Finite Group Databasebeta

← #10801325

Subgroups — #10801325

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82401C211yesmaximal, characteristic
2C2×#5012C211yesmaximal, characteristic, complemented
3C2×#4729C212yesmaximal, complemented
4C2×#4708C211yesmaximal, characteristic, complemented
5C3×#1955C211yesmaximal, characteristic, complemented
6#70476C211yesmaximal, characteristic, complemented
7#70465C211yesmaximal, characteristic, complemented
8#69957C212yesmaximal, complemented
9#69949C212yesmaximal, complemented
10#69913C212yesmaximal, complemented
11#69836C211yesmaximal, characteristic, complemented
12#21494—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×C4×#67C2212yes—
15C22×#208C2212yes—
16C22×ΓL(1,𝔽25)C2211yescharacteristic
17C22×D3×C8C2211yescharacteristic
18C2×#872C2211yescharacteristic
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×C22⋊C8C2212yes—
21C2×C3×C22⋊C8C2212yes—
22C2×C3×#97C2211yescharacteristic
23#5223C411yescharacteristic
24#5222C411yescharacteristic
25C4×#217C2212yes—
26C2×#830C2212yes—
27C2×(C2×D3)⋊C8C2212yes—
28C2×#815C2211yescharacteristic
29C2×Dic3⋊C8C2212yes—
30C2×C8×Dic3C2212yes—
31C2×C8×Dic3C2211yescharacteristic
32#5012—24no—
33#4992C411yescharacteristic
34#4983C411yescharacteristic
35#4961C412yes—
36#4948C412yes—
37C4×C4⋊Dic3C2211yescharacteristic
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39C2×C4×C3⋊C8C2211yescharacteristic
40#4729—28no—
41#4708—24no—
42#4550C2212yes—
43#4550C2212yes—
44#4550C2212yes—
45C2×#436—32no—
46C2×#436—31no—
47C2×#405—31no—
48#2392—31no—
49#1955D311yescharacteristic, complemented
50#1940—31no—
51#1923—32no—
52#1869—32no—
53#1798—31no—
54C8×C4⋊C4—31no—
55#1753—32no—
56C22×C3×D4C2×C411yescharacteristic
57C23×C3×C4C2312yes—
58C22×#67C2×C412yes—
59C22×C4×D3C2311yescharacteristic
60C22×C3×C8C2312yes—
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×C4×D4—22no—
64C3×C4×D4—22no—
65C2×C3×C4⋊C4C2×C411yescharacteristic
66C2×C3×C22⋊C4C2×C412yes—
67C2×C3×C42C2311yescharacteristic
68C2×#217C2×C412yes—
69C2×#217C2×C412yes—
70C2×#217C2×C412yes—
71C4×#67—24no—
72C4×#67—24no—
73C2×(C2×D3)⋊C4C2×C411yescharacteristic
74C2×(C2×D3)⋊C4C2×C411yescharacteristic
75C2×(C2×D3)⋊C4C2×C412yes—
76C2×C4⋊Dic3C2×C412yes—
77C2×Dic3⋊C4C2×C411yescharacteristic
78C2×Dic3⋊C4C2×C412yes—
79C2×Dic3⋊C4C2×C411yescharacteristic
80C2×C4×Dic3C2312yes—
81C2×C4×Dic3C2311yescharacteristic
82C2×#208—24no—
83C2×#208—24no—
84C2×#208—24no—
85C22×C3⋊C8C2312yes—
86C22×C3⋊C8C2311yescharacteristic
87C22×C3⋊C8C2311yescharacteristic
88C2×ΓL(1,𝔽25)—22no—
89C2×ΓL(1,𝔽25)—22no—
90C2×ΓL(1,𝔽25)D414yescomplemented
91C2×D3×C8—22no—
92C2×D3×C8—22no—
93C2×D3×C8—22no—
94#872—22no—
95#872—22no—
96C3×C4⋊C8—22no—
97C3×C22⋊C8—24no—
98C3×C22⋊C8—24no—
99C3×#97—22no—
100C3⋊#95C2×C411yescharacteristic
101C3⋊#95C2×C411yescharacteristic
102C3⋊#95C2×C412yes—
103C3⋊#95C2×C412yes—
104#830—24no—
105(C2×D3)⋊C8—24no—
106#815—22no—
107Dic3⋊C8—24no—
108C8×Dic3—22no—
109C8×Dic3—24no—
110C4⋊(C3⋊C8)—22no—
111C4×C3⋊C8D414yescomplemented
112C22×M16—32no—
113C22×M16—31no—
114C23×C8—31no—
115C2×C4×D4—32no—
116C2×C4×D4C2×D311yescharacteristic
117C2×#117—31no—
118#436—64no—
119#436—68no—
120C2×C4⋊C8—32no—
121C2×C4⋊C8—31no—
122C2×C4⋊C8C2×D311yescharacteristic
123C2×C22⋊C8—32no—
124C2×C22⋊C8—32no—
125C2×C22⋊C8C2×D312yes—
126C2×C22⋊C8C2×D312yes—
127#405—64no—
128C2×#97—31no—
129C2×#97C2×D311yescharacteristic
130C2×C4×C8—31no—
131C2×C4×C8—31no—
132C2×C4×C8—32no—
133#398—31no—
134#397—31no—
135#396—32no—
136#394—31no—
137#394—32no—
138#392—31no—
139C4×C4⋊C4—31no—
140C4×C22⋊C4—32no—
141#336—32no—
142#336—32no—
143#336—32no—
144C24×C3C22×C412yes—
145C23×D3C22×C411yescharacteristic
146C2×C3×D4—22no—
147C2×C3×D4—42no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4C22×C412yes—
155C22×C3×C4C22×C411yescharacteristic
156C22×C3×C4C22×C411yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C2×#67—24no—
159C2×#67—24no—
160C2×#67—42no—
161C22×Dic3C22×C411yescharacteristic
162C22×Dic3C22×C411yescharacteristic
163C22×Dic3C22×C412yes—
164C22×Dic3C22×C411yescharacteristic
165C22×Dic3C22×C412yes—
166C2×C4×D3—21no—
167C2×C4×D3—21no—
168C2×C4×D3—21no—
169C2×C4×D3—21no—
170C2×C4×D3C2×D412yes—
171C2×C4×D3#4112yes—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8C2×D412yes—
180C2×C3×C8#4112yes—
181C3×C4⋊C4—22no—
182C3×C22⋊C4—24no—
183C3×C42C2×D412yes—
184C3×C42#4112yes—
185#217—24no—
186#217—24no—
187#217—42no—
188(C2×D3)⋊C4—24no—
189(C2×D3)⋊C4—22no—
190(C2×D3)⋊C4—22no—
191C4⋊Dic3—24no—
192Dic3⋊C4—22no—
193Dic3⋊C4—22no—
194Dic3⋊C4—24no—
195C4×Dic3—22no—
196C4×Dic3—21no—
197C4×Dic3—21no—
198C4×Dic3—22no—
199#208—48no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8C2×D412yes—
206C2×C3⋊C8C2×D412yes—
207C2×C3⋊C8C2×D412yes—
208C2×C3⋊C8#4112yes—
209ΓL(1,𝔽25)—28no—
210D3×C8—28no—
211C22×D4—32no—
212C22×D4C4×D311yescharacteristic
213C23×C4—31no—
214C23×C4C22×D312yes—
215C2×M16—34no—
216C2×M16—62no—
217C2×M16—62no—
218C2×M16—64no—
219C2×M16—64no—
220C2×M16—64no—
221C22×C8—31no—
222C22×C8—31no—
223C22×C8—32no—
224C22×C8—62no—
225C22×C8—62no—
226C22×C8—62no—
227C22×C8C22×D312yes—
228C22×C8C22×D311yescharacteristic
229C22×C8C22×D311yescharacteristic
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—64no—
233C4×D4—64no—
234#117—62no—
235#117—62no—
236C2×C4⋊C4—31no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4C4×D311yescharacteristic
241C2×C22⋊C4—32no—
242C2×C22⋊C4—32no—
243C2×C22⋊C4—31no—
244C2×C22⋊C4—31no—
245C2×C22⋊C4—32no—
246C2×C22⋊C4—32no—
247C2×C22⋊C4C4×D312yes—
248C2×C42—31no—
249C2×C42—32no—
250C2×C42C22×D311yescharacteristic
251C4⋊C8—22no—
252C4⋊C8—64no—
253C4⋊C8—62no—
254C22⋊C8—24no—
255C22⋊C8—24no—
256C22⋊C8—64no—
257C22⋊C8—64no—
258#97—22no—
259#97—62no—
260C4×C8—34no—
261C4×C8—62no—
262C4×C8—64no—
263#95—32no—
264#95—32no—
265#95—31no—
266#95—31no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3C23×C411yescharacteristic
275C22×D3—21no—
276C22×D3—21no—
277C22×D3—22no—
278C22×D3—22no—
279C22×D3C4×D412yes—
280C3×D4—44no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4C4×D412yes—
291C2×C3×C4C4×D412yes—
292C2×C3×C4C22×D411yescharacteristic
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C2×#4111yescharacteristic
295C2×C3×C4C2×#4111yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4C2×#4111yescharacteristic
298#67—48no—
299C2×Dic3—22no—
300C2×Dic3—21no—
301C2×Dic3—21no—
302C2×Dic3—22no—
303C2×Dic3—21no—
304C2×Dic3—21no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3C4×D412yes—
311C2×Dic3#13114yes—
312C4×D3—22no—
313C4×D3—22no—
314C4×D3—22no—
315C4×D3—22no—
316C3×C8—22no—
317C3×C8—22no—
318C3×C8—42no—
319C3⋊C8—22no—
320C3⋊C8—42no—
321C3⋊C8#12114yes—
322C24—31no—
323C24C2×C4×D312yes—
324C2×D4—22no—
325C2×D4—42no—
326C2×D4—64no—
327C2×D4—64no—
328C2×D4—122no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—32no—
336C22×C4—31no—
337C22×C4—32no—
338C22×C4—32no—
339C22×C4—31no—
340C22×C4—31no—
341C22×C4—32no—
342C22×C4—61no—
343C22×C4—61no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4C2×C4×D312yes—
347C22×C4C2×C4×D311yescharacteristic
348C22×C4C2×C4×D311yescharacteristic
349C22×C4C23×D311yescharacteristic
350M16—68no—
351M16—128no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—32no—
360C2×C8—32no—
361C2×C8—32no—
362C2×C8—32no—
363C2×C8—68no—
364C2×C8—61no—
365C2×C8—61no—
366C2×C8—62no—
367C2×C8—62no—
368C2×C8—62no—
369C2×C8D3×D412yes—
370C2×C8#23712yes—
371C4⋊C4—22no—
372C4⋊C4—64no—
373C4⋊C4—64no—
374C4⋊C4—62no—
375C4⋊C4—62no—
376C22⋊C4—24no—
377C22⋊C4—64no—
378C22⋊C4—64no—
379C22⋊C4—64no—
380C22⋊C4—62no—
381C22⋊C4—62no—
382C22⋊C4—122no—
383C42—61no—
384C42—62no—
385C42—61no—
386C42—62no—
387C42D3×D412yes—
388C42#23712yes—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3C2×C4×D411yescharacteristic
397C22×C3C2×C4×D411yescharacteristic
398C22×C3C4×#4111yescharacteristic
399C22×C3C2×#13112yes—
400C22×C3#56812yes—
401C2×D3—24no—
402C2×D3—22no—
403C2×D3—41no—
404C2×D3—41no—
405C2×D3—42no—
406C3×C4—22no—
407C3×C4—22no—
408C3×C4—42no—
409C3×C4C2×#12111yescharacteristic
410C3×C4#52511yescharacteristic
411C3×C4#52911yescharacteristic
412C3×C4#53211yescharacteristic
413Dic3—24no—
414Dic3—22no—
415Dic3—41no—
416Dic3—41no—
417Dic3—42no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—32no—
426C23—61no—
427C23—62no—
428C23—62no—
429C23—61no—
430C23C22×C4×D311yescharacteristic
431D4—44no—
432D4—128no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—34no—
439C2×C4—32no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—61no—
445C2×C4—62no—
446C2×C4—61no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4#67⋊C412yes—
460C2×C4#67⋊C412yes—
461C2×C4C2×D3×D411yescharacteristic
462C2×C4C2×D3×D411yescharacteristic
463C2×C4C2×#23711yescharacteristic
464C2×C4C2×#23711yescharacteristic
465C2×C4D3×#4111yescharacteristic
466C2×C4D3×#4111yescharacteristic
467C8—22no—
468C8—22no—
469C8—34no—
470C8—42no—
471C8—62no—
472C8—122no—
473C2×C3—82no—
474C2×C3C4×#12111yescharacteristic
475C2×C3#293011yescharacteristic
476C2×C3#293111yescharacteristic
477C2×C3#293312yes—
478C2×C3#296912yes—
479D3—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—62no—
488C22—64no—
489C22—121no—
490C22—121no—
491C22—122no—
492C22C2×#67⋊C411yescharacteristic
493C22C4×#23711yescharacteristic
494C22C4×D3×D411yescharacteristic
495C22#574512yes—
496C22D3×#13112yes—
497C4—22no—
498C4—22no—
499C4—42no—
500C4—62no—
501C4—64no—
502C4—121no—
503C4—121no—
504C4—122no—
505C4D3×#12111yescharacteristic
506C4#560611yescharacteristic
507C4#568011yescharacteristic
508C4#571611yescharacteristic
509C3#2149411yescharacteristic, complemented, Sylow
510C2—82no—
511C2—122no—
512C2#7629111yescharacteristic
513C2#7862711yescharacteristic
514C2#7865711yescharacteristic
515C2#7945912yes—
516C2#7948212yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1955C2
prime radical#5223C4
cosocleC22×C3×C4C24
hypercenter#1955D3
hypocenterC3#21494
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1955

Complements

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

NH ≅ G/Ntype
2. C2×#5012C2semidirect
3. C2×#4729C2semidirect
4. C2×#4708C2semidirect
5. C3×#1955C2semidirect
6. #70476C2semidirect
7. #70465C2semidirect
8. #69957C2semidirect
9. #69949C2semidirect
10. #69913C2semidirect
11. #69836C2semidirect
49. #1955D3semidirect
90. C2×ΓL(1,𝔽25)D4semidirect
111. C4×C3⋊C8D4semidirect
509. C3#21494semidirect

Maximal subgroups

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

subgroupindex
#824012
C2×#50122
C2×#47292
C2×#47082
C3×#19552
#704762
#704652
#699572
#699492
#699132
#698362
#214943