Finite Group Databasebeta

← C2×#73655 #10954792

Subgroups — C2×#73655 #10954792

Order 768.

3678 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#875C211yesmaximal, characteristic
2C22×C4×D12C211yesmaximal, characteristic
3C22×C4×Dic6C211yesmaximal, characteristic, complemented
4C2×C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
5C2×#4989C212yesmaximal, complemented
6C2×#4990C212yesmaximal, complemented
7C2×#4929C211yesmaximal, characteristic, complemented
8C2×#4939C211yesmaximal, characteristic
9C2×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
10#73655C2116yesmaximal, direct product factor
11C2×#4669C212yesmaximal, complemented
12C2×#4666C212yesmaximal, complemented
13C2×#2367—31nomaximal, Sylow
14C23×D12C2211yescharacteristic
15C23×C4×D3C2212yes—
16C23×Dic6C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C22×C3×C42C2211yescharacteristic
21C22×C3×C42C2211yescharacteristic
22C22×(C2×D3)⋊C4C2212yes—
23C22×(C2×D3)⋊C4C2212yes—
24C22×(C2×D3)⋊C4C2212yes—
25C22×(C2×D3)⋊C4C2212yes—
26C22×(C2×D3)⋊C4C2212yes—
27C22×C4⋊Dic3C2211yescharacteristic
28C22×C4⋊Dic3C2211yescharacteristic
29C22×Dic3⋊C4C2212yes—
30C22×C4×Dic3C2212yes—
31C22×C4×Dic3C2212yes—
32C2×#875C4116yescomplemented
33C2×#875C2218yes—
34C2×C4×D12—28no—
35C2×C4×D12C2218yes—
36C2×C4×Dic6—28no—
37C2×C4×Dic6C2218yescomplemented
38C3×C4×C4⋊C4C2218yescomplemented
39C2×C3×#95C2211yescharacteristic
40C2×C3×#95C2211yescharacteristic
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43C2×C3⋊#95C2212yes—
44#4990C22116yescomplemented
45#4989C22116yescomplemented
46#4939C2218yes—
47C4×C4⋊Dic3C2218yescomplemented
48#4929C2218yescomplemented
49#4669C22116yescomplemented
50#4666C22116yescomplemented
51C22×#124—31no—
52C22×C4×Q8—31no—
53C22×C4×D4—31no—
54#2367—316no—
55C2×#391—31no—
56C2×#390—31no—
57C2×#388—32no—
58C2×#388—32no—
59C2×#385—32no—
60C2×C4×C4⋊C4—31no—
61C2×C4×C4⋊C4D311yescharacteristic, complemented
62C2×C4×C22⋊C4—32no—
63C24×D3C2312yes—
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C23×Dic3C2312yes—
72C23×Dic3C2312yes—
73C23×Dic3C2312yes—
74C22×D12—24no—
75C22×D12—24no—
76C22×D12C2×C418yes—
77C22×D12C2314yes—
78C22×C4×D3—28no—
79C22×C4×D3—28no—
80C22×C4×D3—44no—
81C22×C4×D3C2318yes—
82C22×Dic6—24no—
83C22×Dic6—24no—
84C22×Dic6C2×C418yescomplemented
85C22×Dic6C2314yes—
86C2×C3×C4⋊C4—24no—
87C2×C3×C4⋊C4—24no—
88C2×C3×C4⋊C4C2314yes—
89C2×C3×C4⋊C4C2314yes—
90C2×C3×C42—24no—
91C2×C3×C42—24no—
92C2×C3×C42C2×C418yescomplemented
93C2×C3×C42C2314yes—
94C2×C3×C42C2314yes—
95C2×C3×C42C2314yes—
96C2×(C2×D3)⋊C4—28no—
97C2×(C2×D3)⋊C4—28no—
98C2×(C2×D3)⋊C4—44no—
99C2×(C2×D3)⋊C4C2×C4116yescomplemented
100C2×(C2×D3)⋊C4C2×C4116yescomplemented
101C2×(C2×D3)⋊C4C2318yes—
102C2×(C2×D3)⋊C4C2318yes—
103C2×(C2×D3)⋊C4C2318yes—
104C2×(C2×D3)⋊C4C2318yes—
105C2×(C2×D3)⋊C4C2318yes—
106C2×C4⋊Dic3—24no—
107C2×C4⋊Dic3D418yes—
108C2×C4⋊Dic3C2314yes—
109C2×C4⋊Dic3C2314yes—
110C2×Dic3⋊C4—28no—
111C2×Dic3⋊C4C2318yes—
112C2×C4×Dic3—28no—
113C2×C4×Dic3—44no—
114C2×C4×Dic3C2318yes—
115C2×C4×Dic3C2318yes—
116#875C2×C4164yescomplemented
117C4×D12—232no—
118C4×Dic6—232no—
119C3×#95C2314yes—
120C3×#95C2314yes—
121C3⋊#95C2318yes—
122C3⋊#95C2318yes—
123C3⋊#95C2318yes—
124C23×Q8—31no—
125C23×D4—31no—
126C24×C4—32no—
127C2×#124—316no—
128C2×#124—38no—
129C2×C4×Q8—38no—
130C2×C4×Q8—68no—
131C2×C4×D4—38no—
132C2×C4×D4—68no—
133C22×C4⋊C4—31no—
134C22×C4⋊C4—32no—
135C22×C4⋊C4—31no—
136C22×C4⋊C4C2×D311yescharacteristic
137C22×C4⋊C4C2×D311yescharacteristic
138C22×C22⋊C4—32no—
139C22×C22⋊C4—32no—
140C22×C22⋊C4—32no—
141C22×C22⋊C4—32no—
142C22×C22⋊C4—32no—
143C22×C42—32no—
144C22×C42—32no—
145C22×C42C2×D311yescharacteristic
146C22×C42C2×D311yescharacteristic
147C22×C42C2×D311yescharacteristic
148#391—38no—
149#390—38no—
150#388—316no—
151#388—316no—
152#385—316no—
153C4×C4⋊C4—38no—
154C4×C4⋊C4C2×D318yescomplemented
155C4×C22⋊C4—316no—
156C2×#95—32no—
157C2×#95—32no—
158C2×#95—32no—
159C2×#95C2×D311yescharacteristic
160C2×#95C2×D311yescharacteristic
161C24×C3C2411yescharacteristic
162C23×D3—24no—
163C23×D3—24no—
164C23×D3—24no—
165C23×D3—24no—
166C23×D3—24no—
167C23×D3—24no—
168C23×D3C22×C418yes—
169C23×D3C2414yes—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4C22×C414yes—
185C22×C3×C4C22×C414yes—
186C22×C3×C4C22×C414yes—
187C22×C3×C4C2×D414yes—
188C22×C3×C4#4114yes—
189C22×C3×C4#4114yes—
190C22×C3×C4#4114yes—
191C22×C3×C4C2412yes—
192C22×C3×C4C2412yes—
193C22×C3×C4C2412yes—
194C22×C3×C4C2412yes—
195C22×C3×C4C2412yes—
196C22×C3×C4C2412yes—
197C22×C3×C4C2412yes—
198C22×Dic3—24no—
199C22×Dic3—24no—
200C22×Dic3—24no—
201C22×Dic3—24no—
202C22×Dic3—24no—
203C22×Dic3—24no—
204C22×Dic3—24no—
205C22×Dic3C22×C418yes—
206C22×Dic3C2×D418yes—
207C22×Dic3C2414yes—
208C22×Dic3C2414yes—
209C22×Dic3C2414yes—
210C2×D12—232no—
211C2×D12—28no—
212C2×D12—28no—
213C2×D12C22×C4116yes—
214C2×C4×D3—216no—
215C2×C4×D3—216no—
216C2×C4×D3—48no—
217C2×C4×D3—432no—
218C2×Dic6—232no—
219C2×Dic6—28no—
220C2×Dic6—28no—
221C2×Dic6C22×C4116yescomplemented
222C3×C4⋊C4—28no—
223C3×C4⋊C4—28no—
224C3×C42—28no—
225C3×C42—28no—
226C3×C42C22×C4116yescomplemented
227(C2×D3)⋊C4—216no—
228(C2×D3)⋊C4—216no—
229(C2×D3)⋊C4—48no—
230(C2×D3)⋊C4C22×C4132yescomplemented
231(C2×D3)⋊C4C22×C4132yescomplemented
232C4⋊Dic3—28no—
233C4⋊Dic3C2×D4116yes—
234Dic3⋊C4—216no—
235C4×Dic3—216no—
236C4×Dic3—48no—
237C25—32no—
238C22×Q8—38no—
239C22×Q8—34no—
240C22×Q8—64no—
241C22×Q8—64no—
242C22×D4—38no—
243C22×D4—34no—
244C22×D4—64no—
245C22×D4—64no—
246C23×C4—32no—
247C23×C4—32no—
248C23×C4—32no—
249C23×C4—38no—
250C23×C4—68no—
251C23×C4—68no—
252C23×C4—124no—
253C23×C4C22×D311yescharacteristic
254C23×C4C22×D311yescharacteristic
255C23×C4C22×D311yescharacteristic
256C23×C4C22×D311yescharacteristic
257C23×C4C22×D311yescharacteristic
258C23×C4C22×D311yescharacteristic
259C23×C4C22×D311yescharacteristic
260#124—364no—
261C4×Q8—632no—
262C4×D4—632no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—38no—
266C2×C4⋊C4—34no—
267C2×C4⋊C4—38no—
268C2×C4⋊C4—34no—
269C2×C4⋊C4—68no—
270C2×C4⋊C4—64no—
271C2×C4⋊C4C22×D314yes—
272C2×C4⋊C4C22×D314yes—
273C2×C22⋊C4—316no—
274C2×C22⋊C4—316no—
275C2×C22⋊C4—38no—
276C2×C22⋊C4—38no—
277C2×C22⋊C4—38no—
278C2×C22⋊C4—38no—
279C2×C22⋊C4—38no—
280C2×C22⋊C4—68no—
281C2×C22⋊C4—68no—
282C2×C22⋊C4—124no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—38no—
286C2×C42—38no—
287C2×C42—68no—
288C2×C42—124no—
289C2×C42C4×D318yescomplemented
290C2×C42C22×D314yes—
291C2×C42C22×D314yes—
292C2×C42C22×D314yes—
293#95—38no—
294#95—38no—
295#95—38no—
296#95C22×D314yes—
297#95C22×D314yes—
298C23×C3C23×C412yes—
299C23×C3C22×D412yes—
300C23×C3C2×#4112yes—
301C23×C3C2×#4112yes—
302C23×C3C2×#4112yes—
303C23×C3H(5,𝔽2)12yes—
304C23×C32−1+412yes—
305C23×C3C2511yescharacteristic
306C22×D3—216no—
307C22×D3—216no—
308C22×D3—216no—
309C22×D3—24no—
310C22×D3—24no—
311C22×D3—24no—
312C22×D3—24no—
313C22×D3—24no—
314C22×D3—24no—
315C22×D3—48no—
316C22×D3—48no—
317C22×D3—48no—
318C22×D3—48no—
319C22×D3C23×C418yes—
320C2×C3×C4—28no—
321C2×C3×C4—28no—
322C2×C3×C4—28no—
323C2×C3×C4—28no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—44no—
339C2×C3×C4—44no—
340C2×C3×C4C4×D4116yes—
341C2×C3×C4C23×C414yes—
342C2×C3×C4C23×C414yes—
343C2×C3×C4C23×C414yes—
344C2×C3×C4C22×D414yes—
345C2×C3×C4C2×#4114yes—
346C2×C3×C4C2×#4114yes—
347C2×C3×C4C2×#4114yes—
348C2×Dic3—216no—
349C2×Dic3—216no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—24no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—48no—
358C2×Dic3C23×C418yes—
359C2×Dic3C22×D418yes—
360D12—232no—
361C4×D3—432no—
362Dic6—232no—
363C24—38no—
364C24—34no—
365C24—64no—
366C24—64no—
367C24—64no—
368C24—64no—
369C24—64no—
370C24—64no—
371C24C23×D311yescharacteristic
372C2×Q8—316no—
373C2×Q8—632no—
374C2×Q8—68no—
375C2×Q8—68no—
376C2×D4—316no—
377C2×D4—632no—
378C2×D4—68no—
379C2×D4—68no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—38no—
395C22×C4—38no—
396C22×C4—34no—
397C22×C4—34no—
398C22×C4—34no—
399C22×C4—616no—
400C22×C4—64no—
401C22×C4—64no—
402C22×C4—64no—
403C22×C4—64no—
404C22×C4—64no—
405C22×C4—64no—
406C22×C4—64no—
407C22×C4—616no—
408C22×C4—128no—
409C22×C4—1232no—
410C22×C4C2×C4×D314yes—
411C22×C4C2×C4×D314yes—
412C22×C4C2×C4×D314yes—
413C22×C4#23514yes—
414C22×C4#23514yes—
415C22×C4D3×D414yes—
416C22×C4#23914yes—
417C22×C4C23×D312yes—
418C22×C4C23×D312yes—
419C22×C4C23×D312yes—
420C22×C4C23×D312yes—
421C22×C4C23×D312yes—
422C22×C4C23×D312yes—
423C22×C4C23×D312yes—
424C4⋊C4—28no—
425C4⋊C4—28no—
426C4⋊C4—316no—
427C4⋊C4—616no—
428C4⋊C4—68no—
429C22⋊C4—332no—
430C22⋊C4—332no—
431C22⋊C4—616no—
432C22⋊C4—616no—
433C22⋊C4—128no—
434C42—28no—
435C42—28no—
436C42—616no—
437C42—128no—
438C42C2×C4×D3116yescomplemented
439C22×C3C2×C4×D414yes—
440C22×C3C4×#4114yes—
441C22×C3#52014yes—
442C22×C3#54614yes—
443C22×C3#54814yes—
444C22×C3#55614yes—
445C22×C3#55914yes—
446C22×C3C24×C411yescharacteristic
447C22×C3C23×D411yescharacteristic
448C22×C3C22×#4111yescharacteristic
449C22×C3C22×#4111yescharacteristic
450C22×C3C22×#4111yescharacteristic
451C22×C3C2×H(5,𝔽2)11yescharacteristic
452C22×C3C2×2−1+411yescharacteristic
453C2×D3—28no—
454C2×D3—28no—
455C2×D3—28no—
456C2×D3—432no—
457C2×D3—44no—
458C2×D3—44no—
459C2×D3—44no—
460C2×D3—44no—
461C3×C4—24no—
462C3×C4—24no—
463C3×C4—24no—
464C3×C4—24no—
465C3×C4—42no—
466C3×C4—42no—
467C3×C4C2×C4×D418yes—
468Dic3—28no—
469Dic3—28no—
470Dic3—44no—
471C23—38no—
472C23—64no—
473C23—64no—
474C23—64no—
475C23—64no—
476C23—64no—
477C23—64no—
478C23—616no—
479C23—616no—
480C23—616no—
481C23—128no—
482C23—128no—
483C23—128no—
484C23—128no—
485C23C22×C4×D312yes—
486C23C2×#23512yes—
487C23C2×#23512yes—
488C23C2×D3×D412yes—
489C23C2×#23912yes—
490C23#100912yes—
491C23#101012yes—
492C23C24×D311yescharacteristic
493Q8—632no—
494D4—632no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—38no—
514C2×C4—38no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—64no—
518C2×C4—64no—
519C2×C4—64no—
520C2×C4—64no—
521C2×C4—616no—
522C2×C4—616no—
523C2×C4—64no—
524C2×C4—64no—
525C2×C4—64no—
526C2×C4—1232no—
527C2×C4—128no—
528C2×C4#893116yes—
529C2×C4C22×C4×D314yes—
530C2×C4C22×C4×D314yes—
531C2×C4C22×C4×D314yes—
532C2×C4C2×#23514yes—
533C2×C4C2×#23514yes—
534C2×C4C2×D3×D414yes—
535C2×C4C2×#23914yes—
536C2×C3#236718yescomplemented
537C2×C3C22×C4×D411yescharacteristic
538C2×C3C2×C4×#4111yescharacteristic
539C2×C3C2×#52011yescharacteristic
540C2×C3C2×#54611yescharacteristic
541C2×C3C2×#54811yescharacteristic
542C2×C3C2×#55611yescharacteristic
543C2×C3C2×#55911yescharacteristic
544D3—48no—
545C22—68no—
546C22—68no—
547C22—68no—
548C22—124no—
549C22—124no—
550C22—124no—
551C22—1232no—
552C22—124no—
553C22C4×#23514yes—
554C22C2×#89314yes—
555C22#552314yes—
556C22#555014yes—
557C22#555214yes—
558C22#558414yes—
559C22#558514yes—
560C22C23×C4×D311yescharacteristic
561C22C22×#23511yescharacteristic
562C22C22×#23511yescharacteristic
563C22C22×D3×D411yescharacteristic
564C22C22×#23911yescharacteristic
565C22C2×#100911yescharacteristic
566C22C2×#101011yescharacteristic
567C4—24no—
568C4—24no—
569C4—24no—
570C4—24no—
571C4—42no—
572C4—42no—
573C4—68no—
574C4—68no—
575C4—124no—
576C4C2×#89318yes—
577C3C2×#236711yescharacteristic, complemented, Sylow
578C2—128no—
579C2#7365518yesdirect product factor
580C2C2×C4×#23511yescharacteristic
581C2C22×#89311yescharacteristic
582C2C2×#552311yescharacteristic
583C2C2×#555211yescharacteristic
584C2C2×#555011yescharacteristic
585C2C2×#558411yescharacteristic
586C2C2×#558511yescharacteristic
subgroupG/H
centerC24C23×D3
commutatorC22×C3C24×C4
FrattiniC23C24×D3
FittingC2×C3×C4×C4⋊C4C2
socleC24×C3C24
prime radicalC23×D12C22
cosocleC23×C3C25
hypercenterC2×C4×C4⋊C4D3
hypocenterC3C2×#2367
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C22×C4×Dic6C2semidirect
4. C2×C3×C4×C4⋊C4C2semidirect
5. C2×#4989C2semidirect
6. C2×#4990C2semidirect
7. C2×#4929C2semidirect
9. C2×C4×C4⋊Dic3C2semidirect
10. #73655C2direct (N × H)
11. C2×#4669C2semidirect
12. C2×#4666C2semidirect
32. C2×#875C4semidirect
37. C2×C4×Dic6C22semidirect
38. C3×C4×C4⋊C4C22semidirect
44. #4990C22semidirect
45. #4989C22semidirect
47. C4×C4⋊Dic3C22semidirect
48. #4929C22semidirect
49. #4669C22semidirect
50. #4666C22semidirect
61. C2×C4×C4⋊C4D3semidirect
84. C22×Dic6C2×C4semidirect
92. C2×C3×C42C2×C4semidirect
99. C2×(C2×D3)⋊C4C2×C4semidirect
100. C2×(C2×D3)⋊C4C2×C4semidirect
116. #875C2×C4semidirect
154. C4×C4⋊C4C2×D3semidirect
221. C2×Dic6C22×C4semidirect
226. C3×C42C22×C4semidirect
230. (C2×D3)⋊C4C22×C4semidirect
231. (C2×D3)⋊C4C22×C4semidirect
289. C2×C42C4×D3semidirect
438. C42C2×C4×D3semidirect
536. C2×C3#2367semidirect
577. C3C2×#2367semidirect
579. C2#73655direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8752
C22×C4×D122
C22×C4×Dic62
C2×C3×C4×C4⋊C42
C2×#49892
C2×#49902
C2×#49292
C2×#49392
C2×C4×C4⋊Dic32
#736552
C2×#46692
C2×#46662
C2×#23673