Finite Group Databasebeta

← C2×#10594505 #11901911

Subgroups — C2×#10594505 #11901911

Order 1152.

2273 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64099C211yesmaximal, characteristic, complemented
2C2×C3×#892C211yesmaximal, characteristic
3C2×C3×#890C211yesmaximal, characteristic, complemented
4C2×#63919C211yesmaximal, characteristic, complemented
5C2×#63918C211yesmaximal, characteristic, complemented
6C2×#63918C212yesmaximal, complemented
7C2×#63899C212yesmaximal
8C2×#63893C212yesmaximal
9C2×D3×C4⋊Dic3C211yesmaximal, characteristic
10C2×#63867C212yesmaximal, complemented
11C2×#63855C211yesmaximal, characteristic
12#10594505C2116yesmaximal, direct product factor
13C2×#5689—31nomaximal
14C2×#5519—31nomaximal
15C22×C3×C4×D3C2211yescharacteristic
16C22×#627C2212yes—
17C22×#627C2212yes—
18C22×D3×Dic3C2212yes—
19C2×C32×C4⋊C4C2211yescharacteristic
20C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C4⋊(C3⋊Dic3)C2212yes—
22C2×#3750C2212yes—
23C2×C4×C3⋊Dic3C2211yescharacteristic
24#64099C2218yescomplemented
25C2×C3×(C2×D3)⋊C4C2212yes—
26C2×C3×C4⋊Dic3C2211yescharacteristic
27C2×C3×C4⋊Dic3C2212yes—
28C2×C3×C4⋊Dic3C2211yescharacteristic
29C2×C3×Dic3⋊C4C2212yes—
30C2×C3×C4×Dic3C2211yescharacteristic
31C2×C3×C4×Dic3C2212yes—
32C3×#892C2218yes—
33C3×#890C2218yescomplemented
34C2×#3724C2212yes—
35C2×Dic3⋊Dic3C2211yescharacteristic
36C2×Dic3⋊Dic3C2211yescharacteristic
37C2×Dic3⋊Dic3C2212yes—
38C2×Dic3⋊Dic3C2212yes—
39C2×(C2×D3)⋊Dic3C2211yescharacteristic
40C2×(C2×D3)⋊Dic3C2211yescharacteristic
41#63919C2218yescomplemented
42#63918C2218yescomplemented
43#63918C22116yescomplemented
44#63899C22116yes—
45#63893C22116yes—
46D3×C4⋊Dic3C2218yes—
47#63867C22116yescomplemented
48#63855C2218yes—
49C2×C3×#125—31no—
50C2×C3×#117—31no—
51C2×#926—31no—
52C22×C4⋊Dic3—31no—
53#5689—316no—
54#5519—316no—
55C2×#897—32no—
56C2×#896—32no—
57C2×#892D311yescharacteristic
58C2×D3×C4⋊C4—31no—
59C2×#890—31no—
60C2×#890—31no—
61C2×#890—61no—
62C2×#890D311yescharacteristic, complemented
63C2×#889—32no—
64C2×#888—31no—
65C2×#888—31no—
66C2×#888—32no—
67C2×#888—31no—
68C2×#878—32no—
69C2×#878—32no—
70C2×#872—31no—
71C2×#870—32no—
72C2×#869—31no—
73C2×#869—32no—
74C23×C3×D3C2311yescharacteristic
75C22×C32×C4C2312yes—
76C22×C32×C4C2311yescharacteristic
77C22×C3⋊Dic3C2312yes—
78C22×C3⋊Dic3C2311yescharacteristic
79C22×C3⋊Dic3C2311yescharacteristic
80C22×C3×Dic3C2311yescharacteristic
81C22×C3×Dic3C2311yescharacteristic
82C22×C3×Dic3C2312yes—
83C22×C3×Dic3C2312yes—
84C22×C3×Dic3C2311yescharacteristic
85C2×C3×C4×D3Q818yescomplemented
86C2×C3×C4×D3C2314yes—
87C2×#627—44no—
88C2×#627—44no—
89C2×#627C2318yes—
90C2×#627C2318yes—
91C2×D3×Dic3—44no—
92C2×D3×Dic3C2318yes—
93C32×C4⋊C4C2314yes—
94C4⋊(C3⋊Dic3)C2314yes—
95C4⋊(C3⋊Dic3)C2318yes—
96#3750C2318yes—
97C4×C3⋊Dic3C2314yes—
98C3×(C2×D3)⋊C4C2318yes—
99C3×C4⋊Dic3C2314yes—
100C3×C4⋊Dic3C2318yes—
101C3×C4⋊Dic3C2314yes—
102C3×Dic3⋊C4C2318yes—
103C3×C4×Dic3C2314yes—
104C3×C4×Dic3C2318yes—
105#3724C2318yes—
106Dic3⋊Dic3C2314yes—
107Dic3⋊Dic3C2314yes—
108Dic3⋊Dic3C2318yes—
109Dic3⋊Dic3C2318yes—
110(C2×D3)⋊Dic3C2314yes—
111(C2×D3)⋊Dic3C2314yes—
112C2×#544—91noSylow
113C23×C3×C4—31no—
114C23×Dic3—32no—
115C22×C4×D3—32no—
116C22×C4×D3C2×D311yescharacteristic
117C22×Dic6—32no—
118C22×Dic6—32no—
119C22×Dic6—32no—
120C22×Dic6—32no—
121C3×#125—38no—
122C3×#117—38no—
123C2×C3×C4⋊C4—21no—
124C2×C3×C4⋊C4—32no—
125C2×C3×C4⋊C4—31no—
126C2×C3×C4⋊C4—31no—
127C2×C3×C4⋊C4—32no—
128C2×C3×C4⋊C4C2×D311yescharacteristic
129C2×C3×C4⋊C4C2×D311yescharacteristic
130C2×C3×C22⋊C4—32no—
131C2×C3×C42—32no—
132C2×C3×C42—31no—
133C2×#217—31no—
134C2×#217—31no—
135C2×(C2×D3)⋊C4—31no—
136C2×(C2×D3)⋊C4—31no—
137C2×(C2×D3)⋊C4C2×D312yes—
138#926—38no—
139C2×C4⋊Dic3—32no—
140C2×C4⋊Dic3—31no—
141C2×C4⋊Dic3—32no—
142C2×C4⋊Dic3—31no—
143C2×C4⋊Dic3—31no—
144C2×C4⋊Dic3—32no—
145C2×C4⋊Dic3—32no—
146C2×C4⋊Dic3—31no—
147C2×C4⋊Dic3—38no—
148C2×C4⋊Dic3—61no—
149C2×C4⋊Dic3—62no—
150C2×C4⋊Dic3C2×D312yes—
151C2×C4⋊Dic3C2×D311yescharacteristic
152C2×C4⋊Dic3C2×D311yescharacteristic
153C2×Dic3⋊C4—32no—
154C2×Dic3⋊C4—32no—
155C2×Dic3⋊C4—32no—
156C2×Dic3⋊C4—32no—
157C2×Dic3⋊C4—31no—
158C2×Dic3⋊C4—31no—
159C2×Dic3⋊C4—32no—
160C2×Dic3⋊C4—32no—
161C2×Dic3⋊C4—62no—
162C2×Dic3⋊C4C2×D312yes—
163C2×C4×Dic3—31no—
164C2×C4×Dic3—31no—
165C2×C4×Dic3—61no—
166C2×C4×Dic3C2×D312yes—
167C2×C4×Dic3C2×D311yescharacteristic
168#897—316no—
169#896—316no—
170#892C2×D318yes—
171D3×C4⋊C4—38no—
172#890—38no—
173#890—38no—
174#890—68no—
175#890C2×D318yescomplemented
176#889—316no—
177#888—38no—
178#888—38no—
179#888—316no—
180#888—38no—
181#878—316no—
182#878—316no—
183#872—38no—
184#870—316no—
185#869—38no—
186#869—316no—
187C23×C32C2411yescharacteristic
188C22×C3×D3—22no—
189C22×C3×D3—22no—
190C22×C3×D3C2×Q814yes—
191C22×C3×D3C2412yes—
192C2×C32×C4—24no—
193C2×C32×C4C2×Q814yes—
194C2×C32×C4C2414yes—
195C2×C32×C4C2412yes—
196C2×C3⋊Dic3—24no—
197C2×C3⋊Dic3—22no—
198C2×C3⋊Dic3—22no—
199C2×C3⋊Dic3C2412yes—
200C2×C3⋊Dic3C2412yes—
201C2×C3⋊Dic3C2414yes—
202C2×C3×Dic3—24no—
203C2×C3×Dic3—22no—
204C2×C3×Dic3—24no—
205C2×C3×Dic3—22no—
206C2×C3×Dic3C2×Q814yes—
207C2×C3×Dic3C2412yes—
208C2×C3×Dic3C2412yes—
209C2×C3×Dic3C2414yes—
210C2×C3×Dic3C2414yes—
211C2×C3×Dic3C2412yes—
212C3×C4×D3C2×Q8116yescomplemented
213#627—48no—
214#627—48no—
215D3×Dic3—48no—
216#544—916no—
217C2×C4⋊Q8—92no—
218C2×C4⋊Q8—91no—
219C2×C4⋊Q8—91no—
220C2×#125—31no—
221C2×#125—92no—
222C2×#125—91no—
223C2×C22⋊Q8—92no—
224C2×C22⋊Q8—92no—
225C2×#117—31no—
226C2×#117—91no—
227C22×C4⋊C4—91no—
228C24×C3—31no—
229C23×D3C22×D311yescharacteristic
230C22×C3×C4—21no—
231C22×C3×C4—22no—
232C22×C3×C4—31no—
233C22×C3×C4—38no—
234C22×C3×C4—34no—
235C22×C3×C4—32no—
236C22×C3×C4—32no—
237C22×C3×C4—31no—
238C22×C3×C4—31no—
239C22×C3×C4C22×D312yes—
240C22×C3×C4C22×D312yes—
241C22×C3×C4C22×D311yescharacteristic
242C22×C3×C4C22×D311yescharacteristic
243C22×Dic3—31no—
244C22×Dic3—31no—
245C22×Dic3—31no—
246C22×Dic3—31no—
247C22×Dic3—32no—
248C22×Dic3—32no—
249C22×Dic3—38no—
250C22×Dic3—61no—
251C22×Dic3—61no—
252C22×Dic3—62no—
253C22×Dic3—124no—
254C22×Dic3C22×D311yescharacteristic
255C22×Dic3C22×D312yes—
256C22×Dic3C22×D311yescharacteristic
257C22×Dic3C22×D312yes—
258C22×Dic3C22×D311yescharacteristic
259C2×C4×D3—38no—
260C2×C4×D3—124no—
261C2×C4×D3Dic618yescomplemented
262C2×C4×D3C22×D314yes—
263C2×Dic6—38no—
264C2×Dic6—38no—
265C2×Dic6—38no—
266C2×Dic6—38no—
267C2×Dic6—124no—
268C2×Dic6—124no—
269C2×Dic6—124no—
270C2×Dic6—124no—
271C3×C4⋊C4—24no—
272C3×C4⋊C4—34no—
273C3×C4⋊C4—34no—
274C3×C4⋊C4—38no—
275C3×C4⋊C4—38no—
276C3×C4⋊C4C22×D314yes—
277C3×C4⋊C4C22×D314yes—
278C3×C22⋊C4—38no—
279C3×C42—38no—
280C3×C42—34no—
281#217—34no—
282#217—34no—
283(C2×D3)⋊C4—34no—
284(C2×D3)⋊C4—34no—
285(C2×D3)⋊C4C22×D318yes—
286C4⋊Dic3—38no—
287C4⋊Dic3—34no—
288C4⋊Dic3—38no—
289C4⋊Dic3—34no—
290C4⋊Dic3—34no—
291C4⋊Dic3—38no—
292C4⋊Dic3—38no—
293C4⋊Dic3—34no—
294C4⋊Dic3—64no—
295C4⋊Dic3—68no—
296C4⋊Dic3C22×D318yes—
297C4⋊Dic3C22×D314yes—
298C4⋊Dic3C22×D314yes—
299Dic3⋊C4—38no—
300Dic3⋊C4—38no—
301Dic3⋊C4—38no—
302Dic3⋊C4—38no—
303Dic3⋊C4—34no—
304Dic3⋊C4—34no—
305Dic3⋊C4—38no—
306Dic3⋊C4—38no—
307Dic3⋊C4—68no—
308Dic3⋊C4C22×D318yes—
309C4×Dic3—34no—
310C4×Dic3—34no—
311C4×Dic3—64no—
312C4×Dic3C22×D318yes—
313C4×Dic3C22×D314yes—
314C22×C32C22×Q812yes—
315C22×C32H(5,𝔽2)12yes—
316C22×C322−1+412yes—
317C22×C32C2511yescharacteristic
318C2×C3×D3—28no—
319C2×C3×D3—22no—
320C2×C3×D3—22no—
321C2×C3×D3C22×Q814yes—
322C32×C4—24no—
323C32×C4C22×Q814yes—
324C3⋊Dic3—24no—
325C3⋊Dic3—22no—
326C3⋊Dic3—22no—
327C3×Dic3—24no—
328C3×Dic3—22no—
329C3×Dic3—24no—
330C3×Dic3—22no—
331C3×Dic3C22×Q814yes—
332C22×Q8—92no—
333C22×Q8—92no—
334C23×C4—31no—
335C23×C4—92no—
336C4⋊Q8—916no—
337C4⋊Q8—98no—
338C4⋊Q8—98no—
339#125—38no—
340#125—916no—
341#125—98no—
342C22⋊Q8—916no—
343C22⋊Q8—916no—
344#117—38no—
345#117—98no—
346C2×C4⋊C4—31no—
347C2×C4⋊C4—31no—
348C2×C4⋊C4—32no—
349C2×C4⋊C4—32no—
350C2×C4⋊C4—98no—
351C2×C4⋊C4—92no—
352C2×C4⋊C4—91no—
353C2×C4⋊C4—91no—
354C2×C4⋊C4—92no—
355C2×C4⋊C4—92no—
356C2×C4⋊C4—91no—
357C2×C4⋊C4—92no—
358C2×C4⋊C4—92no—
359C2×C4⋊C4D3211yescharacteristic
360C2×C22⋊C4—32no—
361C2×C22⋊C4—91no—
362C2×C22⋊C4—91no—
363C2×C42—32no—
364C2×C42—31no—
365C2×C42—91no—
366C23×C3—21no—
367C23×C3—34no—
368C23×C3—32no—
369C23×C3—62no—
370C23×C3—62no—
371C23×C3C23×D311yescharacteristic
372C23×C3C23×D311yescharacteristic
373C22×D3—22no—
374C22×D3—22no—
375C22×D3C2×Dic614yes—
376C22×D3C23×D312yes—
377C2×C3×C4—24no—
378C2×C3×C4—24no—
379C2×C3×C4—22no—
380C2×C3×C4—24no—
381C2×C3×C4—24no—
382C2×C3×C4—34no—
383C2×C3×C4—32no—
384C2×C3×C4—316no—
385C2×C3×C4—34no—
386C2×C3×C4—34no—
387C2×C3×C4—32no—
388C2×C3×C4—32no—
389C2×C3×C4—44no—
390C2×C3×C4—64no—
391C2×C3×C4—62no—
392C2×C3×C4—64no—
393C2×C3×C4—62no—
394C2×C3×C4C2×Dic614yes—
395C2×C3×C4D3×Q814yes—
396C2×C3×C4C23×D314yes—
397C2×C3×C4C23×D314yes—
398C2×C3×C4C23×D312yes—
399C2×C3×C4C23×D312yes—
400C2×Dic3—22no—
401C2×Dic3—24no—
402C2×Dic3—24no—
403C2×Dic3—22no—
404C2×Dic3—32no—
405C2×Dic3—32no—
406C2×Dic3—32no—
407C2×Dic3—32no—
408C2×Dic3—34no—
409C2×Dic3—34no—
410C2×Dic3—62no—
411C2×Dic3—62no—
412C2×Dic3—64no—
413C2×Dic3—62no—
414C2×Dic3—62no—
415C2×Dic3—64no—
416C2×Dic3—64no—
417C2×Dic3—62no—
418C2×Dic3—62no—
419C2×Dic3—122no—
420C2×Dic3—124no—
421C2×Dic3—122no—
422C2×Dic3—128no—
423C2×Dic3C2×Dic614yes—
424C2×Dic3C23×D312yes—
425C2×Dic3C23×D314yes—
426C2×Dic3C23×D312yes—
427C2×Dic3C23×D314yes—
428C2×Dic3C23×D312yes—
429C4×D3—128no—
430C4×D3C2×Dic6116yescomplemented
431Dic6—128no—
432Dic6—128no—
433Dic6—128no—
434Dic6—128no—
435C2×C32#54414yescomplemented
436C2×C32C23×Q811yescharacteristic
437C2×C32C2×H(5,𝔽2)11yescharacteristic
438C2×C32C2×2−1+411yescharacteristic
439C3×D3—24no—
440C24—31no—
441C2×Q8—98no—
442C2×Q8—98no—
443C2×Q8—364no—
444C2×Q8—364no—
445C22×C4—31no—
446C22×C4—34no—
447C22×C4—38no—
448C22×C4—32no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—32no—
452C22×C4—98no—
453C22×C4—91no—
454C22×C4—91no—
455C22×C4—92no—
456C22×C4—364no—
457C22×C4C2×D3212yes—
458C22×C4C2×D3211yescharacteristic
459C4⋊C4—34no—
460C4⋊C4—34no—
461C4⋊C4—38no—
462C4⋊C4—38no—
463C4⋊C4—98no—
464C4⋊C4—94no—
465C4⋊C4—94no—
466C4⋊C4—98no—
467C4⋊C4—98no—
468C4⋊C4—94no—
469C4⋊C4—98no—
470C4⋊C4—98no—
471C4⋊C4C2×D3214yes—
472C22⋊C4—38no—
473C22⋊C4—94no—
474C22⋊C4—94no—
475C42—38no—
476C42—34no—
477C42—94no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—21no—
482C22×C3—34no—
483C22×C3—68no—
484C22×C3—62no—
485C22×C3—62no—
486C22×C3C22×Dic612yes—
487C22×C3C2×D3×Q812yes—
488C22×C3#100912yes—
489C22×C3#100912yes—
490C22×C3#101012yes—
491C22×C3#101012yes—
492C22×C3C24×D311yescharacteristic
493C22×C3C24×D311yescharacteristic
494C2×D3—22no—
495C2×D3—22no—
496C2×D3—28no—
497C2×D3C22×Dic614yes—
498C3×C4—24no—
499C3×C4—24no—
500C3×C4—24no—
501C3×C4—34no—
502C3×C4—44no—
503C3×C4—64no—
504C3×C4—62no—
505C3×C4—64no—
506C3×C4—62no—
507C3×C4C22×Dic614yes—
508C3×C4C2×D3×Q814yes—
509Dic3—22no—
510Dic3—24no—
511Dic3—24no—
512Dic3—22no—
513Dic3—62no—
514Dic3—62no—
515Dic3—64no—
516Dic3—64no—
517Dic3—62no—
518Dic3—62no—
519Dic3—122no—
520Dic3—124no—
521Dic3—122no—
522Dic3C22×Dic614yes—
523C32C2×#54411yescharacteristic, complemented, Sylow
524C23—34no—
525C23—32no—
526C23—62no—
527C23—62no—
528C23C22×D3211yescharacteristic
529Q8—368no—
530Q8—368no—
531C2×C4—24no—
532C2×C4—34no—
533C2×C4—32no—
534C2×C4—316no—
535C2×C4—34no—
536C2×C4—32no—
537C2×C4—32no—
538C2×C4—34no—
539C2×C4—62no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—62no—
543C2×C4—92no—
544C2×C4—92no—
545C2×C4—94no—
546C2×C4—182no—
547C2×C4—184no—
548C2×C4—182no—
549C2×C4—368no—
550C2×C4D3×Dic614yes—
551C2×C4C22×D3214yes—
552C2×C4C22×D3212yes—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—24no—
557C2×C3—64no—
558C2×C3#551914yescomplemented
559C2×C3#568914yescomplemented
560C2×C3C23×Dic611yescharacteristic
561C2×C3C22×D3×Q811yescharacteristic
562C2×C3C2×#100911yescharacteristic
563C2×C3C2×#100911yescharacteristic
564C2×C3C2×#101011yescharacteristic
565C2×C3C2×#101011yescharacteristic
566D3—24no—
567C22—34no—
568C22—62no—
569C22—62no—
570C22—68no—
571C22C2×D3×Dic612yes—
572C22#6430912yes—
573C22#6432012yes—
574C22C23×D3211yescharacteristic
575C4—24no—
576C4—34no—
577C4—62no—
578C4—64no—
579C4—64no—
580C4—62no—
581C4—182no—
582C4—184no—
583C4—182no—
584C4C2×D3×Dic614yes—
585C3—21no—
586C3C2×#551911yescharacteristic, complemented
587C3C2×#568911yescharacteristic, complemented
588C2—64no—
589C2#1059450514yesdirect product factor
590C2C22×D3×Dic611yescharacteristic
591C2C2×#6430911yescharacteristic
592C2C2×#6432011yescharacteristic
subgroupG/H
centerC23C22×D32
commutator, cosocleC22×C32C25
FrattiniC22C23×D32
FittingC2×C32×C4⋊C4C22
socleC23×C32C24
prime radicalC23×C3×D3C23
hypercenterC2×C4⋊C4D32
hypocenterC32C2×#544
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#64099C2semidirect
3. C2×C3×#890C2semidirect
4. C2×#63919C2semidirect
5. C2×#63918C2semidirect
6. C2×#63918C2semidirect
10. C2×#63867C2semidirect
12. #10594505C2direct (N × H)
24. #64099C22semidirect
33. C3×#890C22semidirect
41. #63919C22semidirect
42. #63918C22semidirect
43. #63918C22semidirect
47. #63867C22semidirect
62. C2×#890D3semidirect
85. C2×C3×C4×D3Q8semidirect
175. #890C2×D3semidirect
212. C3×C4×D3C2×Q8semidirect
261. C2×C4×D3Dic6semidirect
430. C4×D3C2×Dic6semidirect
435. C2×C32#544semidirect
523. C32C2×#544semidirect
558. C2×C3#5519semidirect
559. C2×C3#5689semidirect
586. C3C2×#5519semidirect
587. C3C2×#5689semidirect
589. C2#10594505direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#640992
C2×C3×#8922
C2×C3×#8902
C2×#639192
C2×#639182
C2×#638992
C2×#638932
C2×D3×C4⋊Dic32
C2×#638672
C2×#638552
#105945052
C2×#56893
C2×#55193