Finite Group Databasebeta

← C3×Q8×#237 #11903734

Subgroups — C3×Q8×#237 #11903734

Order 1152.

2276 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×D4×Q8C211yesmaximal, characteristic, complemented
2C3×Q8×#67C212yesmaximal, complemented
3C2×C3×Q8×Dic3C212yesmaximal, complemented
4C3×#5728C213yesmaximal, complemented
5C3×#5721C213yesmaximal, complemented
6C3×#5635C216yesmaximal, complemented
7C3×#5622C216yesmaximal, complemented
8C3×C4×D3×Q8C211yesmaximal, characteristic, complemented
9C3×Q8×Dic6C211yesmaximal, characteristic, complemented
10C3×#5543C213yesmaximal, complemented
11C3×C4×#237C213yesmaximal
12C3×Q8×#41—31nomaximal
13Q8×#237C311yesmaximal, characteristic, direct product factor
14C22×C32×Q8C2212yes—
15C2×C3×D3×Q8C2211yescharacteristic
16C2×C3×#237C2211yescharacteristic
17C32×C4⋊Q8C2213yes—
18C32×C22⋊Q8C2216yes—
19C32×C4×Q8C2211yescharacteristic
20C32×C4×D4C2213yes—
21C3×#946C2216yes—
22C3×Q8×Dic3C2211yescharacteristic
23C3×Q8×Dic3C2212yes—
24C3×Q8×Dic3C2216yes—
25C3×Q8×Dic3C2216yes—
26C3×#944C2216yes—
27C3×D4×Dic3C2213yes—
28C3×C4×#67C2216yes—
29C2×C3×C4⋊Dic3C2216yes—
30C3×#924C2216yes—
31C2×C3×C4×Dic3C2216yes—
32C3×#897C2216yes—
33C3×#892C2213yes—
34C3×D3×C4⋊C4C2213yes—
35C3×#890C2216yes—
36C3×#889C22112yes—
37C3×#888C2216yes—
38C3×#887C2213yes—
39C3×#887C2213yes—
40C3×#887C2216yes—
41C3×#67⋊C4C2216yes—
42C3×#878C22112yes—
43C3×#877C2216yes—
44C3×C42×D3C2213yes—
45C3×#869C2213yes—
46C3×C4×Dic6C2213yes—
47C3×C4×Dic6C2213yes—
48C3×Q82—31no—
49C3×#557—33no—
50C3×#554—36no—
51C3×#554—33no—
52C3×D4×Q8—21no—
53C3×D4×Q8—32no—
54C3×D4×Q8D311yescharacteristic, complemented
55C3×D4×Q8C2×C311yescharacteristic, complemented
56C3×#533—33no—
57C3×#533—36no—
58C3×C4×#41—33no—
59C2×C3×C4×Q8—32no—
60C2×C3×C4×Q8—31no—
61Q8×#67C2×C312yescomplemented
62C2×Q8×Dic3C2×C312yescomplemented
63#5728C2×C313yescomplemented
64#5721C2×C313yescomplemented
65#5635C2×C316yescomplemented
66#5622C2×C316yescomplemented
67C4×D3×Q8C2×C311yescharacteristic, complemented
68Q8×Dic6C2×C311yescharacteristic, complemented
69#5543C2×C313yescomplemented
70C4×#237C2×C313yes—
71C2×C32×Q8—28no—
72C2×C32×Q8C2316yes—
73C2×C32×Q8C2311yescharacteristic
74C2×C32×D4C2311yescharacteristic
75C22×C32×C4C2316yes—
76C2×C3×#67C2312yes—
77C22×C3×Dic3C2316yes—
78C22×C3×Dic3C2312yes—
79C3×D3×Q8—24no—
80C3×#237Q818yesdirect product factor
81C2×C3×C4×D3C2313yes—
82C2×C3×C4×D3C2313yes—
83C2×C3×C4×D3C2311yescharacteristic
84C2×C3×Dic6C2313yes—
85C2×C3×Dic6C2316yes—
86C2×C3×Dic6C2311yescharacteristic
87C32×C4⋊C4C2313yes—
88C32×C4⋊C4C2316yes—
89C32×C4⋊C4C2313yes—
90C32×C22⋊C4C2316yes—
91C32×C42C2313yes—
92C3×#217C2316yes—
93C3×(C2×D3)⋊C4C2316yes—
94C3×C4⋊Dic3C23112yes—
95C3×C4⋊Dic3C2316yes—
96C3×C4⋊Dic3C2313yes—
97C3×C4⋊Dic3C2313yes—
98C3×Dic3⋊C4C2316yes—
99C3×Dic3⋊C4C2316yes—
100C3×Dic3⋊C4C2316yes—
101C3×Dic3⋊C4C23112yes—
102C3×C4×Dic3C2313yes—
103C3×C4×Dic3C2316yes—
104C3×C4×Dic3C2316yes—
105C3×C4×Dic3C2313yes—
106C3×C4×Dic3C2313yes—
107C3×C4×Dic3C2316yes—
108Q8×#41—31noSylow
109C2×C3×#41—31no—
110C22×C3×Q8—22no—
111C22×C3×Q8—31no—
112C22×C3×Q8C2×D312yes—
113C22×C3×Q8C22×C312yes—
114C2×D3×Q8C22×C311yescharacteristic
115C2×#237C22×C311yescharacteristic
116C3×C4⋊Q8—23no—
117C3×C4⋊Q8—36no—
118C3×C4⋊Q8—36no—
119C3×C4⋊Q8—33no—
120C3×C4⋊Q8C2×D313yes—
121C3×C4⋊Q8C22×C313yes—
122C3×#125—312no—
123C3×#125—36no—
124C3×C22⋊Q8—26no—
125C3×C22⋊Q8—36no—
126C3×C22⋊Q8—36no—
127C3×C22⋊Q8—36no—
128C3×C22⋊Q8—312no—
129C3×C22⋊Q8C2×D316yes—
130C3×C22⋊Q8C22×C316yes—
131C3×C4×Q8—21no—
132C3×C4×Q8—32no—
133C3×C4×Q8—31no—
134C3×C4×Q8—36no—
135C3×C4×Q8—36no—
136C3×C4×Q8—33no—
137C3×C4×Q8—33no—
138C3×C4×Q8—36no—
139C3×C4×Q8—33no—
140C3×C4×Q8—33no—
141C3×C4×Q8C2×D311yescharacteristic
142C3×C4×Q8C22×C311yescharacteristic
143C3×C4×D4—23no—
144C3×C4×D4—33no—
145C3×C4×D4—36no—
146C3×C4×D4—36no—
147C3×C4×D4C2×D313yes—
148C3×C4×D4C22×C313yes—
149C3×#117—33no—
150C3×#117—36no—
151C2×C3×C4⋊C4—36no—
152C2×C3×C4⋊C4—33no—
153C2×C3×C42—36no—
154C2×C3×C42—33no—
155#946C22×C316yes—
156Q8×Dic3C22×C316yes—
157Q8×Dic3C22×C316yes—
158Q8×Dic3C22×C311yescharacteristic
159Q8×Dic3C22×C312yes—
160#944C22×C316yes—
161D4×Dic3C22×C313yes—
162C4×#67C22×C316yes—
163C2×C4⋊Dic3C22×C316yes—
164#924C22×C316yes—
165C2×C4×Dic3C22×C316yes—
166#897C22×C316yes—
167#892C22×C313yes—
168D3×C4⋊C4C22×C313yes—
169#890C22×C316yes—
170#889C22×C3112yes—
171#888C22×C316yes—
172#887C22×C313yes—
173#887C22×C316yes—
174#887C22×C313yes—
175#67⋊C4C22×C316yes—
176#878C22×C3112yes—
177#877C22×C316yes—
178C42×D3C22×C313yes—
179#869C22×C313yes—
180C4×Dic6C22×C313yes—
181C4×Dic6C22×C313yes—
182C23×C32C2412yes—
183C22×C3×D3C2411yescharacteristic
184C32×Q8—212no—
185C32×Q8#4114yescomplemented
186C32×D4C2×Q814yescomplemented
187C2×C32×C4—212no—
188C2×C32×C4C2416yes—
189C2×C32×C4C2413yes—
190C2×C32×C4C2411yescharacteristic
191C2×C32×C4C2413yes—
192C3×#67C2×Q818yescomplemented
193C2×C3×Dic3—46no—
194C2×C3×Dic3C2×Q818yescomplemented
195C2×C3×Dic3C2413yes—
196C2×C3×Dic3C2413yes—
197C2×C3×Dic3C2416yes—
198C2×C3×Dic3C2411yescharacteristic
199C2×C3×Dic3C2412yes—
200C3×C4×D3—26no—
201C3×C4×D3—43no—
202C3×C4×D3C2×Q814yescomplemented
203C3×Dic6—26no—
204C3×Dic6—46no—
205C3×Dic6C2×Q814yescomplemented
206Q82—31no—
207#557—33no—
208#554—36no—
209#554—33no—
210D4×Q8—32no—
211D4×Q8C3×D311yescharacteristic, complemented
212#533—33no—
213#533—36no—
214C4×#41—33no—
215C2×C4×Q8—32no—
216C2×C4×Q8—31no—
217C3×#41—38no—
218C2×C3×Q8—28no—
219C2×C3×Q8—28no—
220C2×C3×Q8—21no—
221C2×C3×Q8—26no—
222C2×C3×Q8—31no—
223C2×C3×Q8—33no—
224C2×C3×Q8—36no—
225C2×C3×Q8—48no—
226C2×C3×Q8—64no—
227C2×C3×Q8C22×D316yes—
228C2×C3×Q8C22×D311yescharacteristic
229C2×C3×Q8C23×C316yes—
230C2×C3×Q8C23×C311yescharacteristic
231C2×C3×D4—21no—
232C2×C3×D4—32no—
233C2×C3×D4C22×D311yescharacteristic
234C2×C3×D4C23×C311yescharacteristic
235C22×C3×C4—26no—
236C22×C3×C4—32no—
237C22×C3×C4—31no—
238C22×C3×C4—36no—
239C22×C3×C4—33no—
240C22×C3×C4—33no—
241C22×C3×C4C22×D316yes—
242C22×C3×C4C23×C316yes—
243C2×#67C23×C312yes—
244C22×Dic3C23×C316yes—
245C22×Dic3C23×C312yes—
246D3×Q8—24no—
247#237C3×Q818yesdirect product factor
248C2×C4×D3C23×C313yes—
249C2×C4×D3C23×C313yes—
250C2×C4×D3C23×C311yescharacteristic
251C2×Dic6C23×C313yes—
252C2×Dic6C23×C316yes—
253C2×Dic6C23×C311yescharacteristic
254C3×C4⋊C4—23no—
255C3×C4⋊C4—23no—
256C3×C4⋊C4—26no—
257C3×C4⋊C4—36no—
258C3×C4⋊C4—36no—
259C3×C4⋊C4—36no—
260C3×C4⋊C4—312no—
261C3×C4⋊C4—33no—
262C3×C4⋊C4—312no—
263C3×C4⋊C4—36no—
264C3×C4⋊C4—33no—
265C3×C4⋊C4C22×D313yes—
266C3×C4⋊C4C22×D313yes—
267C3×C4⋊C4C22×D316yes—
268C3×C4⋊C4C23×C313yes—
269C3×C4⋊C4C23×C316yes—
270C3×C4⋊C4C23×C313yes—
271C3×C22⋊C4—26no—
272C3×C22⋊C4—36no—
273C3×C22⋊C4—36no—
274C3×C22⋊C4C22×D316yes—
275C3×C22⋊C4C23×C316yes—
276C3×C42—23no—
277C3×C42—33no—
278C3×C42—36no—
279C3×C42—33no—
280C3×C42—36no—
281C3×C42—33no—
282C3×C42—36no—
283C3×C42C22×D313yes—
284C3×C42C23×C313yes—
285#217C23×C316yes—
286(C2×D3)⋊C4C23×C316yes—
287C4⋊Dic3C23×C313yes—
288C4⋊Dic3C23×C313yes—
289C4⋊Dic3C23×C316yes—
290C4⋊Dic3C23×C3112yes—
291Dic3⋊C4C23×C316yes—
292Dic3⋊C4C23×C316yes—
293Dic3⋊C4C23×C316yes—
294Dic3⋊C4C23×C3112yes—
295C4×Dic3C23×C313yes—
296C4×Dic3C23×C316yes—
297C4×Dic3C23×C313yes—
298C4×Dic3C23×C313yes—
299C4×Dic3C23×C316yes—
300C4×Dic3C23×C316yes—
301C22×C32—22no—
302C22×C32—22no—
303C22×C32C22×Q814yes—
304C22×C32C2511yescharacteristic
305C2×C3×D3—21no—
306C2×C3×D3—21no—
307C2×C3×D3C22×Q812yes—
308C32×C4—26no—
309C32×C4—23no—
310C32×C4C22×Q812yes—
311C32×C4C2×#4116yes—
312C3×Dic3—23no—
313C3×Dic3—26no—
314C3×Dic3—23no—
315C3×Dic3C22×Q814yes—
316C3×Dic3C22×Q812yes—
317C2×#41—31no—
318C22×Q8—31no—
319C22×Q8C2×C3×D312yes—
320C4⋊Q8—36no—
321C4⋊Q8—36no—
322C4⋊Q8—33no—
323C4⋊Q8C2×C3×D313yes—
324#125—312no—
325#125—36no—
326C22⋊Q8—36no—
327C22⋊Q8—36no—
328C22⋊Q8—36no—
329C22⋊Q8—312no—
330C22⋊Q8C2×C3×D316yes—
331C4×Q8—31no—
332C4×Q8—36no—
333C4×Q8—32no—
334C4×Q8—36no—
335C4×Q8—33no—
336C4×Q8—33no—
337C4×Q8—36no—
338C4×Q8—33no—
339C4×Q8—33no—
340C4×Q8C2×C3×D311yescharacteristic
341C4×D4—33no—
342C4×D4—36no—
343C4×D4—36no—
344C4×D4C2×C3×D313yes—
345#117—33no—
346#117—36no—
347C2×C4⋊C4—36no—
348C2×C4⋊C4—33no—
349C2×C42—36no—
350C2×C42—33no—
351C23×C3—22no—
352C23×C3—31no—
353C23×C3C23×D312yes—
354C23×C3C24×C312yes—
355C22×D3C24×C311yescharacteristic
356C3×Q8—212no—
357C3×Q8—212no—
358C3×Q8—24no—
359C3×Q8—34no—
360C3×Q8—412no—
361C3×Q8—66no—
362C3×Q8—126no—
363C3×Q8#23714yesdirect product factor
364C3×Q8C3×#4114yescomplemented
365C3×D4—24no—
366C3×D4—38no—
367C3×D4D3×Q814yescomplemented
368C3×D4C2×C3×Q814yescomplemented
369C2×C3×C4—212no—
370C2×C3×C4—212no—
371C2×C3×C4—21no—
372C2×C3×C4—23no—
373C2×C3×C4—23no—
374C2×C3×C4—26no—
375C2×C3×C4—38no—
376C2×C3×C4—31no—
377C2×C3×C4—32no—
378C2×C3×C4—34no—
379C2×C3×C4—33no—
380C2×C3×C4—33no—
381C2×C3×C4—36no—
382C2×C3×C4—412no—
383C2×C3×C4—66no—
384C2×C3×C4—123no—
385C2×C3×C4—126no—
386C2×C3×C4C23×D316yes—
387C2×C3×C4C23×D313yes—
388C2×C3×C4C23×D313yes—
389C2×C3×C4C23×D311yescharacteristic
390C2×C3×C4C24×C316yes—
391C2×C3×C4C24×C313yes—
392C2×C3×C4C24×C311yescharacteristic
393C2×C3×C4C24×C313yes—
394#67C2×C3×Q818yescomplemented
395C2×Dic3—46no—
396C2×Dic3C2×C3×Q818yescomplemented
397C2×Dic3C24×C313yes—
398C2×Dic3C24×C313yes—
399C2×Dic3C24×C316yes—
400C2×Dic3C24×C311yescharacteristic
401C2×Dic3C24×C312yes—
402C4×D3—26no—
403C4×D3—43no—
404C4×D3C2×C3×Q814yescomplemented
405Dic6—26no—
406Dic6—46no—
407Dic6C2×C3×Q814yescomplemented
408C2×C32—24no—
409C2×C32C23×Q811yescharacteristic
410C2×C32C22×#4111yescharacteristic
411C2×C32#58511yescharacteristic
412C3×D3—22no—
413#41—38no—
414C2×Q8—28no—
415C2×Q8—31no—
416C2×Q8—33no—
417C2×Q8—36no—
418C2×Q8—64no—
419C2×Q8C22×C3×D316yes—
420C2×Q8C22×C3×D311yescharacteristic
421C2×D4—32no—
422C2×D4C22×C3×D311yescharacteristic
423C22×C4—32no—
424C22×C4—31no—
425C22×C4—36no—
426C22×C4—33no—
427C22×C4—33no—
428C22×C4C22×C3×D316yes—
429C4⋊C4—33no—
430C4⋊C4—36no—
431C4⋊C4—312no—
432C4⋊C4—36no—
433C4⋊C4—36no—
434C4⋊C4—36no—
435C4⋊C4—33no—
436C4⋊C4—312no—
437C4⋊C4C22×C3×D313yes—
438C4⋊C4C22×C3×D313yes—
439C4⋊C4C22×C3×D316yes—
440C22⋊C4—36no—
441C22⋊C4—36no—
442C22⋊C4C22×C3×D316yes—
443C42—33no—
444C42—36no—
445C42—33no—
446C42—36no—
447C42—33no—
448C42—36no—
449C42C22×C3×D313yes—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—24no—
455C22×C3—21no—
456C22×C3—32no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—61no—
460C22×C3—61no—
461C22×C3C2×D3×Q814yes—
462C22×C3C22×C3×Q814yes—
463C22×C3C24×D311yescharacteristic
464C22×C3C25×C311yescharacteristic
465C2×D3—21no—
466C2×D3—21no—
467C2×D3C22×C3×Q812yes—
468C3×C4—26no—
469C3×C4—23no—
470C3×C4—26no—
471C3×C4—23no—
472C3×C4—26no—
473C3×C4—22no—
474C3×C4—32no—
475C3×C4—34no—
476C3×C4—46no—
477C3×C4—43no—
478C3×C4—63no—
479C3×C4—63no—
480C3×C4—66no—
481C3×C4C2×#23716yes—
482C3×C4C2×D3×Q812yes—
483C3×C4C22×C3×Q812yes—
484C3×C4C2×C3×#4116yes—
485Dic3—23no—
486Dic3—26no—
487Dic3—23no—
488Dic3C22×C3×Q814yes—
489Dic3C22×C3×Q812yes—
490C32Q8×#4111yescharacteristic, complemented, Sylow
491C23—31no—
492C23C23×C3×D312yes—
493Q8—212no—
494Q8—34no—
495Q8—66no—
496Q8—126no—
497Q8C3×#23714yesdirect product factor
498D4—38no—
499D4C3×D3×Q814yescomplemented
500C2×C4—212no—
501C2×C4—38no—
502C2×C4—31no—
503C2×C4—32no—
504C2×C4—34no—
505C2×C4—33no—
506C2×C4—33no—
507C2×C4—36no—
508C2×C4—66no—
509C2×C4—123no—
510C2×C4—126no—
511C2×C4C23×C3×D316yes—
512C2×C4C23×C3×D313yes—
513C2×C4C23×C3×D313yes—
514C2×C4C23×C3×D311yescharacteristic
515C2×C3—24no—
516C2×C3—24no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—44no—
521C2×C3—62no—
522C2×C3C22×#23711yescharacteristic
523C2×C3C22×D3×Q811yescharacteristic
524C2×C3#596811yescharacteristic
525C2×C3C23×C3×Q811yescharacteristic
526C2×C3C22×C3×#4111yescharacteristic
527C2×C3C3×#58511yescharacteristic
528D3—22no—
529C22—22no—
530C22—22no—
531C22—32no—
532C22—61no—
533C22—61no—
534C22C2×C3×D3×Q814yes—
535C22C24×C3×D311yescharacteristic
536C4—26no—
537C4—23no—
538C4—32no—
539C4—34no—
540C4—63no—
541C4—63no—
542C4—66no—
543C4C2×C3×#23716yes—
544C4C2×C3×D3×Q812yes—
545C3—21no—
546C3Q8×#23711yescharacteristic, direct product factor
547C3C3×Q8×#4111yescharacteristic, complemented
548C2—24no—
549C2—62no—
550C2C22×C3×#23711yescharacteristic
551C2C22×C3×D3×Q811yescharacteristic
552C2C3×#596811yescharacteristic
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×D4×Q8C2
socleC22×C32C25
prime radicalC2×C3×#237C22
hypercenterC3×D4×Q8D3
hypocenterC3C3×Q8×#41
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C32×D4×Q8C2semidirect
2. C3×Q8×#67C2semidirect
3. C2×C3×Q8×Dic3C2semidirect
4. C3×#5728C2semidirect
5. C3×#5721C2semidirect
6. C3×#5635C2semidirect
7. C3×#5622C2semidirect
8. C3×C4×D3×Q8C2semidirect
9. C3×Q8×Dic6C2semidirect
10. C3×#5543C2semidirect
13. Q8×#237C3direct (N × H)
54. C3×D4×Q8D3semidirect
55. C3×D4×Q8C2×C3semidirect
61. Q8×#67C2×C3semidirect
62. C2×Q8×Dic3C2×C3semidirect
63. #5728C2×C3semidirect
64. #5721C2×C3semidirect
65. #5635C2×C3semidirect
66. #5622C2×C3semidirect
67. C4×D3×Q8C2×C3semidirect
68. Q8×Dic6C2×C3semidirect
69. #5543C2×C3semidirect
80. C3×#237Q8direct (N × H)
185. C32×Q8#41semidirect
186. C32×D4C2×Q8semidirect
192. C3×#67C2×Q8semidirect
194. C2×C3×Dic3C2×Q8semidirect
202. C3×C4×D3C2×Q8semidirect
205. C3×Dic6C2×Q8semidirect
211. D4×Q8C3×D3semidirect
247. #237C3×Q8direct (N × H)
363. C3×Q8#237direct (N × H)
364. C3×Q8C3×#41semidirect
367. C3×D4D3×Q8semidirect
368. C3×D4C2×C3×Q8semidirect
394. #67C2×C3×Q8semidirect
396. C2×Dic3C2×C3×Q8semidirect
404. C4×D3C2×C3×Q8semidirect
407. Dic6C2×C3×Q8semidirect
490. C32Q8×#41semidirect
497. Q8C3×#237direct (N × H)
499. D4C3×D3×Q8semidirect
546. C3Q8×#237direct (N × H)
547. C3C3×Q8×#41semidirect

Maximal subgroups

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

subgroupindex
C32×D4×Q82
C3×Q8×#672
C2×C3×Q8×Dic32
C3×#57282
C3×#57212
C3×#56352
C3×#56222
C3×C4×D3×Q82
C3×Q8×Dic62
C3×#55432
C3×C4×#2372
C3×Q8×#413
Q8×#2373