Finite Group Databasebeta

← C2×#77867 #11430705

Subgroups — C2×#77867 #11430705

Order 768.

2246 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#889C211yesmaximal, characteristic
2C22×#889C211yesmaximal, characteristic
3C2×C3×#398C211yesmaximal, characteristic
4C2×#4983C211yesmaximal, characteristic
5C2×#4982C211yesmaximal, characteristic
6C2×#4982C211yesmaximal, characteristic
7C2×#4976C211yesmaximal, characteristic
8C2×#4972C211yesmaximal, characteristic
9C2×#4972C211yesmaximal, characteristic
10#77867C2116yesmaximal, direct product factor
11C2×#4657C211yesmaximal, characteristic
12C2×#4657C211yesmaximal, characteristic
13C2×#4647C211yesmaximal, characteristic
14C2×#4647C211yesmaximal, characteristic
15C2×#4654C211yesmaximal, characteristic
16C2×#4653C211yesmaximal, characteristic
17C2×#2726—31nomaximal, Sylow
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C22×C4⋊Dic3C2211yescharacteristic
25C22×C4⋊Dic3C2211yescharacteristic
26C22×Dic3⋊C4C2211yescharacteristic
27C22×Dic3⋊C4C2211yescharacteristic
28C22×Dic3⋊C4C2211yescharacteristic
29C22×Dic3⋊C4C2211yescharacteristic
30C22×Dic3⋊C4C2211yescharacteristic
31C22×Dic3⋊C4C2211yescharacteristic
32C22×Dic3⋊C4C2211yescharacteristic
33C22×Dic3⋊C4C2211yescharacteristic
34C22×Dic3⋊C4C2211yescharacteristic
35C22×Dic3⋊C4C2211yescharacteristic
36C22×Dic3⋊C4C2211yescharacteristic
37C22×Dic3⋊C4C2211yescharacteristic
38C22×C4×Dic3C2211yescharacteristic
39C22×C4×Dic3C2211yescharacteristic
40C22×C4×Dic3C2211yescharacteristic
41C2×#889—28no—
42C2×#889—28no—
43C2×#889C2218yes—
44C2×#889C2218yes—
45C3×#398C2218yes—
46C2×C3×#95C2211yescharacteristic
47C2×C3×#95C2211yescharacteristic
48C2×C3×#95C2211yescharacteristic
49C2×C3⋊#95C2211yescharacteristic
50C2×C3⋊#95C2211yescharacteristic
51C2×C3⋊#95C2211yescharacteristic
52C2×C3⋊#95C2211yescharacteristic
53C2×C3⋊#95C2211yescharacteristic
54C2×C3⋊#95C2211yescharacteristic
55C2×C3⋊#95C2211yescharacteristic
56C2×C3⋊#95C2211yescharacteristic
57C2×C3⋊#95C2211yescharacteristic
58#4983C2218yes—
59#4982C2218yes—
60#4982C2218yes—
61#4976C2218yes—
62#4972C2218yes—
63#4972C2218yes—
64#4657C2218yes—
65#4657C2218yes—
66#4654C2218yes—
67#4653C2218yes—
68#4647C2218yes—
69#4647C2218yes—
70C22×#125—31no—
71C22×#125—31no—
72#2726—316no—
73C2×#398—31no—
74C2×#398—31no—
75C2×#398—31no—
76C2×#398—31no—
77C2×#398—31no—
78C2×#398D311yescharacteristic
79C2×#389—31no—
80C2×#389—31no—
81C2×#389—31no—
82C2×#389—31no—
83C2×#389—31no—
84C2×#387—31no—
85C2×#387—31no—
86C23×C3×C4C2311yescharacteristic
87C23×C3×C4C2311yescharacteristic
88C23×C3×C4C2311yescharacteristic
89C23×C3×C4C2311yescharacteristic
90C23×C3×C4C2311yescharacteristic
91C23×C3×C4C2311yescharacteristic
92C23×C3×C4C2311yescharacteristic
93C23×Dic3C2311yescharacteristic
94C23×Dic3C2311yescharacteristic
95C23×Dic3C2311yescharacteristic
96C23×Dic3C2311yescharacteristic
97C23×Dic3C2311yescharacteristic
98C23×Dic3C2311yescharacteristic
99C23×Dic3C2311yescharacteristic
100C23×Dic3C2311yescharacteristic
101C2×C3×C4⋊C4—24no—
102C2×C3×C4⋊C4—24no—
103C2×C3×C4⋊C4—42no—
104C2×C3×C4⋊C4—42no—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C4⋊C4C2314yes—
107C2×C3×C4⋊C4C2314yes—
108C2×C3×C4⋊C4C2314yes—
109C2×C4⋊Dic3—24no—
110C2×C4⋊Dic3—24no—
111C2×C4⋊Dic3—24no—
112C2×C4⋊Dic3Q818yes—
113C2×C4⋊Dic3C2314yes—
114C2×C4⋊Dic3C2314yes—
115C2×C4⋊Dic3C2314yes—
116C2×C4⋊Dic3C2314yes—
117C2×Dic3⋊C4—24no—
118C2×Dic3⋊C4—24no—
119C2×Dic3⋊C4—24no—
120C2×Dic3⋊C4—24no—
121C2×Dic3⋊C4—24no—
122C2×Dic3⋊C4—24no—
123C2×Dic3⋊C4—24no—
124C2×Dic3⋊C4—24no—
125C2×Dic3⋊C4—42no—
126C2×Dic3⋊C4—42no—
127C2×Dic3⋊C4—42no—
128C2×Dic3⋊C4D418yes—
129C2×Dic3⋊C4C2314yes—
130C2×Dic3⋊C4C2314yes—
131C2×Dic3⋊C4C2314yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2314yes—
134C2×Dic3⋊C4C2314yes—
135C2×Dic3⋊C4C2314yes—
136C2×Dic3⋊C4C2314yes—
137C2×Dic3⋊C4C2314yes—
138C2×Dic3⋊C4C2314yes—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2314yes—
141C2×C4×Dic3—24no—
142C2×C4×Dic3—24no—
143C2×C4×Dic3—42no—
144C2×C4×Dic3C2314yes—
145C2×C4×Dic3C2314yes—
146C2×C4×Dic3C2314yes—
147#889—232no—
148#889—232no—
149C3×#95C2314yes—
150C3×#95C2314yes—
151C3×#95C2314yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C2×#125—38no—
162C2×#125—38no—
163C2×#125—68no—
164C2×#125—68no—
165C22×C4⋊C4—31no—
166C22×C4⋊C4—31no—
167C22×C4⋊C4—31no—
168C22×C4⋊C4—31no—
169C22×C4⋊C4—31no—
170C22×C4⋊C4—31no—
171C22×C4⋊C4—31no—
172C22×C4⋊C4—31no—
173C22×C4⋊C4—31no—
174C22×C4⋊C4—31no—
175C22×C4⋊C4—31no—
176C22×C4⋊C4—31no—
177C22×C4⋊C4—31no—
178C22×C4⋊C4—31no—
179C22×C4⋊C4—31no—
180C22×C4⋊C4—31no—
181C22×C4⋊C4C2×D311yescharacteristic
182C22×C4⋊C4C2×D311yescharacteristic
183C22×C4⋊C4C2×D311yescharacteristic
184C22×C4⋊C4C2×D311yescharacteristic
185C22×C42—31no—
186C22×C42—31no—
187C22×C42—31no—
188#398—38no—
189#398—38no—
190#398—38no—
191#398—38no—
192#398—38no—
193#398C2×D318yes—
194#389—38no—
195#389—38no—
196#389—38no—
197#389—38no—
198#389—38no—
199#387—38no—
200#387—38no—
201C2×#95—31no—
202C2×#95—31no—
203C2×#95—31no—
204C2×#95—31no—
205C2×#95—31no—
206C2×#95—31no—
207C2×#95—31no—
208C2×#95—31no—
209C2×#95—31no—
210C2×#95C2×D311yescharacteristic
211C2×#95C2×D311yescharacteristic
212C2×#95C2×D311yescharacteristic
213C24×C3C2411yescharacteristic
214C22×C3×C4—22no—
215C22×C3×C4—22no—
216C22×C3×C4—22no—
217C22×C3×C4—22no—
218C22×C3×C4—22no—
219C22×C3×C4—22no—
220C22×C3×C4—22no—
221C22×C3×C4—22no—
222C22×C3×C4—22no—
223C22×C3×C4—22no—
224C22×C3×C4—22no—
225C22×C3×C4—22no—
226C22×C3×C4—22no—
227C22×C3×C4—22no—
228C22×C3×C4—22no—
229C22×C3×C4—22no—
230C22×C3×C4—22no—
231C22×C3×C4—22no—
232C22×C3×C4—22no—
233C22×C3×C4C2×D414yes—
234C22×C3×C4C2×Q814yes—
235C22×C3×C4C2412yes—
236C22×C3×C4C2412yes—
237C22×C3×C4C2412yes—
238C22×C3×C4C2412yes—
239C22×C3×C4C2412yes—
240C22×C3×C4C2412yes—
241C22×C3×C4C2412yes—
242C22×Dic3—22no—
243C22×Dic3—22no—
244C22×Dic3—22no—
245C22×Dic3—22no—
246C22×Dic3—22no—
247C22×Dic3—22no—
248C22×Dic3—22no—
249C22×Dic3—22no—
250C22×Dic3—22no—
251C22×Dic3—22no—
252C22×Dic3—22no—
253C22×Dic3—22no—
254C22×Dic3—22no—
255C22×Dic3—22no—
256C22×Dic3—22no—
257C22×Dic3—22no—
258C22×Dic3—22no—
259C22×Dic3—22no—
260C22×Dic3—22no—
261C22×Dic3C2×D414yes—
262C22×Dic3C2×D414yes—
263C22×Dic3C2×Q814yes—
264C22×Dic3C2×Q814yes—
265C22×Dic3#4114yes—
266C22×Dic3C2412yes—
267C22×Dic3C2412yes—
268C22×Dic3C2412yes—
269C22×Dic3C2412yes—
270C22×Dic3C2412yes—
271C22×Dic3C2412yes—
272C22×Dic3C2412yes—
273C22×Dic3C2412yes—
274C3×C4⋊C4—28no—
275C3×C4⋊C4—28no—
276C3×C4⋊C4—44no—
277C3×C4⋊C4—44no—
278C4⋊Dic3—28no—
279C4⋊Dic3—28no—
280C4⋊Dic3—28no—
281C4⋊Dic3C2×Q8116yes—
282Dic3⋊C4—28no—
283Dic3⋊C4—28no—
284Dic3⋊C4—28no—
285Dic3⋊C4—28no—
286Dic3⋊C4—28no—
287Dic3⋊C4—28no—
288Dic3⋊C4—28no—
289Dic3⋊C4—28no—
290Dic3⋊C4—44no—
291Dic3⋊C4—44no—
292Dic3⋊C4—44no—
293Dic3⋊C4C2×D4116yes—
294C4×Dic3—28no—
295C4×Dic3—28no—
296C4×Dic3—44no—
297C23×C4—31no—
298C23×C4—31no—
299C23×C4—31no—
300C23×C4—31no—
301C23×C4—31no—
302C23×C4—31no—
303C23×C4—31no—
304C23×C4—31no—
305C23×C4C22×D311yescharacteristic
306C23×C4C22×D311yescharacteristic
307C23×C4C22×D311yescharacteristic
308C23×C4C22×D311yescharacteristic
309C23×C4C22×D311yescharacteristic
310C23×C4C22×D311yescharacteristic
311C23×C4C22×D311yescharacteristic
312#125—632no—
313#125—632no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—38no—
317C2×C4⋊C4—38no—
318C2×C4⋊C4—34no—
319C2×C4⋊C4—34no—
320C2×C4⋊C4—34no—
321C2×C4⋊C4—34no—
322C2×C4⋊C4—34no—
323C2×C4⋊C4—34no—
324C2×C4⋊C4—34no—
325C2×C4⋊C4—34no—
326C2×C4⋊C4—34no—
327C2×C4⋊C4—34no—
328C2×C4⋊C4—34no—
329C2×C4⋊C4—34no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—34no—
333C2×C4⋊C4—34no—
334C2×C4⋊C4—42no—
335C2×C4⋊C4—42no—
336C2×C4⋊C4—64no—
337C2×C4⋊C4—64no—
338C2×C4⋊C4—64no—
339C2×C4⋊C4—64no—
340C2×C4⋊C4—64no—
341C2×C4⋊C4—64no—
342C2×C4⋊C4—64no—
343C2×C4⋊C4—64no—
344C2×C4⋊C4—64no—
345C2×C4⋊C4—64no—
346C2×C4⋊C4—64no—
347C2×C4⋊C4—122no—
348C2×C4⋊C4—122no—
349C2×C4⋊C4—122no—
350C2×C4⋊C4C22×D314yes—
351C2×C4⋊C4C22×D314yes—
352C2×C4⋊C4C22×D314yes—
353C2×C4⋊C4C22×D314yes—
354C2×C42—34no—
355C2×C42—34no—
356C2×C42—34no—
357C2×C42—64no—
358C2×C42—64no—
359C2×C42—122no—
360#95—34no—
361#95—34no—
362#95—34no—
363#95—34no—
364#95—34no—
365#95—34no—
366#95—34no—
367#95—34no—
368#95—34no—
369#95C22×D314yes—
370#95C22×D314yes—
371#95C22×D314yes—
372C23×C3C22×D412yes—
373C23×C3C22×Q812yes—
374C23×C3C2×#4112yes—
375C23×C3H(5,𝔽2)12yes—
376C23×C32−1+412yes—
377C23×C32−1+412yes—
378C23×C32−1+412yes—
379C23×C3C2511yescharacteristic
380C2×C3×C4—28no—
381C2×C3×C4—28no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—44no—
402C2×C3×C4—44no—
403C2×C3×C4—44no—
404C2×C3×C4—44no—
405C2×C3×C4—44no—
406C2×C3×C4C22×D414yes—
407C2×C3×C4C22×Q814yes—
408C2×Dic3—28no—
409C2×Dic3—28no—
410C2×Dic3—28no—
411C2×Dic3—28no—
412C2×Dic3—28no—
413C2×Dic3—22no—
414C2×Dic3—22no—
415C2×Dic3—22no—
416C2×Dic3—22no—
417C2×Dic3—22no—
418C2×Dic3—22no—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—44no—
433C2×Dic3—44no—
434C2×Dic3—44no—
435C2×Dic3C22×D414yes—
436C2×Dic3C22×D414yes—
437C2×Dic3C22×Q814yes—
438C2×Dic3C22×Q814yes—
439C2×Dic3C2×#4114yes—
440C24C23×D311yescharacteristic
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—34no—
461C22×C4—34no—
462C22×C4—34no—
463C22×C4—34no—
464C22×C4—34no—
465C22×C4—32no—
466C22×C4—32no—
467C22×C4—32no—
468C22×C4—32no—
469C22×C4—32no—
470C22×C4—32no—
471C22×C4—32no—
472C22×C4—32no—
473C22×C4—62no—
474C22×C4—62no—
475C22×C4—62no—
476C22×C4—62no—
477C22×C4—62no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4—62no—
483C22×C4—62no—
484C22×C4—62no—
485C22×C4—62no—
486C22×C4—62no—
487C22×C4—62no—
488C22×C4—62no—
489C22×C4—62no—
490C22×C4—62no—
491C22×C4—62no—
492C22×C4D3×D414yes—
493C22×C4D3×Q814yes—
494C22×C4C23×D312yes—
495C22×C4C23×D312yes—
496C22×C4C23×D312yes—
497C22×C4C23×D312yes—
498C22×C4C23×D312yes—
499C22×C4C23×D312yes—
500C22×C4C23×D312yes—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—316no—
504C4⋊C4—316no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—68no—
508C4⋊C4—68no—
509C4⋊C4—68no—
510C4⋊C4—68no—
511C4⋊C4—68no—
512C4⋊C4—68no—
513C4⋊C4—68no—
514C4⋊C4—68no—
515C4⋊C4—68no—
516C4⋊C4—68no—
517C4⋊C4—68no—
518C4⋊C4—124no—
519C4⋊C4—124no—
520C4⋊C4—124no—
521C42—68no—
522C42—68no—
523C42—124no—
524C22×C3#53714yes—
525C22×C3#53914yes—
526C22×C3#54414yes—
527C22×C3#54714yes—
528C22×C3D4×Q814yes—
529C22×C3#55714yes—
530C22×C3#56414yes—
531C22×C3C23×D411yescharacteristic
532C22×C3C23×Q811yescharacteristic
533C22×C3C22×#4111yescharacteristic
534C22×C3C2×H(5,𝔽2)11yescharacteristic
535C22×C3C2×2−1+411yescharacteristic
536C22×C3C2×2−1+411yescharacteristic
537C22×C3C2×2−1+411yescharacteristic
538C3×C4—24no—
539C3×C4—24no—
540C3×C4—42no—
541C3×C4—42no—
542C3×C4—42no—
543C3×C4—42no—
544C3×C4—42no—
545Dic3—24no—
546Dic3—24no—
547Dic3—24no—
548Dic3—24no—
549Dic3—24no—
550Dic3—42no—
551Dic3—42no—
552Dic3—42no—
553C23C2×D3×D412yes—
554C23#100412yes—
555C23C2×D3×Q812yes—
556C23#100712yes—
557C23D3×#4112yes—
558C23#101012yes—
559C23#101012yes—
560C23C24×D311yescharacteristic
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—34no—
583C2×C4—34no—
584C2×C4—34no—
585C2×C4—34no—
586C2×C4—34no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—68no—
593C2×C4—68no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—68no—
605C2×C4—68no—
606C2×C4—68no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—62no—
616C2×C4—124no—
617C2×C4—124no—
618C2×C4—124no—
619C2×C4C2×D3×D414yes—
620C2×C4C2×D3×Q814yes—
621C2×C3#272618yescomplemented
622C2×C3C2×#53711yescharacteristic
623C2×C3C2×#53911yescharacteristic
624C2×C3C2×#54411yescharacteristic
625C2×C3C2×#54711yescharacteristic
626C2×C3C2×D4×Q811yescharacteristic
627C2×C3C2×#55711yescharacteristic
628C2×C3C2×#56411yescharacteristic
629C22#560814yes—
630C22#562314yes—
631C22#563814yes—
632C22#565514yes—
633C22#568514yes—
634C22C3⋊#56414yes—
635C22#572714yes—
636C22C22×D3×D411yescharacteristic
637C22C2×#100411yescharacteristic
638C22C22×D3×Q811yescharacteristic
639C22C2×#100711yescharacteristic
640C22C2×D3×#4111yescharacteristic
641C22C2×#101011yescharacteristic
642C22C2×#101011yescharacteristic
643C4—24no—
644C4—24no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—64no—
651C4—64no—
652C4—64no—
653C4—64no—
654C4—64no—
655C4—122no—
656C4—122no—
657C4—122no—
658C3C2×#272611yescharacteristic, complemented, Sylow
659C2#7786718yesdirect product factor
660C2C2×#560811yescharacteristic
661C2C2×#562311yescharacteristic
662C2C2×#563811yescharacteristic
663C2C2×#565511yescharacteristic
664C2C2×#568511yescharacteristic
665C2C2×C3⋊#56411yescharacteristic
666C2C2×#572711yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#398C2
socle, prime radicalC24×C3C24
hypercenterC2×#398D3
hypocenterC3C2×#2726
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
10. #77867C2direct (N × H)
621. C2×C3#2726semidirect
658. C3C2×#2726semidirect
659. C2#77867direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8892
C2×C3×#3982
C2×#49832
C2×#49822
C2×#49762
C2×#49722
#778672
C2×#46572
C2×#46472
C2×#46542
C2×#46532
C2×#27263