Finite Group Databasebeta

← C2×#10595348 #11901581

Subgroups — C2×#10595348 #11901581

Order 1152.

2153 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64153C211yesmaximal, characteristic, complemented
2C2×C3×#946C211yesmaximal, characteristic
3C2×C3×Q8×Dic3C211yesmaximal, characteristic, complemented
4#10595348C2116yesmaximal, direct product factor
5C2×#63872C211yesmaximal, characteristic, complemented
6C2×#63867C212yesmaximal, complemented
7C2×#63859C212yesmaximal
8C2×#63858C212yesmaximal
9C2×#63857C211yesmaximal, characteristic
10C2×#63855C211yesmaximal, characteristic
11C2×#63846C212yesmaximal, complemented
12C2×#63844C211yesmaximal, characteristic, complemented
13C2×#5625—31nomaximal
14C2×#5583—31nomaximal
15C22×C32×Q8C2211yescharacteristic
16C22×#634C2211yescharacteristic
17C22×C3×C4×D3C2211yescharacteristic
18#64153C2218yescomplemented
19C2×#3750C2212yes—
20C2×#3750C2212yes—
21C2×C4×C3⋊Dic3C2211yescharacteristic
22C3×#946C2218yes—
23C3×Q8×Dic3C2218yescomplemented
24C2×C3×(C2×D3)⋊C4C2212yes—
25C2×C3×C4⋊Dic3C2211yescharacteristic
26C2×C3×C4⋊Dic3C2212yes—
27C2×C3×C4⋊Dic3C2211yescharacteristic
28C2×C3×Dic3⋊C4C2212yes—
29C2×C3×C4×Dic3C2211yescharacteristic
30C2×C3×C4×Dic3C2212yes—
31C2×#3724C2212yes—
32C2×Dic3⋊Dic3C2211yescharacteristic
33C2×Dic3⋊Dic3C2212yes—
34C2×Dic3⋊Dic3C2212yes—
35C2×Dic3⋊Dic3C2211yescharacteristic
36C2×Dic3⋊Dic3C2212yes—
37C2×(C2×D3)⋊Dic3C2211yescharacteristic
38C2×(C2×D3)⋊Dic3C2211yescharacteristic
39C2×(C2×D3)⋊Dic3C2212yes—
40C2×Dic32C2212yes—
41#63872C2218yescomplemented
42#63867C22116yescomplemented
43#63859C22116yes—
44#63858C22116yes—
45#63857C2218yes—
46#63855C2218yes—
47#63846C22116yescomplemented
48#63844C2218yescomplemented
49C2×C3×C22⋊Q8—31no—
50C2×C3×C4×Q8—31no—
51C2×#946D311yescharacteristic
52C2×Q8×Dic3D311yescharacteristic, complemented
53C2×#944—31no—
54C2×#944—31no—
55C2×#944—61no—
56C2×#926—31no—
57C2×#924—31no—
58#5625—316no—
59#5583—316no—
60C2×#898—32no—
61C2×#897—31no—
62C2×#890—32no—
63C2×#890—31no—
64C2×#890—31no—
65C2×#889—32no—
66C2×#889—32no—
67C2×#887—31no—
68C2×#879—32no—
69C2×#879—32no—
70C2×#876—32no—
71C2×#872—31no—
72C2×#870—32no—
73C2×C4×Dic6—31no—
74C23×C3×D3C2311yescharacteristic
75C2×C32×Q8—24no—
76C2×C32×Q8C2314yes—
77C22×C32×C4C2312yes—
78C22×C32×C4C2311yescharacteristic
79C22×C3⋊Dic3C2312yes—
80C22×C3⋊Dic3C2311yescharacteristic
81C22×C3⋊Dic3C2311yescharacteristic
82C2×#634—24no—
83C2×#634C2314yes—
84C22×C3×Dic3C2311yescharacteristic
85C22×C3×Dic3C2311yescharacteristic
86C22×C3×Dic3C2312yes—
87C22×C3×Dic3C2311yescharacteristic
88C22×C3×Dic3C2312yes—
89C2×C3×C4×D3—24no—
90C2×C3×C4×D3C2314yes—
91#3750C2318yes—
92#3750C2318yes—
93C4×C3⋊Dic3C2314yes—
94C3×(C2×D3)⋊C4C2318yes—
95C3×C4⋊Dic3C2318yes—
96C3×C4⋊Dic3C2314yes—
97C3×C4⋊Dic3C2314yes—
98C3×Dic3⋊C4C2318yes—
99C3×C4×Dic3C2314yes—
100C3×C4×Dic3C2318yes—
101#3724C2318yes—
102Dic3⋊Dic3C2314yes—
103Dic3⋊Dic3C2318yes—
104Dic3⋊Dic3C2318yes—
105Dic3⋊Dic3C2314yes—
106Dic3⋊Dic3C2318yes—
107(C2×D3)⋊Dic3C2314yes—
108(C2×D3)⋊Dic3C2314yes—
109(C2×D3)⋊Dic3C2318yes—
110Dic32C2318yes—
111C2×#541—91noSylow
112C22×C3×Q8—21no—
113C22×C3×Q8C2×D311yescharacteristic
114C22×C3×Q8C2×D311yescharacteristic
115C23×C3×C4—31no—
116C22×C4×D3C2×D311yescharacteristic
117C22×Dic6—31no—
118C22×Dic6—31no—
119C22×Dic6—61no—
120C3×C22⋊Q8—38no—
121C3×C4×Q8—38no—
122C2×C3×C4⋊C4—32no—
123C2×C3×C4⋊C4—31no—
124C2×C3×C4⋊C4—32no—
125C2×C3×C4⋊C4—31no—
126C2×C3×C22⋊C4—32no—
127C2×C3×C42—31no—
128C2×C3×C42—32no—
129C2×#217—32no—
130C2×#217—31no—
131C2×#217—31no—
132#946C2×D318yes—
133Q8×Dic3C2×D318yescomplemented
134#944—38no—
135#944—38no—
136#944—68no—
137C2×(C2×D3)⋊C4—31no—
138C2×(C2×D3)⋊C4—31no—
139C2×(C2×D3)⋊C4—32no—
140C2×(C2×D3)⋊C4C2×D312yes—
141#926—38no—
142C2×C4⋊Dic3—32no—
143C2×C4⋊Dic3—32no—
144C2×C4⋊Dic3—32no—
145C2×C4⋊Dic3—31no—
146C2×C4⋊Dic3—31no—
147C2×C4⋊Dic3C2×D311yescharacteristic
148C2×C4⋊Dic3C2×D312yes—
149C2×C4⋊Dic3C2×D311yescharacteristic
150#924—38no—
151C2×Dic3⋊C4—32no—
152C2×Dic3⋊C4—32no—
153C2×Dic3⋊C4—32no—
154C2×Dic3⋊C4—32no—
155C2×Dic3⋊C4—32no—
156C2×Dic3⋊C4—31no—
157C2×Dic3⋊C4—31no—
158C2×Dic3⋊C4—32no—
159C2×Dic3⋊C4—32no—
160C2×Dic3⋊C4—32no—
161C2×Dic3⋊C4—32no—
162C2×Dic3⋊C4—62no—
163C2×Dic3⋊C4—62no—
164C2×Dic3⋊C4C2×D312yes—
165C2×C4×Dic3—31no—
166C2×C4×Dic3—31no—
167C2×C4×Dic3—32no—
168C2×C4×Dic3—32no—
169C2×C4×Dic3—61no—
170C2×C4×Dic3C2×D312yes—
171C2×C4×Dic3C2×D311yescharacteristic
172#898—316no—
173#897—38no—
174#890—316no—
175#890—38no—
176#890—38no—
177#889—316no—
178#889—316no—
179#887—38no—
180#879—316no—
181#879—316no—
182#876—316no—
183#872—38no—
184#870—316no—
185C4×Dic6—38no—
186C23×C32C2411yescharacteristic
187C22×C3×D3—22no—
188C22×C3×D3—22no—
189C22×C3×D3—22no—
190C22×C3×D3C2412yes—
191C32×Q8—28no—
192C2×C32×C4—24no—
193C2×C32×C4#4114yes—
194C2×C32×C4C2414yes—
195C2×C32×C4C2412yes—
196C2×C3⋊Dic3—22no—
197C2×C3⋊Dic3—22no—
198C2×C3⋊Dic3—24no—
199C2×C3⋊Dic3C2414yes—
200C2×C3⋊Dic3C2412yes—
201C2×C3⋊Dic3C2412yes—
202#634—28no—
203C2×C3×Dic3—22no—
204C2×C3×Dic3—22no—
205C2×C3×Dic3—22no—
206C2×C3×Dic3—24no—
207C2×C3×Dic3—24no—
208C2×C3×Dic3C2412yes—
209C2×C3×Dic3C2414yes—
210C2×C3×Dic3C2414yes—
211C2×C3×Dic3C2412yes—
212C2×C3×Dic3C2412yes—
213C3×C4×D3—28no—
214#541—916no—
215C2×C4⋊Q8—91no—
216C2×#126—92no—
217C2×#126—92no—
218C2×#125—92no—
219C2×#125—92no—
220C2×#125—91no—
221C2×C22⋊Q8—31no—
222C2×C22⋊Q8—91no—
223C2×C4×Q8—31no—
224C2×C4×Q8—91no—
225C2×#117—91no—
226C24×C3—31no—
227C23×D3C22×D311yescharacteristic
228C2×C3×Q8—24no—
229C2×C3×Q8—24no—
230C2×C3×Q8—24no—
231C2×C3×Q8—44no—
232C2×C3×Q8C22×D314yes—
233C2×C3×Q8C22×D314yes—
234C22×C3×C4—21no—
235C22×C3×C4—22no—
236C22×C3×C4—31no—
237C22×C3×C4—32no—
238C22×C3×C4—31no—
239C22×C3×C4—31no—
240C22×C3×C4—32no—
241C22×C3×C4—34no—
242C22×C3×C4—64no—
243C22×C3×C4C22×D312yes—
244C22×C3×C4C22×D312yes—
245C22×C3×C4C22×D311yescharacteristic
246C22×C3×C4C22×D311yescharacteristic
247C22×Dic3—31no—
248C22×Dic3—31no—
249C22×Dic3—32no—
250C22×Dic3—31no—
251C22×Dic3—32no—
252C22×Dic3—31no—
253C22×Dic3—61no—
254C22×Dic3—61no—
255C22×Dic3—62no—
256C22×Dic3C22×D311yescharacteristic
257C22×Dic3C22×D312yes—
258C22×Dic3C22×D311yescharacteristic
259C22×Dic3C22×D312yes—
260C22×Dic3C22×D311yescharacteristic
261C2×C4×D3—24no—
262C2×C4×D3C22×D314yes—
263C2×Dic6—34no—
264C2×Dic6—34no—
265C2×Dic6—64no—
266C2×Dic6—64no—
267C2×Dic6—64no—
268C2×Dic6—124no—
269C3×C4⋊C4—34no—
270C3×C4⋊C4—38no—
271C3×C4⋊C4—38no—
272C3×C4⋊C4—34no—
273C3×C22⋊C4—38no—
274C3×C42—34no—
275C3×C42—38no—
276#217—38no—
277#217—34no—
278#217—34no—
279(C2×D3)⋊C4—34no—
280(C2×D3)⋊C4—34no—
281(C2×D3)⋊C4—38no—
282(C2×D3)⋊C4C22×D318yes—
283C4⋊Dic3—38no—
284C4⋊Dic3—38no—
285C4⋊Dic3—38no—
286C4⋊Dic3—34no—
287C4⋊Dic3—34no—
288C4⋊Dic3C22×D314yes—
289C4⋊Dic3C22×D318yes—
290C4⋊Dic3C22×D314yes—
291Dic3⋊C4—38no—
292Dic3⋊C4—38no—
293Dic3⋊C4—38no—
294Dic3⋊C4—38no—
295Dic3⋊C4—38no—
296Dic3⋊C4—34no—
297Dic3⋊C4—34no—
298Dic3⋊C4—38no—
299Dic3⋊C4—38no—
300Dic3⋊C4—38no—
301Dic3⋊C4—38no—
302Dic3⋊C4—68no—
303Dic3⋊C4—68no—
304Dic3⋊C4C22×D318yes—
305C4×Dic3—34no—
306C4×Dic3—34no—
307C4×Dic3—38no—
308C4×Dic3—38no—
309C4×Dic3—64no—
310C4×Dic3C22×D318yes—
311C4×Dic3C22×D314yes—
312C22×C32C2×#4112yes—
313C22×C322−1+412yes—
314C22×C322−1+412yes—
315C22×C32C2511yescharacteristic
316C2×C3×D3—22no—
317C2×C3×D3—22no—
318C2×C3×D3—22no—
319C2×C3×D3—44no—
320C32×C4—24no—
321C32×C4C2×#4114yes—
322C3⋊Dic3—22no—
323C3⋊Dic3—22no—
324C3⋊Dic3—24no—
325C3×Dic3—22no—
326C3×Dic3—22no—
327C3×Dic3—22no—
328C3×Dic3—24no—
329C3×Dic3—24no—
330C22×Q8—91no—
331C22×Q8D3211yescharacteristic
332C23×C4—31no—
333C4⋊Q8—98no—
334#126—916no—
335#126—916no—
336#125—916no—
337#125—916no—
338#125—98no—
339C22⋊Q8—38no—
340C22⋊Q8—98no—
341C4×Q8—38no—
342C4×Q8—98no—
343#117—98no—
344C2×C4⋊C4—31no—
345C2×C4⋊C4—32no—
346C2×C4⋊C4—31no—
347C2×C4⋊C4—32no—
348C2×C4⋊C4—91no—
349C2×C4⋊C4—91no—
350C2×C4⋊C4—92no—
351C2×C4⋊C4—92no—
352C2×C4⋊C4—92no—
353C2×C4⋊C4—92no—
354C2×C4⋊C4—92no—
355C2×C4⋊C4—92no—
356C2×C22⋊C4—32no—
357C2×C22⋊C4—91no—
358C2×C22⋊C4—91no—
359C2×C22⋊C4—92no—
360C2×C42—31no—
361C2×C42—32no—
362C2×C42—92no—
363C2×C42—91no—
364C23×C3—21no—
365C23×C3—32no—
366C23×C3—62no—
367C23×C3—62no—
368C23×C3—62no—
369C23×C3C23×D311yescharacteristic
370C23×C3C23×D311yescharacteristic
371C22×D3—22no—
372C22×D3—22no—
373C22×D3—22no—
374C22×D3C23×D312yes—
375C3×Q8—28no—
376C3×Q8—28no—
377C3×Q8—48no—
378C2×C3×C4—24no—
379C2×C3×C4—24no—
380C2×C3×C4—22no—
381C2×C3×C4—24no—
382C2×C3×C4—24no—
383C2×C3×C4—32no—
384C2×C3×C4—34no—
385C2×C3×C4—32no—
386C2×C3×C4—32no—
387C2×C3×C4—34no—
388C2×C3×C4—44no—
389C2×C3×C4—62no—
390C2×C3×C4—64no—
391C2×C3×C4—62no—
392C2×C3×C4—64no—
393C2×C3×C4—62no—
394C2×C3×C4—68no—
395C2×C3×C4#23514yes—
396C2×C3×C4#23714yes—
397C2×C3×C4C23×D314yes—
398C2×C3×C4C23×D314yes—
399C2×C3×C4C23×D312yes—
400C2×C3×C4C23×D312yes—
401C2×Dic3—22no—
402C2×Dic3—22no—
403C2×Dic3—22no—
404C2×Dic3—24no—
405C2×Dic3—24no—
406C2×Dic3—32no—
407C2×Dic3—32no—
408C2×Dic3—34no—
409C2×Dic3—32no—
410C2×Dic3—34no—
411C2×Dic3—32no—
412C2×Dic3—62no—
413C2×Dic3—62no—
414C2×Dic3—64no—
415C2×Dic3—64no—
416C2×Dic3—64no—
417C2×Dic3—64no—
418C2×Dic3—64no—
419C2×Dic3—62no—
420C2×Dic3—62no—
421C2×Dic3—62no—
422C2×Dic3—62no—
423C2×Dic3—124no—
424C2×Dic3C23×D312yes—
425C2×Dic3C23×D314yes—
426C2×Dic3C23×D312yes—
427C2×Dic3C23×D314yes—
428C2×Dic3C23×D312yes—
429C4×D3—28no—
430Dic6—68no—
431Dic6—68no—
432Dic6—128no—
433C2×C32#54114yescomplemented
434C2×C32C22×#4111yescharacteristic
435C2×C32C2×2−1+411yescharacteristic
436C2×C32C2×2−1+411yescharacteristic
437C3×D3—42no—
438C24—31no—
439C2×Q8—24no—
440C2×Q8—94no—
441C2×Q8—184no—
442C2×Q8C2×D3214yes—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—32no—
446C22×C4—32no—
447C22×C4—31no—
448C22×C4—34no—
449C22×C4—64no—
450C22×C4—91no—
451C22×C4—91no—
452C22×C4—92no—
453C22×C4C2×D3212yes—
454C22×C4C2×D3211yescharacteristic
455C4⋊C4—34no—
456C4⋊C4—38no—
457C4⋊C4—34no—
458C4⋊C4—38no—
459C4⋊C4—94no—
460C4⋊C4—94no—
461C4⋊C4—98no—
462C4⋊C4—98no—
463C4⋊C4—98no—
464C4⋊C4—98no—
465C4⋊C4—98no—
466C4⋊C4—98no—
467C22⋊C4—38no—
468C22⋊C4—94no—
469C22⋊C4—94no—
470C22⋊C4—98no—
471C42—34no—
472C42—38no—
473C42—98no—
474C42—94no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—21no—
479C22×C3—62no—
480C22×C3—62no—
481C22×C3—62no—
482C22×C3—124no—
483C22×C3C2×#23512yes—
484C22×C3C2×#23712yes—
485C22×C3#100712yes—
486C22×C3#100712yes—
487C22×C3#101012yes—
488C22×C3#101012yes—
489C22×C3C24×D311yescharacteristic
490C22×C3C24×D311yescharacteristic
491C2×D3—22no—
492C2×D3—22no—
493C2×D3—22no—
494C2×D3—44no—
495C3×C4—24no—
496C3×C4—24no—
497C3×C4—24no—
498C3×C4—44no—
499C3×C4—62no—
500C3×C4—64no—
501C3×C4—62no—
502C3×C4—64no—
503C3×C4—62no—
504C3×C4C2×#23514yes—
505C3×C4C2×#23714yes—
506Dic3—22no—
507Dic3—22no—
508Dic3—22no—
509Dic3—24no—
510Dic3—24no—
511Dic3—64no—
512Dic3—64no—
513Dic3—64no—
514Dic3—64no—
515Dic3—62no—
516Dic3—62no—
517Dic3—62no—
518Dic3—62no—
519Dic3—124no—
520C32C2×#54111yescharacteristic, complemented, Sylow
521C23—32no—
522C23—62no—
523C23—62no—
524C23—62no—
525C23C22×D3211yescharacteristic
526Q8—28no—
527Q8—188no—
528C2×C4—24no—
529C2×C4—32no—
530C2×C4—32no—
531C2×C4—34no—
532C2×C4—34no—
533C2×C4—32no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—64no—
537C2×C4—64no—
538C2×C4—62no—
539C2×C4—68no—
540C2×C4—92no—
541C2×C4—92no—
542C2×C4—94no—
543C2×C4—182no—
544C2×C4—182no—
545C2×C4—184no—
546C2×C4#380414yes—
547C2×C4C22×D3214yes—
548C2×C4C22×D3212yes—
549C2×C3—24no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—122no—
554C2×C3#558314yescomplemented
555C2×C3#562514yescomplemented
556C2×C3C22×#23511yescharacteristic
557C2×C3C22×#23711yescharacteristic
558C2×C3C2×#100711yescharacteristic
559C2×C3C2×#100711yescharacteristic
560C2×C3C2×#101011yescharacteristic
561C2×C3C2×#101011yescharacteristic
562D3—42no—
563C22—62no—
564C22—62no—
565C22—62no—
566C22—124no—
567C22#6431512yes—
568C22#6431512yes—
569C22C2×#380412yes—
570C22C23×D3211yescharacteristic
571C4—24no—
572C4—62no—
573C4—62no—
574C4—64no—
575C4—64no—
576C4—62no—
577C4—182no—
578C4—182no—
579C4—184no—
580C4C2×#380414yes—
581C3—21no—
582C3C2×#558311yescharacteristic, complemented
583C3C2×#562511yescharacteristic, complemented
584C2—122no—
585C2#1059534814yesdirect product factor
586C2C2×#6431511yescharacteristic
587C2C2×#6431511yescharacteristic
588C2C22×#380411yescharacteristic
subgroupG/H
centerC23C22×D32
commutator, cosocleC22×C32C25
FrattiniC22C23×D32
FittingC22×C32×Q8C22
socleC23×C32C24
prime radicalC23×C3×D3C23
hypercenterC22×Q8D32
hypocenterC32C2×#541
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#64153C2semidirect
3. C2×C3×Q8×Dic3C2semidirect
4. #10595348C2direct (N × H)
5. C2×#63872C2semidirect
6. C2×#63867C2semidirect
11. C2×#63846C2semidirect
12. C2×#63844C2semidirect
18. #64153C22semidirect
23. C3×Q8×Dic3C22semidirect
41. #63872C22semidirect
42. #63867C22semidirect
47. #63846C22semidirect
48. #63844C22semidirect
52. C2×Q8×Dic3D3semidirect
133. Q8×Dic3C2×D3semidirect
433. C2×C32#541semidirect
520. C32C2×#541semidirect
554. C2×C3#5583semidirect
555. C2×C3#5625semidirect
582. C3C2×#5583semidirect
583. C3C2×#5625semidirect
585. C2#10595348direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#641532
C2×C3×#9462
C2×C3×Q8×Dic32
#105953482
C2×#638722
C2×#638672
C2×#638592
C2×#638582
C2×#638572
C2×#638552
C2×#638462
C2×#638442
C2×#56253
C2×#55833