Finite Group Databasebeta

← #13156438

Subgroups — #13156438

Order 1728.

698 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 96 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#11723531C211yesmaximal, characteristic
2#11723413C211yesmaximal, characteristic, complemented
3#11722080C211yesmaximal, characteristic, complemented
4#11722024C211yesmaximal, characteristic, complemented
5C3×#63573C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic, complemented
8#10591425—31nomaximal
9#10590652—31nomaximal
10#10590513—31nomaximal
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6291—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88804—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21Dic3×C3⋊Dic3—21no—
22#64147—31no—
23#64101—31no—
24#63916—31no—
25#63916—31no—
26#63916—61no—
27#63849—31no—
28#63849—31no—
29#63849—61no—
30#63647—31no—
31C3×#832—31no—
32C3×#832—31no—
33C3×#821—31no—
34C3×#817—31no—
35C3×#809—31no—
36C3×#808—31no—
37#63576—31no—
38#63575—31no—
39#63575—31no—
40#63573D311yescharacteristic, complemented
41#63563D311yescharacteristic, complemented
42#63562D311yescharacteristic, complemented
43#63557—31no—
44#63557—31no—
45#63557—61no—
46C22×C3⋊(C3⋊D3)D411yescharacteristic
47C22×C32×D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)D411yescharacteristic
50C4×C3⋊(C3⋊D3)—21no—
51C4×C3⋊(C3⋊D3)—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C32×Dic3—21no—
55C32×D12—21no—
56C32×D12—21no—
57#6291—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60#4866—91no—
61#4784—91no—
62#4779—91no—
63C2×C32×D4—31no—
64C2×#638—61no—
65C2×C4×C3⋊D3—31no—
66C2×C4×C3⋊D3—31no—
67C2×C4×C3⋊D3—31no—
68C2×C4×C3⋊D3—61no—
69C2×C4×C3⋊D3—61no—
70C2×C4×C3⋊D3—61no—
71C2×C4×C3⋊D3—121no—
72C2×C3×D12—21no—
73C2×C3×D12C2×D311yescharacteristic
74C2×C3×D12C2×D311yescharacteristic
75C2×#626—61no—
76C2×#626—61no—
77C2×#626—62no—
78C2×C32×C8—31no—
79C32×C4⋊C4—31no—
80#3757—61no—
81#3752—61no—
82C4⋊(C3⋊Dic3)C2×D311yescharacteristic
83C4⋊(C3⋊Dic3)C2×D311yescharacteristic
84C4×C3⋊Dic3—61no—
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3—61no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C2×D311yescharacteristic
95C2×C3×C3⋊C8—21no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—61no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101#3722—61no—
102#3722—61no—
103#3722—62no—
104(C2×D3)⋊Dic3—61no—
105(C2×D3)⋊Dic3—61no—
106(C2×D3)⋊Dic3—62no—
107Dic32—61no—
108Dic32—61no—
109Dic32—62no—
110C22×C33C2×D411yescharacteristic
111C2×C3⋊(C3⋊D3)—21no—
112C2×C3⋊(C3⋊D3)—21no—
113C2×C3⋊(C3⋊D3)—21no—
114C2×C32×D3—41no—
115C2×C32×D3—41no—
116C2×C32×D3—41no—
117C33×C4#4111yescharacteristic
118C33×C4#4111yescharacteristic
119C3⋊(C3⋊Dic3)—21no—
120C3×C3⋊Dic3—41no—
121C3×C3⋊Dic3—41no—
122C32×Dic3—41no—
123#938—91no—
124#938—91no—
125#938—181no—
126#895—91no—
127#892—91no—
128#892—91no—
129#892—91no—
130#892—181no—
131C3×#106—91no—
132C3×#102—91no—
133C3×#102—91no—
134#832—31no—
135#832—31no—
136#821—31no—
137(C2×D3)⋊C8—91no—
138(C2×D3)⋊C8—91no—
139(C2×D3)⋊C8—91no—
140(C2×D3)⋊C8—181no—
141#818—91no—
142#817—31no—
143#809—31no—
144#808—31no—
145#807—91no—
146#807—91no—
147C23×C32—61no—
148C22×C3⋊D3—31no—
149C22×C3⋊D3—31no—
150C22×C3⋊D3—31no—
151C22×C3⋊D3—61no—
152C22×C3⋊D3—61no—
153C22×C3⋊D3—61no—
154C22×C3⋊D3—121no—
155C22×C3×D3—22no—
156C22×C3×D3—21no—
157C22×C3×D3—21no—
158C32×D4—61no—
159C32×D4—61no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—41no—
164C2×C32×C4—61no—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168#638—122no—
169C2×C3⋊Dic3—21no—
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3—31no—
172C2×C3⋊Dic3—31no—
173C2×C3⋊Dic3—31no—
174C2×C3⋊Dic3—61no—
175C2×C3⋊Dic3—61no—
176C2×C3⋊Dic3—61no—
177C2×C3⋊Dic3—121no—
178C4×C3⋊D3—61no—
179C4×C3⋊D3—61no—
180C4×C3⋊D3—62no—
181C4×C3⋊D3—62no—
182C4×C3⋊D3—61no—
183C4×C3⋊D3—61no—
184C4×C3⋊D3—61no—
185C4×C3⋊D3—61no—
186C4×C3⋊D3—121no—
187C4×C3⋊D3—121no—
188C4×C3⋊D3—121no—
189C4×C3⋊D3—121no—
190C4×C3⋊D3—122no—
191C4×C3⋊D3—122no—
192C2×C3×Dic3—22no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—62no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—61no—
201C3×D12—21no—
202C3×D12—21no—
203C3×D12—21no—
204C3×D12—21no—
205C3×D12—41no—
206C3×D12—41no—
207#626—124no—
208#626—122no—
209#626—122no—
210C32×C8—61no—
211C3⋊(C3⋊C8)—21no—
212C3×C3⋊C8—21no—
213C3×C3⋊C8—21no—
214C3×C3⋊C8—41no—
215C3×C3⋊C8—61no—
216C3×C3⋊C8—61no—
217C3×C3⋊C8—121no—
218#482—271noSylow
219C2×C33#12311yescharacteristic
220C2×C33#13511yescharacteristic
221C2×C33AGL(1,ℤ/8)11yescharacteristic
222C3⋊(C3⋊D3)—41no—
223C32×D3—81no—
224C2×C3×D4—31no—
225C2×C3×D4—31no—
226C2×C3×D4—61no—
227C2×#67—181no—
228C2×#67—181no—
229C2×#67—182no—
230C2×D12—181no—
231C2×D12D3211yescharacteristic
232C2×C4×D3—91no—
233C2×C4×D3—91no—
234C2×C4×D3—91no—
235C2×C4×D3—181no—
236C2×C4×D3—181no—
237C2×C4×D3—181no—
238C2×C4×D3—361no—
239C2×C3×C8—31no—
240C2×C3×C8—31no—
241C2×C3×C8—61no—
242C2×C3×C8—91no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—61no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—91no—
248#217—181no—
249#217—181no—
250#217—182no—
251(C2×D3)⋊C4—181no—
252(C2×D3)⋊C4—181no—
253(C2×D3)⋊C4—181no—
254(C2×D3)⋊C4—182no—
255(C2×D3)⋊C4—181no—
256C4⋊Dic3—31no—
257C4⋊Dic3—31no—
258C4⋊Dic3—31no—
259C4⋊Dic3—31no—
260C4⋊Dic3—61no—
261C4⋊Dic3—61no—
262C4⋊Dic3D3211yescharacteristic
263C4×Dic3—181no—
264C4×Dic3—181no—
265C4×Dic3—181no—
266C4×Dic3—182no—
267C2×C3⋊C8—31no—
268C2×C3⋊C8—31no—
269C2×C3⋊C8—61no—
270C2×C3⋊C8D3211yescharacteristic
271C22×C32—21no—
272C22×C32—21no—
273C22×C32—21no—
274C22×C32—41no—
275C22×C32—121no—
276C22×C32—121no—
277C22×C32—121no—
278C22×C32D3×D411yescharacteristic
279C22×C32D3×D411yescharacteristic
280C22×C32D3×D411yescharacteristic
281C2×C3⋊D3—61no—
282C2×C3⋊D3—61no—
283C2×C3⋊D3—61no—
284C2×C3⋊D3—62no—
285C2×C3⋊D3—62no—
286C2×C3⋊D3—62no—
287C2×C3⋊D3—61no—
288C2×C3⋊D3—61no—
289C2×C3⋊D3—61no—
290C2×C3⋊D3—61no—
291C2×C3⋊D3—61no—
292C2×C3⋊D3—61no—
293C2×C3⋊D3—121no—
294C2×C3⋊D3—121no—
295C2×C3⋊D3—121no—
296C2×C3⋊D3—121no—
297C2×C3⋊D3—121no—
298C2×C3⋊D3—121no—
299C2×C3⋊D3—122no—
300C2×C3⋊D3—122no—
301C2×C3⋊D3—122no—
302C2×C3×D3—42no—
303C2×C3×D3—42no—
304C2×C3×D3—42no—
305C2×C3×D3—41no—
306C2×C3×D3—41no—
307C2×C3×D3—41no—
308C2×C3×D3—41no—
309C2×C3×D3—41no—
310C2×C3×D3—41no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—41no—
318C32×C4—41no—
319C32×C4—121no—
320C32×C4#23511yescharacteristic
321C32×C4#23511yescharacteristic
322C32×C4#23511yescharacteristic
323C32×C4#23711yescharacteristic
324C32×C4#23711yescharacteristic
325C32×C4#23911yescharacteristic
326C3⋊Dic3—41no—
327C3⋊Dic3—41no—
328C3⋊Dic3—61no—
329C3⋊Dic3—62no—
330C3⋊Dic3—61no—
331C3⋊Dic3—61no—
332C3⋊Dic3—121no—
333C3⋊Dic3—121no—
334C3⋊Dic3—122no—
335C3×Dic3—42no—
336C3×Dic3—41no—
337C3×Dic3—41no—
338C3×Dic3—122no—
339C3×Dic3—122no—
340C3×Dic3—121no—
341C3×Dic3—121no—
342C3×Dic3—121no—
343C3×Dic3—121no—
344#121—271no—
345#117—271no—
346#107—271no—
347#106—91no—
348#102—91no—
349#102—91no—
350C22⋊C8—271no—
351C33#48211yescharacteristic, complemented, Sylow
352C23×C3—62no—
353C23×C3—61no—
354C23×C3—61no—
355C22×D3—21no—
356C22×D3—91no—
357C22×D3—91no—
358C22×D3—91no—
359C22×D3—181no—
360C22×D3—181no—
361C22×D3—181no—
362C22×D3—361no—
363C3×D4—61no—
364C3×D4—61no—
365C3×D4—61no—
366C3×D4—61no—
367C3×D4—121no—
368C3×D4—121no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—41no—
373C2×C3×C4—62no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—181no—
377C2×C3×C4—181no—
378C2×C3×C4C2×D3211yescharacteristic
379C2×C3×C4C2×D3211yescharacteristic
380C2×C3×C4C2×D3211yescharacteristic
381#67—364no—
382#67—362no—
383#67—362no—
384C2×Dic3—21no—
385C2×Dic3—62no—
386C2×Dic3—62no—
387C2×Dic3—61no—
388C2×Dic3—61no—
389C2×Dic3—61no—
390C2×Dic3—61no—
391C2×Dic3—91no—
392C2×Dic3—91no—
393C2×Dic3—91no—
394C2×Dic3—181no—
395C2×Dic3—181no—
396C2×Dic3—181no—
397C2×Dic3—361no—
398D12—21no—
399D12—21no—
400D12—362no—
401C4×D3—182no—
402C4×D3—182no—
403C4×D3—181no—
404C4×D3—181no—
405C4×D3—181no—
406C4×D3—181no—
407C4×D3—181no—
408C4×D3—181no—
409C4×D3—361no—
410C4×D3—361no—
411C4×D3—361no—
412C4×D3—361no—
413C4×D3—362no—
414C4×D3—362no—
415C3×C8—61no—
416C3×C8—61no—
417C3×C8—121no—
418C3×C8—181no—
419C3⋊C8—21no—
420C3⋊C8—61no—
421C3⋊C8—61no—
422C3⋊C8—121no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—41no—
433C2×C32—41no—
434C2×C32—41no—
435C2×C32—241no—
436C2×C32#88311yescharacteristic
437C2×C32#88311yescharacteristic
438C2×C32#89411yescharacteristic
439C2×C32#91111yescharacteristic
440C2×C32#91211yescharacteristic
441C2×C32#91411yescharacteristic
442C2×C32#91411yescharacteristic
443C2×C32#91611yescharacteristic
444C2×C32#91611yescharacteristic
445C3⋊D3—122no—
446C3⋊D3—122no—
447C3⋊D3—122no—
448C3⋊D3—124no—
449C3⋊D3—121no—
450C3⋊D3—121no—
451C3⋊D3—121no—
452C3×D3—82no—
453C3×D3—81no—
454C3×D3—81no—
455C2×D4—31no—
456C2×D4—541no—
457C22×C4—271no—
458C2×C8—31no—
459C2×C8—91no—
460C4⋊C4—31no—
461C4⋊C4—91no—
462C4⋊C4—91no—
463C22⋊C4—541no—
464C22⋊C4—541no—
465C42—541no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—41no—
470C22×C3—122no—
471C22×C3—122no—
472C22×C3—122no—
473C22×C3—121no—
474C22×C3—121no—
475C22×C3—121no—
476C22×C3—121no—
477C22×C3—121no—
478C22×C3—121no—
479C22×C3#380211yescharacteristic
480C22×C3#381111yescharacteristic
481C22×C3#381111yescharacteristic
482C2×D3—41no—
483C2×D3—41no—
484C2×D3—41no—
485C2×D3—182no—
486C2×D3—182no—
487C2×D3—182no—
488C2×D3—181no—
489C2×D3—181no—
490C2×D3—181no—
491C2×D3—181no—
492C2×D3—181no—
493C2×D3—181no—
494C2×D3—181no—
495C2×D3—181no—
496C2×D3—181no—
497C2×D3—361no—
498C2×D3—361no—
499C2×D3—361no—
500C2×D3—361no—
501C2×D3—361no—
502C2×D3—361no—
503C2×D3—362no—
504C2×D3—362no—
505C2×D3—362no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4—122no—
515C3×C4—121no—
516C3×C4—121no—
517C3×C4—361no—
518C3×C4—361no—
519C3×C4#379611yescharacteristic
520C3×C4#379811yescharacteristic
521C3×C4#379911yescharacteristic
522C3×C4#380411yescharacteristic
523C3×C4#380411yescharacteristic
524C3×C4#380411yescharacteristic
525Dic3—41no—
526Dic3—122no—
527Dic3—122no—
528Dic3—121no—
529Dic3—121no—
530Dic3—121no—
531Dic3—121no—
532Dic3—182no—
533Dic3—181no—
534Dic3—181no—
535Dic3—181no—
536Dic3—361no—
537Dic3—361no—
538Dic3—362no—
539C32—21no—
540C32—21no—
541C32—21no—
542C32—41no—
543C32#477911yescharacteristic, complemented
544C32#478411yescharacteristic, complemented
545C32#486611yescharacteristic, complemented
546C23—61no—
547C23—271no—
548D4—61no—
549D4—61no—
550D4—1082no—
551C2×C4—61no—
552C2×C4—181no—
553C2×C4—181no—
554C2×C4—271no—
555C2×C4—541no—
556C2×C4—541no—
557C2×C4D3311yescharacteristic
558C8—61no—
559C8—181no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—41no—
570C2×C3—41no—
571C2×C3—41no—
572C2×C3—242no—
573C2×C3—241no—
574C2×C3—241no—
575C2×C3#6379811yescharacteristic
576C2×C3#6380911yescharacteristic
577C2×C3#6385311yescharacteristic
578C2×C3#6385411yescharacteristic
579C2×C3#6393611yescharacteristic
580C2×C3#6393611yescharacteristic
581C2×C3#6393711yescharacteristic
582C2×C3#6394111yescharacteristic
583C2×C3#6397511yescharacteristic
584D3—81no—
585D3—362no—
586D3—362no—
587D3—362no—
588D3—364no—
589D3—361no—
590D3—361no—
591D3—361no—
592C22—121no—
593C22—121no—
594C22—121no—
595C22—541no—
596C22—541no—
597C22—541no—
598C22#8895411yescharacteristic
599C4—121no—
600C4—361no—
601C4—361no—
602C4—541no—
603C4#8895911yescharacteristic
604C4#8896211yescharacteristic
605C3—21no—
606C3—21no—
607C3—21no—
608C3—41no—
609C3#1059051311yescharacteristic, complemented
610C3#1059065211yescharacteristic, complemented
611C3#1059142511yescharacteristic, complemented
612C2—241no—
613C2—1081no—
614C2#1172286111yescharacteristic
615C2#1172287811yescharacteristic
616C2#1172299611yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723531C2
hypocenterC33#482
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723413C2semidirect
3. #11722080C2semidirect
4. #11722024C2semidirect
5. C3×#63573C2semidirect
6. C3×#63563C2semidirect
7. C3×#63562C2semidirect
40. #63573D3semidirect
41. #63563D3semidirect
42. #63562D3semidirect
351. C33#482semidirect
543. C32#4779semidirect
544. C32#4784semidirect
545. C32#4866semidirect
609. C3#10590513semidirect
610. C3#10590652semidirect
611. C3#10591425semidirect

Maximal subgroups

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

subgroupindex
#117235312
#117234132
#117220802
#117220242
C3×#635732
C3×#635632
C3×#635622
#105914253
#105906523
#105905133