Finite Group Databasebeta

← C3×Q8×#235 #11903746

Subgroups — C3×Q8×#235 #11903746

Order 1152.

2285 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Q8C211yesmaximal, characteristic, complemented
2C3×Q8×#67C212yesmaximal, complemented
3C3×#5579C213yesmaximal, complemented
4C3×#5578C216yesmaximal, complemented
5C3×Q8×D12C211yesmaximal, characteristic, complemented
6C3×C4×D3×Q8C212yesmaximal, complemented
7C3×#5567C213yesmaximal, complemented
8C3×Q8×Dic6C211yesmaximal, characteristic, complemented
9C3×#5511C216yesmaximal, complemented
10C3×C4×#235C213yesmaximal
11C3×#5470C213yesmaximal, complemented
12C3×Q8×#41—31nomaximal
13Q8×#235C311yesmaximal, characteristic, direct product factor
14C22×C32×Q8C2211yescharacteristic
15C2×C3×D3×Q8C2212yes—
16C2×C3×#235C2211yescharacteristic
17C32×C4×Q8C2211yescharacteristic
18C32×C4×Q8C2213yes—
19C32×C4×Q8C2211yescharacteristic
20C32×C4×Q8C2213yes—
21C2×C32×C4⋊C4C2213yes—
22C2×C32×C42C2213yes—
23C3×#946C2216yes—
24C3×Q8×Dic3C2212yes—
25C3×#944C2216yes—
26C3×C4×#67C2216yes—
27C3×C4×#67C2216yes—
28C3×#926C2213yes—
29C3×#924C2216yes—
30C3×#924C2216yes—
31C3×#897C2216yes—
32C3×#896C22112yes—
33C3×D3×C4⋊C4C2216yes—
34C3×#889C22112yes—
35C3×#888C2216yes—
36C3×#887C2216yes—
37C3×C4×D12C2213yes—
38C3×C4×D12C2213yes—
39C3×#872C2216yes—
40C3×C42×D3C2216yes—
41C3×#870C2216yes—
42C3×#869C2213yes—
43C3×#869C2213yes—
44C3×C4×Dic6C2213yes—
45C3×C4×Dic6C2213yes—
46C3×C4×Dic6C2216yes—
47C3×C4×Dic6C2216yes—
48C3×Q82—31no—
49C3×#557—33no—
50C3×#554—33no—
51C3×#554—36no—
52C3×D4×Q8—32no—
53C3×D4×Q8—31no—
54C3×#533—36no—
55C3×#533—33no—
56C3×C4×#41—33no—
57C2×C3×C4×Q8—21no—
58C2×C3×C4×Q8—32no—
59C2×C3×C4×Q8D311yescharacteristic, complemented
60C2×C3×C4×Q8C2×C311yescharacteristic, complemented
61Q8×#67C2×C312yescomplemented
62#5579C2×C313yescomplemented
63#5578C2×C316yescomplemented
64Q8×D12C2×C311yescharacteristic, complemented
65C4×D3×Q8C2×C312yescomplemented
66#5567C2×C313yescomplemented
67Q8×Dic6C2×C311yescharacteristic, complemented
68#5511C2×C316yescomplemented
69C4×#235C2×C313yes—
70#5470C2×C313yescomplemented
71C2×C32×Q8—24no—
72C2×C32×Q8C2313yes—
73C2×C32×Q8C2311yescharacteristic
74C22×C32×C4C2313yes—
75C22×C32×C4C2313yes—
76C22×C32×C4C2311yescharacteristic
77C2×C3×#67C2312yes—
78C3×D3×Q8—28no—
79C3×#235Q818yesdirect product factor
80C2×C3×D12C2311yescharacteristic
81C2×C3×C4×D3C2316yes—
82C2×C3×C4×D3C2316yes—
83C2×C3×C4×D3C2312yes—
84C2×C3×Dic6C2316yes—
85C2×C3×Dic6C2316yes—
86C2×C3×Dic6C2311yescharacteristic
87C32×C4⋊C4C2313yes—
88C32×C4⋊C4C2313yes—
89C32×C4⋊C4C2316yes—
90C32×C42C2313yes—
91C32×C42C2313yes—
92C32×C42C2313yes—
93C32×C42C2313yes—
94C3×#217C2316yes—
95C3×(C2×D3)⋊C4C2316yes—
96C3×(C2×D3)⋊C4C2316yes—
97C3×C4⋊Dic3C2316yes—
98C3×C4⋊Dic3C2316yes—
99C3×C4⋊Dic3C2313yes—
100C3×C4⋊Dic3C2313yes—
101C3×Dic3⋊C4C23112yes—
102C3×Dic3⋊C4C2316yes—
103C3×Dic3⋊C4C2316yes—
104C3×Dic3⋊C4C23112yes—
105C3×C4×Dic3C2316yes—
106C3×C4×Dic3C2316yes—
107C3×C4×Dic3C2316yes—
108Q8×#41—31noSylow
109C2×C3×#41—31no—
110C22×C3×Q8—21no—
111C22×C3×Q8—32no—
112C22×C3×Q8C2×D311yescharacteristic
113C22×C3×Q8C22×C311yescharacteristic
114C2×D3×Q8C22×C312yes—
115C2×#235C22×C311yescharacteristic
116C3×C4⋊Q8—36no—
117C3×C4⋊Q8—36no—
118C3×C4⋊Q8—33no—
119C3×C4⋊Q8—33no—
120C3×#125—312no—
121C3×#125—36no—
122C3×C22⋊Q8—36no—
123C3×C22⋊Q8—36no—
124C3×C22⋊Q8—36no—
125C3×C22⋊Q8—312no—
126C3×C22⋊Q8—36no—
127C3×C4×Q8—21no—
128C3×C4×Q8—23no—
129C3×C4×Q8—21no—
130C3×C4×Q8—23no—
131C3×C4×Q8—32no—
132C3×C4×Q8—36no—
133C3×C4×Q8—33no—
134C3×C4×Q8—36no—
135C3×C4×Q8—33no—
136C3×C4×Q8—36no—
137C3×C4×Q8C2×D311yescharacteristic
138C3×C4×Q8C2×D311yescharacteristic
139C3×C4×Q8C2×D313yes—
140C3×C4×Q8C2×D313yes—
141C3×C4×Q8C22×C311yescharacteristic
142C3×C4×Q8C22×C313yes—
143C3×C4×Q8C22×C311yescharacteristic
144C3×C4×Q8C22×C313yes—
145C3×C4×D4—36no—
146C3×C4×D4—36no—
147C3×C4×D4—33no—
148C3×C4×D4—33no—
149C3×#117—33no—
150C3×#117—36no—
151C2×C3×C4⋊C4—23no—
152C2×C3×C4⋊C4—36no—
153C2×C3×C4⋊C4C2×D313yes—
154C2×C3×C4⋊C4C22×C313yes—
155C2×C3×C42—23no—
156C2×C3×C42—36no—
157C2×C3×C42C2×D313yes—
158C2×C3×C42C22×C313yes—
159#946C22×C316yes—
160Q8×Dic3C22×C312yes—
161#944C22×C316yes—
162C4×#67C22×C316yes—
163C4×#67C22×C316yes—
164#926C22×C313yes—
165#924C22×C316yes—
166#924C22×C316yes—
167#897C22×C316yes—
168#896C22×C3112yes—
169D3×C4⋊C4C22×C316yes—
170#889C22×C3112yes—
171#888C22×C316yes—
172#887C22×C316yes—
173C4×D12C22×C313yes—
174C4×D12C22×C313yes—
175#872C22×C316yes—
176C42×D3C22×C316yes—
177#870C22×C316yes—
178#869C22×C313yes—
179#869C22×C313yes—
180C4×Dic6C22×C316yes—
181C4×Dic6C22×C316yes—
182C4×Dic6C22×C313yes—
183C4×Dic6C22×C313yes—
184C23×C32C2411yescharacteristic
185C22×C3×D3C2412yes—
186C32×Q8—26no—
187C32×Q8#4114yescomplemented
188C2×C32×C4—26no—
189C2×C32×C4—43no—
190C2×C32×C4C2×Q814yescomplemented
191C2×C32×C4C2413yes—
192C2×C32×C4C2413yes—
193C2×C32×C4C2413yes—
194C2×C32×C4C2411yescharacteristic
195C2×C32×C4C2413yes—
196C2×C32×C4C2411yescharacteristic
197C3×#67C2×Q818yescomplemented
198C2×C3×Dic3C2416yes—
199C2×C3×Dic3C2416yes—
200C2×C3×Dic3C2412yes—
201C3×D12C2×Q814yescomplemented
202C3×C4×D3—212no—
203C3×C4×D3—46no—
204C3×C4×D3C2×Q818yescomplemented
205C3×Dic6—212no—
206C3×Dic6—46no—
207C3×Dic6C2×Q814yescomplemented
208Q82—31no—
209#557—33no—
210#554—33no—
211#554—36no—
212D4×Q8—32no—
213D4×Q8—31no—
214#533—36no—
215#533—33no—
216C4×#41—33no—
217C2×C4×Q8—32no—
218C2×C4×Q8C3×D311yescharacteristic, complemented
219C3×#41—38no—
220C2×C3×Q8—28no—
221C2×C3×Q8—24no—
222C2×C3×Q8—24no—
223C2×C3×Q8—23no—
224C2×C3×Q8—21no—
225C2×C3×Q8—31no—
226C2×C3×Q8—36no—
227C2×C3×Q8—36no—
228C2×C3×Q8—68no—
229C2×C3×Q8C22×D313yes—
230C2×C3×Q8C22×D311yescharacteristic
231C2×C3×Q8C23×C313yes—
232C2×C3×Q8C23×C311yescharacteristic
233C2×C3×D4—32no—
234C2×C3×D4—31no—
235C22×C3×C4—21no—
236C22×C3×C4—23no—
237C22×C3×C4—23no—
238C22×C3×C4—32no—
239C22×C3×C4—36no—
240C22×C3×C4—36no—
241C22×C3×C4C22×D313yes—
242C22×C3×C4C22×D313yes—
243C22×C3×C4C22×D311yescharacteristic
244C22×C3×C4C23×C313yes—
245C22×C3×C4C23×C313yes—
246C22×C3×C4C23×C311yescharacteristic
247C2×#67C23×C312yes—
248D3×Q8—28no—
249#235C3×Q818yesdirect product factor
250C2×D12C23×C311yescharacteristic
251C2×C4×D3C23×C316yes—
252C2×C4×D3C23×C316yes—
253C2×C4×D3C23×C312yes—
254C2×Dic6C23×C316yes—
255C2×Dic6C23×C316yes—
256C2×Dic6C23×C311yescharacteristic
257C3×C4⋊C4—23no—
258C3×C4⋊C4—26no—
259C3×C4⋊C4—23no—
260C3×C4⋊C4—312no—
261C3×C4⋊C4—36no—
262C3×C4⋊C4—36no—
263C3×C4⋊C4—312no—
264C3×C4⋊C4—36no—
265C3×C4⋊C4—33no—
266C3×C4⋊C4—33no—
267C3×C4⋊C4—36no—
268C3×C4⋊C4C22×D313yes—
269C3×C4⋊C4C22×D316yes—
270C3×C4⋊C4C22×D313yes—
271C3×C4⋊C4C23×C313yes—
272C3×C4⋊C4C23×C313yes—
273C3×C4⋊C4C23×C316yes—
274C3×C22⋊C4—36no—
275C3×C22⋊C4—36no—
276C3×C22⋊C4—36no—
277C3×C42—23no—
278C3×C42—23no—
279C3×C42—23no—
280C3×C42—23no—
281C3×C42—36no—
282C3×C42—36no—
283C3×C42—36no—
284C3×C42C22×D313yes—
285C3×C42C22×D313yes—
286C3×C42C22×D313yes—
287C3×C42C22×D313yes—
288C3×C42C23×C313yes—
289C3×C42C23×C313yes—
290C3×C42C23×C313yes—
291C3×C42C23×C313yes—
292#217C23×C316yes—
293(C2×D3)⋊C4C23×C316yes—
294(C2×D3)⋊C4C23×C316yes—
295C4⋊Dic3C23×C313yes—
296C4⋊Dic3C23×C313yes—
297C4⋊Dic3C23×C316yes—
298C4⋊Dic3C23×C316yes—
299Dic3⋊C4C23×C3112yes—
300Dic3⋊C4C23×C3112yes—
301Dic3⋊C4C23×C316yes—
302Dic3⋊C4C23×C316yes—
303C4×Dic3C23×C316yes—
304C4×Dic3C23×C316yes—
305C4×Dic3C23×C316yes—
306C22×C32—21no—
307C22×C32—21no—
308C22×C32C22×Q812yes—
309C22×C32C2511yescharacteristic
310C2×C3×D3—22no—
311C2×C3×D3—22no—
312C2×C3×D3C22×Q814yes—
313C32×C4—23no—
314C32×C4—23no—
315C32×C4—23no—
316C32×C4C22×Q812yes—
317C32×C4C22×Q812yes—
318C32×C4C2×#4116yes—
319C3×Dic3—26no—
320C3×Dic3—26no—
321C3×Dic3C22×Q814yes—
322C2×#41—31no—
323C22×Q8—32no—
324C22×Q8C2×C3×D311yescharacteristic
325C4⋊Q8—36no—
326C4⋊Q8—36no—
327C4⋊Q8—33no—
328C4⋊Q8—33no—
329#125—312no—
330#125—36no—
331C22⋊Q8—36no—
332C22⋊Q8—36no—
333C22⋊Q8—36no—
334C22⋊Q8—36no—
335C22⋊Q8—312no—
336C4×Q8—32no—
337C4×Q8—36no—
338C4×Q8—36no—
339C4×Q8—33no—
340C4×Q8—33no—
341C4×Q8—36no—
342C4×Q8C2×C3×D311yescharacteristic
343C4×Q8C2×C3×D313yes—
344C4×Q8C2×C3×D311yescharacteristic
345C4×Q8C2×C3×D313yes—
346C4×D4—36no—
347C4×D4—36no—
348C4×D4—33no—
349C4×D4—33no—
350#117—33no—
351#117—36no—
352C2×C4⋊C4—36no—
353C2×C4⋊C4C2×C3×D313yes—
354C2×C42—36no—
355C2×C42C2×C3×D313yes—
356C23×C3—21no—
357C23×C3—32no—
358C23×C3C23×D311yescharacteristic
359C23×C3C24×C311yescharacteristic
360C22×D3C24×C312yes—
361C3×Q8—212no—
362C3×Q8—26no—
363C3×Q8—26no—
364C3×Q8—24no—
365C3×Q8—34no—
366C3×Q8—612no—
367C3×Q8—126no—
368C3×Q8#23514yesdirect product factor
369C3×Q8C3×#4114yescomplemented
370C3×D4—38no—
371C3×D4—34no—
372C2×C3×C4—212no—
373C2×C3×C4—26no—
374C2×C3×C4—26no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—23no—
378C2×C3×C4—23no—
379C2×C3×C4—23no—
380C2×C3×C4—23no—
381C2×C3×C4—24no—
382C2×C3×C4—32no—
383C2×C3×C4—38no—
384C2×C3×C4—36no—
385C2×C3×C4—36no—
386C2×C3×C4—43no—
387C2×C3×C4—43no—
388C2×C3×C4—46no—
389C2×C3×C4—612no—
390C2×C3×C4—126no—
391C2×C3×C4D3×Q814yescomplemented
392C2×C3×C4C2×C3×Q814yescomplemented
393C2×C3×C4C23×D313yes—
394C2×C3×C4C23×D313yes—
395C2×C3×C4C23×D313yes—
396C2×C3×C4C23×D313yes—
397C2×C3×C4C23×D311yescharacteristic
398C2×C3×C4C23×D311yescharacteristic
399C2×C3×C4C24×C313yes—
400C2×C3×C4C24×C313yes—
401C2×C3×C4C24×C313yes—
402C2×C3×C4C24×C311yescharacteristic
403C2×C3×C4C24×C311yescharacteristic
404C2×C3×C4C24×C313yes—
405#67C2×C3×Q818yescomplemented
406C2×Dic3C24×C316yes—
407C2×Dic3C24×C316yes—
408C2×Dic3C24×C312yes—
409D12C2×C3×Q814yescomplemented
410C4×D3—212no—
411C4×D3—46no—
412C4×D3C2×C3×Q818yescomplemented
413Dic6—212no—
414Dic6—46no—
415Dic6C2×C3×Q814yescomplemented
416C2×C32—22no—
417C2×C32C23×Q811yescharacteristic
418C2×C32C22×#4111yescharacteristic
419C2×C32#58511yescharacteristic
420C3×D3—24no—
421#41—38no—
422C2×Q8—24no—
423C2×Q8—31no—
424C2×Q8—36no—
425C2×Q8—36no—
426C2×Q8—68no—
427C2×Q8C22×C3×D313yes—
428C2×Q8C22×C3×D311yescharacteristic
429C2×D4—32no—
430C2×D4—31no—
431C22×C4—32no—
432C22×C4—36no—
433C22×C4—36no—
434C22×C4C22×C3×D313yes—
435C22×C4C22×C3×D313yes—
436C22×C4C22×C3×D311yescharacteristic
437C4⋊C4—312no—
438C4⋊C4—36no—
439C4⋊C4—36no—
440C4⋊C4—36no—
441C4⋊C4—33no—
442C4⋊C4—33no—
443C4⋊C4—312no—
444C4⋊C4—36no—
445C4⋊C4C22×C3×D313yes—
446C4⋊C4C22×C3×D316yes—
447C4⋊C4C22×C3×D313yes—
448C22⋊C4—36no—
449C22⋊C4—36no—
450C22⋊C4—36no—
451C42—36no—
452C42—36no—
453C42—36no—
454C42C22×C3×D313yes—
455C42C22×C3×D313yes—
456C42C22×C3×D313yes—
457C42C22×C3×D313yes—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—22no—
465C22×C3—21no—
466C22×C3—34no—
467C22×C3—62no—
468C22×C3—62no—
469C22×C3C2×D3×Q812yes—
470C22×C3C22×C3×Q812yes—
471C22×C3C24×D311yescharacteristic
472C22×C3C25×C311yescharacteristic
473C2×D3—22no—
474C2×D3—22no—
475C2×D3C22×C3×Q814yes—
476C3×C4—26no—
477C3×C4—23no—
478C3×C4—23no—
479C3×C4—23no—
480C3×C4—23no—
481C3×C4—23no—
482C3×C4—23no—
483C3×C4—26no—
484C3×C4—22no—
485C3×C4—22no—
486C3×C4—34no—
487C3×C4—43no—
488C3×C4—43no—
489C3×C4—66no—
490C3×C4—66no—
491C3×C4C2×#23516yes—
492C3×C4C2×D3×Q812yes—
493C3×C4C2×D3×Q812yes—
494C3×C4C22×C3×Q812yes—
495C3×C4C22×C3×Q812yes—
496C3×C4C2×C3×#4116yes—
497Dic3—26no—
498Dic3—26no—
499Dic3C22×C3×Q814yes—
500C32Q8×#4111yescharacteristic, complemented, Sylow
501C23—32no—
502C23C23×C3×D311yescharacteristic
503Q8—26no—
504Q8—34no—
505Q8—612no—
506Q8—126no—
507Q8C3×#23514yesdirect product factor
508D4—38no—
509D4—34no—
510C2×C4—26no—
511C2×C4—32no—
512C2×C4—38no—
513C2×C4—36no—
514C2×C4—36no—
515C2×C4—43no—
516C2×C4—612no—
517C2×C4—126no—
518C2×C4C3×D3×Q814yescomplemented
519C2×C4C23×C3×D313yes—
520C2×C4C23×C3×D313yes—
521C2×C4C23×C3×D313yes—
522C2×C4C23×C3×D313yes—
523C2×C4C23×C3×D311yescharacteristic
524C2×C4C23×C3×D311yescharacteristic
525C2×C3—24no—
526C2×C3—22no—
527C2×C3—22no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—64no—
532C2×C3C22×#23511yescharacteristic
533C2×C3C22×D3×Q811yescharacteristic
534C2×C3#596411yescharacteristic
535C2×C3C23×C3×Q811yescharacteristic
536C2×C3C22×C3×#4111yescharacteristic
537C2×C3C3×#58511yescharacteristic
538D3—24no—
539C22—21no—
540C22—21no—
541C22—34no—
542C22—62no—
543C22—62no—
544C22C2×C3×D3×Q812yes—
545C22C24×C3×D311yescharacteristic
546C4—23no—
547C4—23no—
548C4—23no—
549C4—34no—
550C4—66no—
551C4—66no—
552C4C2×C3×#23516yes—
553C4C2×C3×D3×Q812yes—
554C4C2×C3×D3×Q812yes—
555C3—21no—
556C3Q8×#23511yescharacteristic, direct product factor
557C3C3×Q8×#4111yescharacteristic, complemented
558C2—22no—
559C2—64no—
560C2C22×C3×#23511yescharacteristic
561C2C22×C3×D3×Q811yescharacteristic
562C2C3×#596411yescharacteristic
subgroupG/H
centerC2×C3×C4C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×C4×Q8C2
socleC22×C32C25
prime radicalC2×C3×#235C22
hypercenterC2×C3×C4×Q8D3
hypocenterC3C3×Q8×#41
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4×Q8C2semidirect
2. C3×Q8×#67C2semidirect
3. C3×#5579C2semidirect
4. C3×#5578C2semidirect
5. C3×Q8×D12C2semidirect
6. C3×C4×D3×Q8C2semidirect
7. C3×#5567C2semidirect
8. C3×Q8×Dic6C2semidirect
9. C3×#5511C2semidirect
11. C3×#5470C2semidirect
13. Q8×#235C3direct (N × H)
59. C2×C3×C4×Q8D3semidirect
60. C2×C3×C4×Q8C2×C3semidirect
61. Q8×#67C2×C3semidirect
62. #5579C2×C3semidirect
63. #5578C2×C3semidirect
64. Q8×D12C2×C3semidirect
65. C4×D3×Q8C2×C3semidirect
66. #5567C2×C3semidirect
67. Q8×Dic6C2×C3semidirect
68. #5511C2×C3semidirect
70. #5470C2×C3semidirect
79. C3×#235Q8direct (N × H)
187. C32×Q8#41semidirect
190. C2×C32×C4C2×Q8semidirect
197. C3×#67C2×Q8semidirect
201. C3×D12C2×Q8semidirect
204. C3×C4×D3C2×Q8semidirect
207. C3×Dic6C2×Q8semidirect
218. C2×C4×Q8C3×D3semidirect
249. #235C3×Q8direct (N × H)
368. C3×Q8#235direct (N × H)
369. C3×Q8C3×#41semidirect
391. C2×C3×C4D3×Q8semidirect
392. C2×C3×C4C2×C3×Q8semidirect
405. #67C2×C3×Q8semidirect
409. D12C2×C3×Q8semidirect
412. C4×D3C2×C3×Q8semidirect
415. Dic6C2×C3×Q8semidirect
500. C32Q8×#41semidirect
507. Q8C3×#235direct (N × H)
518. C2×C4C3×D3×Q8semidirect
556. C3Q8×#235direct (N × H)
557. C3C3×Q8×#41semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×Q82
C3×Q8×#672
C3×#55792
C3×#55782
C3×Q8×D122
C3×C4×D3×Q82
C3×#55672
C3×Q8×Dic62
C3×#55112
C3×C4×#2352
C3×#54702
C3×Q8×#413
Q8×#2353