Finite Group Databasebeta

← #10801293

Subgroups — #10801293

Order 768.

1058 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#82273C211yesmaximal, characteristic
2C2×#4997C211yesmaximal, characteristic, complemented
3C2×#4834C211yesmaximal, characteristic, complemented
4C2×#4728C212yesmaximal, complemented
5C3×#1798C211yesmaximal, characteristic, complemented
6#70471C211yesmaximal, characteristic, complemented
7#70469C211yesmaximal, characteristic, complemented
8#70466C211yesmaximal, characteristic, complemented
9#69952C212yesmaximal, complemented
10#68657C212yesmaximal, complemented
11#68648C212yesmaximal, complemented
12#21492—31nomaximal, Sylow
13C2×C3×#117C2211yescharacteristic
14C2×C4×#67C2212yes—
15C22×#208C2211yescharacteristic
16C23×C3⋊C8C2211yescharacteristic
17C22×D3×C8C2212yes—
18C2×C4×D12C2211yescharacteristic
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C22⋊C8C2212yes—
22C2×(C2×D3)⋊C8C2212yes—
23C2×(C2×D3)⋊C8C2212yes—
24C2×(C2×D3)⋊C8C2212yes—
25C2×#815C2212yes—
26C2×Dic3⋊C8C2212yes—
27#4997—24no—
28#4994C412yes—
29#4991C411yescharacteristic
30#4981C411yescharacteristic
31#4961C412yes—
32#4960C411yescharacteristic
33#4953C411yescharacteristic
34C4×(C2×D3)⋊C4C2212yes—
35C4×C4⋊Dic3C2211yescharacteristic
36C2×#803C2211yescharacteristic
37C2×#803C2211yescharacteristic
38C2×C4×C3⋊C8C2211yescharacteristic
39C2×C4×C3⋊C8C2211yescharacteristic
40#4834—24no—
41#4728—28no—
42C3×#336C2212yes—
43#4550C2212yes—
44#4550C2212yes—
45C2×#435—32no—
46C2×C8×D4—31no—
47C2×#405—31no—
48#2392—31no—
49#1955—31no—
50#1940—31no—
51#1923—32no—
52#1869—32no—
53#1798D311yescharacteristic, complemented
54#1770—31no—
55#1752—32no—
56C23×C3×C4C2311yescharacteristic
57C22×#67C2×C412yes—
58C22×D12C2×C411yescharacteristic
59C22×C4×D3C2312yes—
60C22×C3×C8C2312yes—
61C22×C3×C8C2312yes—
62C3×#117—22no—
63C3×#117—22no—
64C2×C3×C4⋊C4C2×C411yescharacteristic
65C2×C3×C4⋊C4C2×C411yescharacteristic
66C2×C3×C22⋊C4C2×C411yescharacteristic
67C2×C3×C22⋊C4C2×C411yescharacteristic
68C2×C3×C42C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×#217C2×C412yes—
71C4×#67—24no—
72C4×#67—24no—
73C2×(C2×D3)⋊C4C2×C412yes—
74C2×(C2×D3)⋊C4C2×C412yes—
75C2×(C2×D3)⋊C4C2×C412yes—
76C2×(C2×D3)⋊C4C2×C412yes—
77C2×C4⋊Dic3C2×C411yescharacteristic
78C2×C4⋊Dic3C2×C412yes—
79C2×Dic3⋊C4C2×C412yes—
80C2×C4×Dic3C2312yes—
81C2×#208—22no—
82C2×#208—22no—
83C2×#208D414yescomplemented
84C22×C3⋊C8—22no—
85C22×C3⋊C8—22no—
86C22×C3⋊C8—22no—
87C22×C3⋊C8C2311yescharacteristic
88C22×C3⋊C8C2311yescharacteristic
89C22×C3⋊C8C2311yescharacteristic
90C22×C3⋊C8C2311yescharacteristic
91C2×D3×C8—24no—
92C2×D3×C8—24no—
93C2×D3×C8—24no—
94C4×D12—22no—
95C4×D12—22no—
96C3×C4⋊C8—22no—
97C3×C4⋊C8—22no—
98C3×C22⋊C8—24no—
99C3⋊#95C2×C411yescharacteristic
100C3⋊#95C2×C411yescharacteristic
101C3⋊#95C2×C412yes—
102C3⋊#95C2×C412yes—
103(C2×D3)⋊C8—24no—
104(C2×D3)⋊C8—24no—
105(C2×D3)⋊C8—24no—
106#815—24no—
107Dic3⋊C8—24no—
108#803—22no—
109#803—22no—
110C4×C3⋊C8—22no—
111C4×C3⋊C8D414yescomplemented
112C22×M16—31no—
113C23×C8—31no—
114C23×C8—32no—
115C2×C4×D4—32no—
116C2×C4×D4—31no—
117C2×#117C2×D311yescharacteristic
118#435—68no—
119C8×D4—64no—
120C2×C4⋊C8—32no—
121C2×C4⋊C8C2×D311yescharacteristic
122C2×C4⋊C8C2×D311yescharacteristic
123C2×C22⋊C8—32no—
124C2×C22⋊C8—32no—
125C2×C22⋊C8—32no—
126C2×C22⋊C8C2×D312yes—
127#405—64no—
128C2×#97—32no—
129C2×#97—31no—
130C2×#97—31no—
131C2×C4×C8—31no—
132C2×C4×C8—31no—
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#336C2×D312yes—
144C24×C3C22×C411yescharacteristic
145C23×D3C22×C412yes—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4C22×C411yescharacteristic
151C22×C3×C4C22×C411yescharacteristic
152C22×C3×C4C22×C411yescharacteristic
153C22×C3×C4C22×C411yescharacteristic
154C22×C3×C4C22×C411yescharacteristic
155C22×C3×C4C2×D412yes—
156C22×C3×C4#4112yes—
157C22×C3×C4C2411yescharacteristic
158C2×#67—24no—
159C2×#67—24no—
160C2×#67—42no—
161C22×Dic3C22×C412yes—
162C22×Dic3C22×C412yes—
163C22×Dic3C22×C412yes—
164C2×D12—24no—
165C2×D12—22no—
166C2×C4×D3—22no—
167C2×C4×D3—22no—
168C2×C4×D3—22no—
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3—22no—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C22⋊C4—22no—
181C3×C22⋊C4—22no—
182C3×C42—21no—
183C3×C42—21no—
184C3×C42C2×D412yes—
185C3×C42#4112yes—
186#217—24no—
187(C2×D3)⋊C4—24no—
188(C2×D3)⋊C4—24no—
189(C2×D3)⋊C4—24no—
190(C2×D3)⋊C4—42no—
191C4⋊Dic3—22no—
192C4⋊Dic3—42no—
193Dic3⋊C4—24no—
194C4×Dic3—22no—
195C4×Dic3—22no—
196#208—28no—
197C2×C3⋊C8—28no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8C2×D412yes—
205C2×C3⋊C8C2×D412yes—
206C2×C3⋊C8C2×D412yes—
207C2×C3⋊C8C2×D412yes—
208C2×C3⋊C8#4112yes—
209C2×C3⋊C8#4112yes—
210D3×C8—48no—
211C22×D4—32no—
212C22×D4—31no—
213C23×C4—32no—
214C23×C4C22×D311yescharacteristic
215C2×M16—34no—
216C2×M16—62no—
217C2×M16—62no—
218C22×C8—31no—
219C22×C8—31no—
220C22×C8—31no—
221C22×C8—31no—
222C22×C8—64no—
223C22×C8—64no—
224C22×C8—64no—
225C22×C8—62no—
226C22×C8—62no—
227C22×C8—62no—
228C22×C8C22×D312yes—
229C22×C8C22×D312yes—
230C4×D4—64no—
231C4×D4—64no—
232C4×D4—62no—
233C4×D4—62no—
234#117—22no—
235#117—22no—
236C2×C4⋊C4—32no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4C4×D311yescharacteristic
240C2×C4⋊C4C4×D311yescharacteristic
241C2×C22⋊C4—32no—
242C2×C22⋊C4—32no—
243C2×C22⋊C4—32no—
244C2×C22⋊C4—32no—
245C2×C22⋊C4—32no—
246C2×C22⋊C4C4×D311yescharacteristic
247C2×C22⋊C4C4×D311yescharacteristic
248C2×C42—32no—
249C2×C42C22×D311yescharacteristic
250C2×C42C22×D311yescharacteristic
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—64no—
254C22⋊C8—24no—
255C22⋊C8—64no—
256C22⋊C8—64no—
257C22⋊C8—64no—
258#97—62no—
259#97—64no—
260#97—62no—
261C4×C8—34no—
262C4×C8—62no—
263#95—32no—
264#95—32no—
265#95—31no—
266#95—31no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3C4×D412yes—
272C23×C3C23×C411yescharacteristic
273C22×D3—22no—
274C22×D3—22no—
275C22×D3—22no—
276C22×D3—22no—
277C22×D3—22no—
278C22×D3—22no—
279C22×D3#13114yes—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4C4×D412yes—
293C2×C3×C4C4×D412yes—
294C2×C3×C4C4×D412yes—
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4C2×#4111yescharacteristic
300C2×C3×C4C2×#4111yescharacteristic
301#67—48no—
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311D12—28no—
312C4×D3—42no—
313C4×D3—42no—
314C4×D3—42no—
315C4×D3—42no—
316C3×C8—42no—
317C3×C8—42no—
318C3⋊C8—22no—
319C3⋊C8—22no—
320C3⋊C8—22no—
321C3⋊C8#12114yes—
322C24—32no—
323C24C2×C4×D311yescharacteristic
324C2×D4—64no—
325C2×D4—64no—
326C2×D4—64no—
327C2×D4—62no—
328C2×D4—122no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—32no—
334C22×C4—32no—
335C22×C4—32no—
336C22×C4—62no—
337C22×C4—62no—
338C22×C4—62no—
339C22×C4—62no—
340C22×C4—62no—
341C22×C4—62no—
342C22×C4C2×C4×D311yescharacteristic
343C22×C4C2×C4×D311yescharacteristic
344C22×C4C2×C4×D311yescharacteristic
345C22×C4C2×C4×D311yescharacteristic
346C22×C4C2×C4×D311yescharacteristic
347C22×C4D3×D412yes—
348C22×C4#23912yes—
349C22×C4C23×D311yescharacteristic
350M16—68no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—32no—
358C2×C8—32no—
359C2×C8—32no—
360C2×C8—32no—
361C2×C8—32no—
362C2×C8—32no—
363C2×C8—68no—
364C2×C8—61no—
365C2×C8—61no—
366C2×C8—61no—
367C2×C8—61no—
368C2×C8—61no—
369C2×C8—61no—
370C2×C8—128no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—62no—
374C4⋊C4—64no—
375C4⋊C4—122no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—64no—
379C22⋊C4—64no—
380C22⋊C4—64no—
381C22⋊C4—64no—
382C22⋊C4—122no—
383C42—21no—
384C42—21no—
385C42—62no—
386C42—62no—
387C42D3×D412yes—
388C42#23712yes—
389C22×C3—24no—
390C22×C3—22no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—42no—
394C22×C3C2×C4×D411yescharacteristic
395C22×C3C2×C4×D411yescharacteristic
396C22×C3C4×#4111yescharacteristic
397C22×C3C2×#13112yes—
398C22×C3#56812yes—
399C2×D3—24no—
400C2×D3—24no—
401C2×D3—24no—
402C2×D3—42no—
403C2×D3—42no—
404C2×D3—42no—
405C2×D3—42no—
406C3×C4—22no—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—41no—
410C3×C4—41no—
411C3×C4C2×#12111yescharacteristic
412C3×C4#52511yescharacteristic
413C3×C4#52911yescharacteristic
414C3×C4#53211yescharacteristic
415Dic3—42no—
416Dic3—42no—
417Dic3—42no—
418C23—21no—
419C23—22no—
420C23—22no—
421C23—21no—
422C23—34no—
423C23—62no—
424C23—62no—
425C23—62no—
426C23—62no—
427C23—62no—
428C23—62no—
429C23#89312yes—
430C23C22×C4×D311yescharacteristic
431D4—68no—
432D4—128no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4#67⋊C412yes—
459C2×C4#67⋊C412yes—
460C2×C4#89312yes—
461C2×C4C2×D3×D411yescharacteristic
462C2×C4C2×D3×D411yescharacteristic
463C2×C4C2×#23711yescharacteristic
464C2×C4C2×#23911yescharacteristic
465C2×C4D3×#4111yescharacteristic
466C2×C4D3×#4111yescharacteristic
467C8—34no—
468C8—42no—
469C8—42no—
470C8—62no—
471C8—62no—
472C8—62no—
473C2×C3—42no—
474C2×C3C4×#12111yescharacteristic
475C2×C3#292911yescharacteristic
476C2×C3#293211yescharacteristic
477C2×C3#293312yes—
478C2×C3#296812yes—
479D3—44no—
480C22—22no—
481C22—24no—
482C22—42no—
483C22—41no—
484C22—41no—
485C22—64no—
486C22—64no—
487C22—64no—
488C22—122no—
489C22—122no—
490C22—122no—
491C22—122no—
492C22C2×#67⋊C411yescharacteristic
493C22C2×#89311yescharacteristic
494C22#552211yescharacteristic
495C22D3×#13112yes—
496C22#575012yes—
497C4—22no—
498C4—22no—
499C4—22no—
500C4—41no—
501C4—41no—
502C4—122no—
503C4—122no—
504C4—122no—
505C4#559611yescharacteristic
506C4D3×#12111yescharacteristic
507C4#562411yescharacteristic
508C4#562711yescharacteristic
509C3#2149211yescharacteristic, complemented, Sylow
510C2—42no—
511C2—124no—
512C2#7386511yescharacteristic
513C2#7865511yescharacteristic
514C2#7865811yescharacteristic
515C2#7907512yes—
516C2#7936812yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1798C2
prime radical#4960C4
cosocleC22×C3×C4C24
hypercenter#1798D3
hypocenterC3#21492
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1798

Complements

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

NH ≅ G/Ntype
2. C2×#4997C2semidirect
3. C2×#4834C2semidirect
4. C2×#4728C2semidirect
5. C3×#1798C2semidirect
6. #70471C2semidirect
7. #70469C2semidirect
8. #70466C2semidirect
9. #69952C2semidirect
10. #68657C2semidirect
11. #68648C2semidirect
53. #1798D3semidirect
83. C2×#208D4semidirect
111. C4×C3⋊C8D4semidirect
509. C3#21492semidirect

Maximal subgroups

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

subgroupindex
#822732
C2×#49972
C2×#48342
C2×#47282
C3×#17982
#704712
#704692
#704662
#699522
#686572
#686482
#214923