Finite Group Databasebeta

← #11778033

Subgroups — #11778033

Order 1152.

672 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10594971C211yesmaximal, characteristic, complemented
2C3×#4731C211yesmaximal, characteristic, complemented
3#10591389C211yesmaximal, characteristic
4#10591172C211yesmaximal, characteristic, complemented
5#10589333C211yesmaximal, characteristic, complemented
6C3×#4468C211yesmaximal, characteristic, complemented
7#10589152C211yesmaximal, characteristic, complemented
8#70976—31nomaximal
9#68114—31nomaximal
10C22×C3×D12C2211yescharacteristic
11C22×#625—21no—
12#64133C2211yescharacteristic
13C2×C3×(C2×D3)⋊C4C2211yescharacteristic
14C3×#924C2211yescharacteristic
15C2×C3×#204—21no—
16D3×#217—21no—
17#63964—21no—
18C2×(C2×D3)⋊Dic3—21no—
19#63899—21no—
20#63896—21no—
21#63893—21no—
22#63887—21no—
23C2×#3716—21no—
24C32×C22⋊C8C2211yescharacteristic
25#63658C2211yescharacteristic
26C3×#830C2211yescharacteristic
27C3×#821—21no—
28#63563—21no—
29#5634—31no—
30#5483—31no—
31C3×#450—31no—
32#5223—31no—
33#5218—31no—
34#4960—31no—
35#4731—31no—
36#4731D311yescharacteristic, complemented
37C3×#323—31no—
38#4538—31no—
39#4468D311yescharacteristic, complemented
40#4467—31no—
41#4467—31no—
42#4467—31no—
43#4467—61no—
44C23×C3×D3—21no—
45C23×C3×D3D411yescharacteristic
46C22×C32×C4C2311yescharacteristic
47C22×C3⋊Dic3—21no—
48C2×#634D411yescharacteristic, complemented
49C22×C3×Dic3D411yescharacteristic
50C2×C3×D12—21no—
51C2×C3×D12—21no—
52C2×C3×D12—42no—
53C2×C3×Dic6D411yescharacteristic, complemented
54C2×#627—21no—
55C2×#625—42no—
56C2×#625—42no—
57C2×#625—42no—
58C2×D3×Dic3—21no—
59C2×D3×Dic3—21no—
60C2×C32×C8—21no—
61#3757—21no—
62C4⋊(C3⋊Dic3)D411yescharacteristic, complemented
63#3750—21no—
64C3×#217—21no—
65C3×(C2×D3)⋊C4—21no—
66C3×(C2×D3)⋊C4—21no—
67C3×C4⋊Dic3D411yescharacteristic, complemented
68C3×Dic3⋊C4—21no—
69C2×C3×C3⋊C8—21no—
70C3×#204—42no—
71#3724—21no—
72Dic3⋊Dic3—21no—
73Dic3⋊Dic3—21no—
74(C2×D3)⋊Dic3—21no—
75(C2×D3)⋊Dic3—21no—
76(C2×D3)⋊Dic3—42no—
77#3716—42no—
78#1596—91noSylow
79C22×C3×D4—31no—
80C22×#67—61no—
81C22×#67—61no—
82C22×D12C2×D311yescharacteristic
83C2×C3×SD16—61no—
84C3×C22⋊Q8—31no—
85C2×C3×C22⋊C4—31no—
86C2×#217—61no—
87C2×#217—61no—
88#946—61no—
89C2×#214—61no—
90C2×(C2×D3)⋊C4—61no—
91C2×(C2×D3)⋊C4C2×D311yescharacteristic
92#924—31no—
93#924—31no—
94#924—61no—
95#924—61no—
96#924C2×D311yescharacteristic
97C2×#204—21no—
98C2×#204—61no—
99#897—61no—
100#896—61no—
101#896—61no—
102D3×C22⋊C4—61no—
103#878—61no—
104#878—61no—
105#878—61no—
106#878—61no—
107#878—61no—
108C3×#102—61no—
109C3×C22⋊C8—21no—
110C3×C22⋊C8—31no—
111C3×C22⋊C8C2×D311yescharacteristic
112C3×C22⋊C8C2×D311yescharacteristic
113#832—61no—
114C3⋊#95—31no—
115C3⋊#95—31no—
116C3⋊#95—61no—
117#830C2×D311yescharacteristic
118#821—21no—
119#821—61no—
120C23×C32C2×D411yescharacteristic
121C22×C3×D3—21no—
122C22×C3×D3—21no—
123C22×C3×D3—22no—
124C22×C3×D3—41no—
125C22×C3×D3—41no—
126C22×C3×D3—41no—
127C22×C3×D3—41no—
128C22×C3×D3—41no—
129C22×C3×D3—41no—
130C22×C3×D3—41no—
131C22×C3×D3—41no—
132C22×C3×D3SD1612yescomplemented
133C2×C32×C4—41no—
134C2×C32×C4C2×D411yescharacteristic
135C2×C32×C4C2×D411yescharacteristic
136C2×C3⋊Dic3—22no—
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—21no—
139C2×C3⋊Dic3—41no—
140C2×C3⋊Dic3—41no—
141#634—41no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—21no—
145C2×C3×Dic3—41no—
146C3×D12—22no—
147C3×D12—22no—
148C3×D12—41no—
149C3×D12—41no—
150C3×Dic6—41no—
151#627—81no—
152#625—48no—
153D3×Dic3—81no—
154D3×Dic3—81no—
155C32×C8—41no—
156C3×C3⋊C8—41no—
157#543—91no—
158#450—31no—
159#450—91no—
160#392—91no—
161#328—91no—
162#328—91no—
163#323—31no—
164C24×C3—31no—
165C24×C3—61no—
166C23×D3—21no—
167C23×D3D1211yescharacteristic
168C2×C3×Q8—31no—
169C2×C3×D4—61no—
170C2×C3×D4—61no—
171C2×C3×D4—122no—
172C22×C3×C4—21no—
173C22×C3×C4—31no—
174C22×C3×C4C22×D311yescharacteristic
175C22×C3×C4C22×D311yescharacteristic
176C2×#67—122no—
177C2×#67—122no—
178C2×#67—122no—
179C2×#67—122no—
180C2×#67—122no—
181C2×#67—122no—
182C22×Dic3—61no—
183C22×Dic3—61no—
184C22×Dic3—61no—
185C22×Dic3—61no—
186C22×Dic3—121no—
187C22×Dic3D1211yescharacteristic
188C2×D12—21no—
189C2×D12—21no—
190C2×D12—42no—
191C2×C4×D3—61no—
192C2×C4×D3—61no—
193C2×Dic6—31no—
194C2×Dic6—31no—
195C2×Dic6—61no—
196C2×Dic6—61no—
197C2×Dic6—61no—
198C2×Dic6#6711yescharacteristic, complemented
199C3×SD16—122no—
200C2×C3×C8—21no—
201C2×C3×C8—21no—
202C2×C3×C8—41no—
203C2×C3×C8—61no—
204C3×C4⋊C4—31no—
205C3×C4⋊C4—61no—
206C3×C22⋊C4—61no—
207C3×C22⋊C4—61no—
208C3×C22⋊C4—61no—
209#217—21no—
210#217—61no—
211#217—61no—
212#217—61no—
213#217—61no—
214#217—121no—
215#217—122no—
216#214—122no—
217(C2×D3)⋊C4—21no—
218(C2×D3)⋊C4—21no—
219(C2×D3)⋊C4—61no—
220(C2×D3)⋊C4—61no—
221(C2×D3)⋊C4—122no—
222C4⋊Dic3—31no—
223C4⋊Dic3—31no—
224C4⋊Dic3—61no—
225C4⋊Dic3—61no—
226C4⋊Dic3—61no—
227C4⋊Dic3#6711yescharacteristic, complemented
228Dic3⋊C4—21no—
229Dic3⋊C4—61no—
230Dic3⋊C4—61no—
231Dic3⋊C4—61no—
232Dic3⋊C4—61no—
233Dic3⋊C4—61no—
234Dic3⋊C4—61no—
235Dic3⋊C4—121no—
236C2×C3⋊C8—21no—
237#204—42no—
238#204—122no—
239C22×C32—21no—
240C22×C32—21no—
241C22×C32#12011yescharacteristic
242C22×C32C2×SD1611yescharacteristic
243C22×C32#13711yescharacteristic
244C2×C3×D3—22no—
245C2×C3×D3—41no—
246C2×C3×D3—41no—
247C2×C3×D3—42no—
248C2×C3×D3—42no—
249C2×C3×D3—42no—
250C2×C3×D3—42no—
251C2×C3×D3—81no—
252C2×C3×D3—81no—
253C2×C3×D3—81no—
254C2×C3×D3—81no—
255C2×C3×D3—81no—
256C32×C4—21no—
257C32×C4—21no—
258C3⋊Dic3—42no—
259C3⋊Dic3—41no—
260C3⋊Dic3—41no—
261C3×Dic3—41no—
262C3×Dic3—41no—
263C3×Dic3—41no—
264C22×D4—31no—
265C22×D4—181no—
266C2×SD16—61no—
267C2×SD16—181no—
268C22⋊Q8—31no—
269C22⋊Q8—91no—
270C22⋊Q8—181no—
271C22⋊Q8—181no—
272C22⋊Q8—181no—
273C22⋊Q8—181no—
274C22⋊Q8—181no—
275C2×C22⋊C4—31no—
276C2×C22⋊C4—181no—
277C2×C22⋊C4—181no—
278#102—61no—
279#102—181no—
280C22⋊C8—31no—
281C22⋊C8D3211yescharacteristic
282#95—91no—
283C23×C3—21no—
284C23×C3—32no—
285C23×C3—62no—
286C23×C3—61no—
287C23×C3—61no—
288C23×C3—121no—
289C23×C3—121no—
290C23×C3—121no—
291C23×C3—121no—
292C23×C3—121no—
293C23×C3—121no—
294C23×C3—121no—
295C23×C3—121no—
296C23×C3C2×D1211yescharacteristic
297C23×C3D3×D411yescharacteristic
298C22×D3—21no—
299C22×D3—21no—
300C22×D3—22no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—41no—
305C22×D3—41no—
306C22×D3—41no—
307C22×D3—41no—
308C22×D3—41no—
309C22×D3#20412yescomplemented
310C3×Q8—121no—
311C3×D4—62no—
312C3×D4—62no—
313C3×D4—121no—
314C3×D4—121no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—81no—
323C2×C3×C4—121no—
324C2×C3×C4D3×D411yescharacteristic
325C2×C3×C4D3×D411yescharacteristic
326C2×C3×C4D3×D411yescharacteristic
327C2×C3×C4C2×#6711yescharacteristic
328#67—128no—
329#67—128no—
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—41no—
334C2×Dic3—62no—
335C2×Dic3—62no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—121no—
341C2×Dic3—121no—
342C2×Dic3—122no—
343C2×Dic3—121no—
344C2×Dic3—121no—
345C2×Dic3—121no—
346C2×Dic3—121no—
347C2×Dic3—241no—
348C2×Dic3—241no—
349C2×Dic3—241no—
350C2×Dic3—241no—
351D12—22no—
352D12—22no—
353D12—41no—
354D12—41no—
355C4×D3—241no—
356C4×D3—241no—
357Dic6—41no—
358Dic6—121no—
359Dic6—121no—
360Dic6—241no—
361Dic6—241no—
362Dic6—241no—
363C3×C8—41no—
364C3×C8—41no—
365C3×C8—81no—
366C3×C8—121no—
367C3⋊C8—41no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32#44811yescharacteristic
371C2×C32#45311yescharacteristic
372C2×C32B(4,𝔽2)11yescharacteristic
373C3×D3—42no—
374C3×D3—82no—
375C24—31no—
376C24—61no—
377C2×Q8—31no—
378C2×Q8—91no—
379C2×Q8—181no—
380C2×D4—61no—
381C2×D4—61no—
382C2×D4—122no—
383C2×D4—362no—
384C2×D4—362no—
385C2×D4—362no—
386C22×C4—31no—
387C22×C4—181no—
388C22×C4—181no—
389C22×C4—181no—
390C22×C4C2×D3211yescharacteristic
391SD16—122no—
392SD16—362no—
393C2×C8—21no—
394C2×C8—61no—
395C4⋊C4—31no—
396C4⋊C4—61no—
397C4⋊C4—91no—
398C4⋊C4—181no—
399C4⋊C4—181no—
400C4⋊C4—181no—
401C4⋊C4—181no—
402C22⋊C4—61no—
403C22⋊C4—61no—
404C22⋊C4—61no—
405C22⋊C4—181no—
406C22⋊C4—181no—
407C22⋊C4—181no—
408C22⋊C4—362no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3—62no—
419C22×C3—122no—
420C22×C3—122no—
421C22×C3—122no—
422C22×C3—121no—
423C22×C3—121no—
424C22×C3—122no—
425C22×C3—241no—
426C22×C3—241no—
427C22×C3—241no—
428C22×C3—241no—
429C22×C3—241no—
430C22×C3#88211yescharacteristic
431C22×C3C2×#20411yescharacteristic
432C22×C3#90911yescharacteristic
433C22×C3D3×SD1611yescharacteristic
434C22×C3#91511yescharacteristic
435C22×C3#93711yescharacteristic
436C2×D3—22no—
437C2×D3—41no—
438C2×D3—41no—
439C2×D3—42no—
440C2×D3—42no—
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—81no—
444C2×D3—81no—
445C2×D3—81no—
446C2×D3—81no—
447C2×D3—81no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—121no—
455C3×C4—121no—
456C3×C4—121no—
457Dic3—41no—
458Dic3—41no—
459Dic3—41no—
460Dic3—122no—
461Dic3—122no—
462Dic3—121no—
463Dic3—121no—
464Dic3—121no—
465Dic3—121no—
466Dic3—241no—
467Dic3—241no—
468Dic3—242no—
469C32#159611yescharacteristic, complemented, Sylow
470C23—32no—
471C23—62no—
472C23—61no—
473C23—61no—
474C23—121no—
475C23—121no—
476C23—121no—
477C23—121no—
478C23—121no—
479C23—121no—
480C23—121no—
481C23—121no—
482C23D3×D1211yescharacteristic
483Q8—121no—
484Q8—361no—
485Q8—721no—
486D4—62no—
487D4—62no—
488D4—121no—
489D4—121no—
490D4—368no—
491C2×C4—41no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—121no—
496C2×C4—182no—
497C2×C4—181no—
498C2×C4—181no—
499C2×C4—361no—
500C2×C4—361no—
501C2×C4—721no—
502C2×C4—721no—
503C2×C4#380211yescharacteristic
504C2×C4D3×#6711yescharacteristic
505C8—41no—
506C8—121no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—41no—
515C2×C3—41no—
516C2×C3—122no—
517C2×C3—242no—
518C2×C3#473811yescharacteristic
519C2×C3#474111yescharacteristic
520C2×C3#474711yescharacteristic
521C2×C3#503211yescharacteristic
522C2×C3#517011yescharacteristic
523C2×C3#519811yescharacteristic
524D3—42no—
525D3—82no—
526C22—21no—
527C22—21no—
528C22—62no—
529C22—122no—
530C22—122no—
531C22—122no—
532C22—121no—
533C22—121no—
534C22—122no—
535C22—241no—
536C22—241no—
537C22—241no—
538C22—241no—
539C22—241no—
540C22D3×#20411yescharacteristic
541C22#6380011yescharacteristic
542C22#6392211yescharacteristic
543C4—21no—
544C4—21no—
545C4—121no—
546C4—121no—
547C4—121no—
548C4—362no—
549C4—361no—
550C4—361no—
551C3—21no—
552C3#6811411yescharacteristic, complemented
553C3#7097611yescharacteristic, complemented
554C2—21no—
555C2—21no—
556C2—122no—
557C2—242no—
558C2#1059040711yescharacteristic
559C2#1059082211yescharacteristic
560C2#1059101411yescharacteristic
subgroupG/H
centerC22#63922
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC32×C22⋊C8C22
socleC22×C32#120
prime radical#10591389C2
hypercenterC22⋊C8D32
hypocenterC32#1596
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
1. #10594971C2semidirect
2. C3×#4731C2semidirect
4. #10591172C2semidirect
5. #10589333C2semidirect
6. C3×#4468C2semidirect
7. #10589152C2semidirect
36. #4731D3semidirect
39. #4468D3semidirect
48. C2×#634D4semidirect
53. C2×C3×Dic6D4semidirect
62. C4⋊(C3⋊Dic3)D4semidirect
67. C3×C4⋊Dic3D4semidirect
132. C22×C3×D3SD16semidirect
198. C2×Dic6#67semidirect
227. C4⋊Dic3#67semidirect
309. C22×D3#204semidirect
469. C32#1596semidirect
552. C3#68114semidirect
553. C3#70976semidirect

Maximal subgroups

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

subgroupindex
#105949712
C3×#47312
#105913892
#105911722
#105893332
C3×#44682
#105891522
#709763
#681143