristic
357C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
358C22×C3C2×#13711yescharacteristic
359C22×C3#57511yescharacteristic
360C2×D3—42no—
361C2×D3—42no—
362C2×D3—42no—
363C2×D3—81no—
364C2×D3—81no—
365C2×D3—81no—
366C2×D3—81no—
367C3×C4—22no—
368C3×C4—22no—
369C3×C4—42no—
370C3×C4—82no—
371C3×C4—81no—
372C3×C4#55412yes—
373C3×C4#55912yes—
374Dic3—22no—
375Dic3—22no—
376Dic3—22no—
377Dic3—44no—
378Dic3—41no—
379Dic3—81no—
380C23—62no—
381C23—121no—
382C23—121no—
383C23—121no—
384C23—121no—
385C23—121no—
386C23—121no—
387C23C2×D3×D411yescharacteristic
388Q8—24no—
389Q8—42no—
390Q8—128no—
391D4—64no—
392D4—122no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—32no—
399C2×C4—32no—
400C2×C4—32no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—61no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—61no—
415C2×C4—64no—
416C2×C4—61no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—244no—
425C2×C4#91412yes—
426C2×C4#91612yes—
427C2×C4#100412yes—
428C2×C4C2×D3×Q811yescharacteristic
429C2×C4D3×#4112yes—
430C2×C4D3×#4111yescharacteristic
431C8—42no—
432C8—122no—
433C2×C3#258011yescharacteristic
434C2×C3#307611yescharacteristic
435C2×C3#307911yescharacteristic
436C2×C3#310811yescharacteristic
437C2×C3#311411yescharacteristic
438C2×C3#320711yescharacteristic
439C2×C3#321511yescharacteristic
440D3—82no—
441C22—122no—
442C22—122no—
443C22—122no—
444C22—241no—
445C22—241no—
446C22—241no—
447C22—241no—
448C22D3×C22⋊Q811yescharacteristic
449C22#565611yescharacteristic
450C22#571511yescharacteristic
451C22C2×#91411yescharacteristic
452C22C2×#91611yescharacteristic
453C22D3×#13711yescharacteristic
454C22#577711yescharacteristic
455C4—22no—
456C4—22no—
457C4—42no—
458C4—62no—
459C4—62no—
460C4—62no—
461C4—82no—
462C4—81no—
463C4—124no—
464C4—121no—
465C4—241no—
466C4#563612yes—
467C4#566412yes—
468C3#2926211yescharacteristic, complemented, Sylow
469C2—242no—
470C2#7804111yescharacteristic
471C2#8101011yescharacteristic
472C2#8101211yescharacteristic
473C2#8141311yescharacteristic
474C2#8141511yescharacteristic
475C2#8176511yescharacteristic
476C2#8187411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2096C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2096D3
hypocenterC3#29262
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2096

Complements

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

NH ≅ G/Ntype
2. #74320C2semidirect
3. C3×#2096C2semidirect
5. #70564C2semidirect
6. #70235C2semidirect
8. #69668C2semidirect
11. #68605C2semidirect
46. #2096D3semidirect
66. #947Q8semidirect
468. C3#29262semidirect

Maximal subgroups

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

subgroupindex
#764802
#743202
C3×#20962
#705842
#705642
#702352
#702302
#696682
#695862
#686782
#686052
#292623