Finite Group Databasebeta

← #11805562

Subgroups — #11805562

Order 1152.

1285 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#634C211yesmaximal, characteristic
2C2×#63844C211yesmaximal, characteristic
3C2×#63844C211yesmaximal, characteristic
4C3×C4×Dic3⋊C4C211yesmaximal, characteristic
5C3×#4928C211yesmaximal, characteristic
6#10591352C211yesmaximal, characteristic
7#10591349C211yesmaximal, characteristic
8#10591328C212yesmaximal
9#10591256C212yesmaximal
10#10591244C212yesmaximal
11#10591241C212yesmaximal
12#75981—31nomaximal
13C4⋊C4×Dic6—31nomaximal
14C22×#634C2211yescharacteristic
15C2×C32×C42C2211yescharacteristic
16C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×#3750C2212yes—
18C2×C4×C3⋊Dic3C2212yes—
19C4×#634—24no—
20C2×C3×C4⋊Dic3C2211yescharacteristic
21C2×C3×C4⋊Dic3C2211yescharacteristic
22C2×C3×Dic3⋊C4C2211yescharacteristic
23C2×C3×Dic3⋊C4C2212yes—
24C2×C3×Dic3⋊C4C2211yescharacteristic
25C2×C3×C4×Dic3C2211yescharacteristic
26C2×C3×C4×Dic3C2211yescharacteristic
27C2×#3724C2212yes—
28C2×#3724C2212yes—
29C2×Dic3⋊Dic3C2212yes—
30C2×Dic3⋊Dic3C2212yes—
31C2×Dic3⋊Dic3C2212yes—
32C2×Dic32C2212yes—
33C2×Dic32C2212yes—
34#63844—24no—
35#63844C418yescomplemented
36C3×C3⋊#95C2212yes—
37C3×C3⋊#95C2211yescharacteristic
38C3×C3⋊#95C2211yescharacteristic
39#63579C2212yes—
40C2×#887—31no—
41C2×#887—31no—
42C2×C4×Dic6—31no—
43C2×C4×Dic6—31no—
44C2×C4×Dic6—31no—
45C2×C4×Dic6—31no—
46C2×C4×Dic6—61no—
47C3×#389—31no—
48C3×C4×C4⋊C4—31no—
49#4984—31no—
50#4976—32no—
51#4975—32no—
52#4975—32no—
53#4975—31no—
54Dic3×C4⋊C4—32no—
55#4935—31no—
56C4×Dic3⋊C4D311yescharacteristic
57#4928—31no—
58#4928D311yescharacteristic
59#4649—32no—
60#4647—32no—
61#4647—32no—
62#4647—32no—
63C22×C32×C4C2311yescharacteristic
64C22×C32×C4C2311yescharacteristic
65C22×C32×C4C2311yescharacteristic
66C22×C3⋊Dic3C2312yes—
67C22×C3⋊Dic3C2312yes—
68C2×#634C2×C414yes—
69C2×#634D414yes—
70C2×#634Q814yes—
71C22×C3×Dic3C2311yescharacteristic
72C22×C3×Dic3C2312yes—
73C22×C3×Dic3C2311yescharacteristic
74C22×C3×Dic3C2312yes—
75C22×C3×Dic3C2311yescharacteristic
76C22×C3×Dic3C2311yescharacteristic
77C32×C42—22no—
78C4⋊(C3⋊Dic3)—22no—
79#3750—24no—
80C4×C3⋊Dic3—24no—
81C3×C4⋊Dic3—22no—
82C3×C4⋊Dic3C2×C414yes—
83C3×Dic3⋊C4—22no—
84C3×Dic3⋊C4—42no—
85C3×Dic3⋊C4Q814yescomplemented
86C3×C4×Dic3—22no—
87C3×C4×Dic3C2×C414yes—
88#3724—24no—
89#3724C2×C418yes—
90Dic3⋊Dic3—24no—
91Dic3⋊Dic3—42no—
92Dic3⋊Dic3—42no—
93Dic32—24no—
94Dic32C2×C418yes—
95Q8×C4⋊C4—91noSylow
96C22×Dic6—31no—
97C22×Dic6—31no—
98C22×Dic6—61no—
99C2×C3×C4⋊C4—31no—
100C2×C3×C4⋊C4—31no—
101C2×C3×C4⋊C4—31no—
102C2×C3×C4⋊C4—31no—
103C2×C3×C4⋊C4—32no—
104C2×C3×C42—21no—
105C2×C3×C42—31no—
106C2×C3×C42—31no—
107C2×C3×C42C2×D311yescharacteristic
108C2×C3×C42C2×D311yescharacteristic
109C2×C4⋊Dic3—31no—
110C2×C4⋊Dic3—31no—
111C2×C4⋊Dic3—32no—
112C2×C4⋊Dic3—32no—
113C2×C4⋊Dic3—32no—
114C2×C4⋊Dic3—61no—
115C2×C4⋊Dic3C2×D311yescharacteristic
116C2×C4⋊Dic3C2×D311yescharacteristic
117C2×Dic3⋊C4—32no—
118C2×Dic3⋊C4—32no—
119C2×Dic3⋊C4—32no—
120C2×Dic3⋊C4—32no—
121C2×Dic3⋊C4—32no—
122C2×Dic3⋊C4—32no—
123C2×Dic3⋊C4—32no—
124C2×Dic3⋊C4—32no—
125C2×Dic3⋊C4—32no—
126C2×Dic3⋊C4—62no—
127C2×Dic3⋊C4C2×D311yescharacteristic
128C2×Dic3⋊C4C2×D312yes—
129C2×Dic3⋊C4C2×D311yescharacteristic
130C2×C4×Dic3—32no—
131C2×C4×Dic3—32no—
132C2×C4×Dic3—32no—
133C2×C4×Dic3—32no—
134C2×C4×Dic3—32no—
135C2×C4×Dic3—32no—
136C2×C4×Dic3—62no—
137C2×C4×Dic3C2×D311yescharacteristic
138C2×C4×Dic3C2×D311yescharacteristic
139#887—38no—
140#887—64no—
141C4×Dic6—38no—
142C4×Dic6—64no—
143C4×Dic6—64no—
144C4×Dic6—64no—
145C4×Dic6—68no—
146C3×#95—31no—
147C3×#95—31no—
148C3×#95—32no—
149C3⋊#95—32no—
150C3⋊#95—32no—
151C3⋊#95C2×D311yescharacteristic
152C3⋊#95C2×D312yes—
153C3⋊#95C2×D311yescharacteristic
154C23×C32C2411yescharacteristic
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4C22×C412yes—
161C2×C32×C4C2×D412yes—
162C2×C32×C4C2×Q812yes—
163C2×C32×C4C2×Q812yes—
164C2×C3⋊Dic3—22no—
165C2×C3⋊Dic3—22no—
166C2×C3⋊Dic3—22no—
167C2×C3⋊Dic3C22×C414yes—
168C2×C3⋊Dic3C2×D414yes—
169C2×C3⋊Dic3C2×Q814yes—
170#634C4⋊C4116yescomplemented
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—22no—
183C2×C3×Dic3C22×C414yes—
184C2×C3×Dic3C22×C412yes—
185C2×C3×Dic3C22×C412yes—
186C2×C3×Dic3C2×Q812yes—
187C2×C3×Dic3C2×Q812yes—
188C2×C3×Dic3#4112yes—
189C2×C4×Q8—91no—
190C2×C4×Q8—91no—
191C2×C4×Q8—91no—
192#389—31no—
193#389—92no—
194#389—92no—
195#389—91no—
196#384—91no—
197#384—92no—
198C4×C4⋊C4—31no—
199C4×C4⋊C4—92no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—31no—
204C22×C3×C4—31no—
205C22×C3×C4—32no—
206C22×C3×C4—31no—
207C22×C3×C4—32no—
208C22×C3×C4—31no—
209C22×C3×C4C22×D311yescharacteristic
210C22×C3×C4C22×D311yescharacteristic
211C22×C3×C4C22×D311yescharacteristic
212C22×C3×C4C22×D311yescharacteristic
213C22×C3×C4C22×D311yescharacteristic
214C22×C3×C4C22×D311yescharacteristic
215C22×Dic3—32no—
216C22×Dic3—32no—
217C22×Dic3—32no—
218C22×Dic3—32no—
219C22×Dic3—62no—
220C22×Dic3—62no—
221C22×Dic3C22×D312yes—
222C22×Dic3C22×D311yescharacteristic
223C22×Dic3C22×D311yescharacteristic
224C22×Dic3C22×D312yes—
225C22×Dic3C22×D311yescharacteristic
226C22×Dic3C22×D311yescharacteristic
227C2×Dic6—34no—
228C2×Dic6—34no—
229C2×Dic6—34no—
230C2×Dic6—34no—
231C2×Dic6—34no—
232C2×Dic6—34no—
233C2×Dic6—64no—
234C2×Dic6—64no—
235C2×Dic6—64no—
236C3×C4⋊C4—34no—
237C3×C4⋊C4—34no—
238C3×C4⋊C4—62no—
239C3×C4⋊C4—62no—
240C3×C4⋊C4—122no—
241C3×C42—24no—
242C3×C42—22no—
243C3×C42—22no—
244C3×C42—34no—
245C3×C42—62no—
246C4⋊Dic3—22no—
247C4⋊Dic3—64no—
248C4⋊Dic3—64no—
249C4⋊Dic3—62no—
250C4⋊Dic3—62no—
251C4⋊Dic3—122no—
252C4⋊Dic3—122no—
253C4⋊Dic3C4×D314yes—
254Dic3⋊C4—22no—
255Dic3⋊C4—38no—
256Dic3⋊C4—38no—
257Dic3⋊C4—42no—
258Dic3⋊C4—64no—
259Dic3⋊C4—64no—
260Dic3⋊C4—64no—
261Dic3⋊C4—64no—
262Dic3⋊C4—68no—
263Dic3⋊C4—64no—
264Dic3⋊C4—122no—
265Dic3⋊C4—122no—
266Dic3⋊C4Dic614yescomplemented
267C4×Dic3—22no—
268C4×Dic3—38no—
269C4×Dic3—38no—
270C4×Dic3—64no—
271C4×Dic3—64no—
272C4×Dic3—64no—
273C4×Dic3—64no—
274C4×Dic3—68no—
275C4×Dic3C4×D314yes—
276C22×C32C23×C411yescharacteristic
277C22×C32C22×D411yescharacteristic
278C22×C32C22×Q811yescharacteristic
279C22×C32C22×Q811yescharacteristic
280C22×C32C2×#4111yescharacteristic
281C22×C32H(5,𝔽2)11yescharacteristic
282C22×C322−1+411yescharacteristic
283C32×C4—22no—
284C32×C4—41no—
285C32×C4C2×C4⋊C414yes—
286C3⋊Dic3—42no—
287C3⋊Dic3C2×C4⋊C418yes—
288C3×Dic3—22no—
289C3×Dic3—24no—
290C3×Dic3—22no—
291C3×Dic3—41no—
292C3×Dic3—42no—
293C3×Dic3C4×Q814yescomplemented
294C22×Q8—91no—
295C4×Q8—98no—
296C4×Q8—184no—
297C4×Q8—184no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4—92no—
304C2×C4⋊C4—92no—
305C2×C4⋊C4—92no—
306C2×C4⋊C4—92no—
307C2×C4⋊C4—91no—
308C2×C4⋊C4—92no—
309C2×C4⋊C4—92no—
310C2×C42—31no—
311C2×C42—31no—
312C2×C42—92no—
313C2×C42—92no—
314C2×C42—92no—
315C2×C42D3211yescharacteristic
316#95—31no—
317#95—31no—
318#95—32no—
319#95—92no—
320C23×C3—21no—
321C23×C3C23×D311yescharacteristic
322C23×C3C23×D311yescharacteristic
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—32no—
341C2×C3×C4—32no—
342C2×C3×C4—32no—
343C2×C3×C4—34no—
344C2×C3×C4—32no—
345C2×C3×C4—32no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—62no—
351C2×C3×C4—61no—
352C2×C3×C4—62no—
353C2×C3×C4—61no—
354C2×C3×C4—62no—
355C2×C3×C4—62no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—62no—
360C2×C3×C4C2×Dic612yes—
361C2×C3×C4C2×C4×D312yes—
362C2×C3×C4C2×C4×D312yes—
363C2×C3×C4D3×D412yes—
364C2×C3×C4D3×D412yes—
365C2×C3×C4D3×Q812yes—
366C2×C3×C4D3×Q812yes—
367C2×C3×C4D3×Q812yes—
368C2×Dic3—21no—
369C2×Dic3—22no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—22no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—21no—
380C2×Dic3—34no—
381C2×Dic3—34no—
382C2×Dic3—34no—
383C2×Dic3—34no—
384C2×Dic3—34no—
385C2×Dic3—34no—
386C2×Dic3—64no—
387C2×Dic3—64no—
388C2×Dic3—64no—
389C2×Dic3—64no—
390C2×Dic3—62no—
391C2×Dic3—62no—
392C2×Dic3—62no—
393C2×Dic3—62no—
394C2×Dic3—62no—
395C2×Dic3—62no—
396C2×Dic3—122no—
397C2×Dic3—122no—
398C2×Dic3C2×Dic612yes—
399C2×Dic3C2×Dic612yes—
400C2×Dic3C2×C4×D314yes—
401C2×Dic3C2×C4×D312yes—
402C2×Dic3C2×C4×D312yes—
403C2×Dic3#23512yes—
404Dic6—316no—
405Dic6—316no—
406Dic6—616no—
407C2×C32C22×C4⋊C411yescharacteristic
408C2×C32C2×C4×Q811yescharacteristic
409C2×C32#52011yescharacteristic
410C2×C32D4×Q811yescharacteristic
411C2×C32#55011yescharacteristic
412C2×C32#55711yescharacteristic
413C2×C32Q8211yescharacteristic
414C2×Q8—94no—
415C2×Q8—94no—
416C2×Q8—94no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—32no—
421C22×C4—31no—
422C22×C4—32no—
423C22×C4—92no—
424C22×C4—92no—
425C22×C4C2×D3211yescharacteristic
426C22×C4C2×D3211yescharacteristic
427C22×C4C2×D3211yescharacteristic
428C4⋊C4—34no—
429C4⋊C4—34no—
430C4⋊C4—62no—
431C4⋊C4—62no—
432C4⋊C4—98no—
433C4⋊C4—122no—
434C4⋊C4—184no—
435C4⋊C4—184no—
436C4⋊C4—184no—
437C4⋊C4—182no—
438C4⋊C4—362no—
439C4⋊C4—362no—
440C42—22no—
441C42—34no—
442C42—62no—
443C42—98no—
444C42—184no—
445C42—184no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3C22×Dic611yescharacteristic
454C22×C3C22×C4×D311yescharacteristic
455C22×C3C22×C4×D311yescharacteristic
456C22×C3C2×#23511yescharacteristic
457C22×C3C2×D3×D411yescharacteristic
458C22×C3C2×D3×D411yescharacteristic
459C22×C3#100411yescharacteristic
460C22×C3C2×D3×Q811yescharacteristic
461C22×C3C2×D3×Q811yescharacteristic
462C22×C3C2×D3×Q811yescharacteristic
463C22×C3D3×#4111yescharacteristic
464C22×C3#100911yescharacteristic
465C22×C3#101011yescharacteristic
466C22×C3#101011yescharacteristic
467C3×C4—24no—
468C3×C4—22no—
469C3×C4—22no—
470C3×C4—24no—
471C3×C4—34no—
472C3×C4—42no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4—62no—
476C3×C4—64no—
477C3×C4—62no—
478C3×C4—122no—
479C3×C4—121no—
480C3×C4D3×C4⋊C414yes—
481C3×C4D3×C4⋊C414yes—
482Dic3—22no—
483Dic3—22no—
484Dic3—24no—
485Dic3—38no—
486Dic3—38no—
487Dic3—41no—
488Dic3—42no—
489Dic3—68no—
490Dic3—124no—
491Dic3—122no—
492Dic3—122no—
493Dic3C4×Dic614yescomplemented
494C32Q8×C4⋊C411yescharacteristic, complemented, Sylow
495C23C22×D3211yescharacteristic
496Q8—916no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—32no—
503C2×C4—32no—
504C2×C4—32no—
505C2×C4—32no—
506C2×C4—32no—
507C2×C4—34no—
508C2×C4—61no—
509C2×C4—62no—
510C2×C4—61no—
511C2×C4—62no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—62no—
520C2×C4—94no—
521C2×C4—94no—
522C2×C4—94no—
523C2×C4—182no—
524C2×C4—182no—
525C2×C4—182no—
526C2×C4D3×Dic612yes—
527C2×C4#379712yes—
528C2×C4C4×D3212yes—
529C2×C4#381112yes—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3C2×C4×Dic611yescharacteristic
538C2×C3C2×D3×C4⋊C411yescharacteristic
539C2×C3C2×D3×C4⋊C411yescharacteristic
540C2×C3#552311yescharacteristic
541C2×C3D4×Dic611yescharacteristic
542C2×C3#554611yescharacteristic
543C2×C3#555311yescharacteristic
544C2×C3Q8×Dic611yescharacteristic
545C2×C3#556711yescharacteristic
546C2×C3C4×D3×Q811yescharacteristic
547C2×C3#559911yescharacteristic
548C2×C3#563211yescharacteristic
549C2×C3#568511yescharacteristic
550C2×C3#572211yescharacteristic
551C22C2×D3×Dic611yescharacteristic
552C22#6429811yescharacteristic
553C22C2×#379711yescharacteristic
554C22C2×C4×D3211yescharacteristic
555C22D3×#23511yescharacteristic
556C22#6430711yescharacteristic
557C22C2×#381111yescharacteristic
558C4—22no—
559C4—34no—
560C4—41no—
561C4—62no—
562C4—64no—
563C4—62no—
564C4—98no—
565C4—121no—
566C4—122no—
567C4—362no—
568C4#6388314yes—
569C3—21no—
570C3C4⋊C4×Dic611yescharacteristic, complemented
571C3#7598111yescharacteristic, complemented
572C2#1059385211yescharacteristic
573C2#1059385611yescharacteristic
574C2C4×D3×Dic611yescharacteristic
575C2#1059388711yescharacteristic
576C2C2×#6388311yescharacteristic
577C2#1059491511yescharacteristic
578C2#1059491911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C42C22
socle, prime radical, cosocleC23×C32C24
hypercenterC2×C42D32
hypocenterC32Q8×C4⋊C4
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
35. #63844C4semidirect
85. C3×Dic3⋊C4Q8semidirect
170. #634C4⋊C4semidirect
266. Dic3⋊C4Dic6semidirect
293. C3×Dic3C4×Q8semidirect
493. Dic3C4×Dic6semidirect
494. C32Q8×C4⋊C4semidirect
570. C3C4⋊C4×Dic6semidirect
571. C3#75981semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#6342
C2×#638442
C3×C4×Dic3⋊C42
C3×#49282
#105913522
#105913492
#105913282
#105912562
#105912442
#105912412
#759813
C4⋊C4×Dic63