Finite Group Databasebeta

← #13156012

Subgroups — #13156012

Order 1728.

710 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C2×#88811C211yesmaximal, characteristic
2#11723501C211yesmaximal, characteristic
3#11723451C211yesmaximal, characteristic, complemented
4C2×C3×#3717C211yesmaximal, characteristic
5#11722050C211yesmaximal, characteristic, complemented
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×#63565C211yesmaximal, characteristic, complemented
8#10591569—31nomaximal
9#10590748—31nomaximal
10#10590565—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×Dic3×C3⋊D3—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88811—22no—
20#88806—21no—
21#88806—21no—
22#88803—21no—
23C3×#3717—22no—
24#64153—31no—
25#64102—31no—
26C2×C3×#215—31no—
27C2×C3×#204—31no—
28#63898—31no—
29#63898—31no—
30#63898—61no—
31#63873—31no—
32#63873—31no—
33#63873—61no—
34C2×#3717—31no—
35C2×#3717D311yescharacteristic
36C2×#3716—31no—
37C2×#3715—31no—
38#63641—31no—
39C3×#835—31no—
40C3×#816—31no—
41C3×#810—31no—
42C3×#810—31no—
43#63572—31no—
44#63572—31no—
45#63572—61no—
46#63567D311yescharacteristic, complemented
47#63565D311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)D411yescharacteristic
51C2×C3⋊(C3⋊Dic3)D411yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)D412yes—
55C3×#634—21no—
56C3×#634—21no—
57C2×C32×Dic3—21no—
58C2×C32×Dic3—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61Dic3×C3⋊D3—42no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64#5162—91no—
65#4818—91no—
66#4818—91no—
67C2×C32×Q8—31no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×C4×C3⋊D3—61no—
70C2×#634C2×D311yescharacteristic
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—61no—
74C2×C3×Dic6—21no—
75C2×C3×Dic6—31no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—61no—
78C2×C3×Dic6C2×D311yescharacteristic
79C2×C3×Dic6C2×D311yescharacteristic
80C2×D3×Dic3—61no—
81C2×D3×Dic3—61no—
82C2×D3×Dic3—62no—
83C2×C32×C8—31no—
84C32×C4⋊C4—31no—
85#3752—61no—
86#3750—61no—
87#3750—61no—
88C4×C3⋊Dic3—31no—
89C4×C3⋊Dic3—31no—
90C4×C3⋊Dic3—31no—
91C4×C3⋊Dic3—61no—
92C4×C3⋊Dic3—61no—
93C4×C3⋊Dic3—61no—
94C4×C3⋊Dic3—121no—
95C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
96C3×#215—62no—
97C3×C4⋊Dic3—21no—
98C3×C4⋊Dic3C2×D311yescharacteristic
99C3×C4⋊Dic3C2×D311yescharacteristic
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—61no—
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C3×#204—62no—
107#3724—61no—
108#3724—61no—
109#3724—61no—
110#3724—61no—
111#3724—62no—
112#3724—62no—
113(C2×D3)⋊Dic3—61no—
114(C2×D3)⋊Dic3—61no—
115(C2×D3)⋊Dic3—62no—
116#3717—22no—
117#3717—62no—
118#3716—62no—
119#3715—62no—
120C22×C33C2×D411yescharacteristic
121C2×C3×C3⋊D3—22no—
122C2×C3×C3⋊D3—41no—
123C2×C3×C3⋊D3—41no—
124C33×C4C2×D411yescharacteristic
125C33×C4#4111yescharacteristic
126C3⋊(C3⋊Dic3)—21no—
127C3⋊(C3⋊Dic3)SD1612yescomplemented
128C3×C3⋊Dic3—41no—
129C32×Dic3—41no—
130C32×Dic3—41no—
131C2×C3×SD16—91no—
132#944—91no—
133#944—91no—
134#944—91no—
135#944—181no—
136C2×#215—31no—
137C2×#215—91no—
138D4×Dic3—91no—
139C2×#214—91no—
140C2×#204—31no—
141C2×#204—91no—
142#893—91no—
143#893—91no—
144#893—181no—
145C3×#103—91no—
146C3×#103—91no—
147#835—31no—
148#816—31no—
149Dic3⋊C8—91no—
150Dic3⋊C8—91no—
151Dic3⋊C8—91no—
152Dic3⋊C8—181no—
153#810—31no—
154#810—31no—
155C22×C3⋊D3—21no—
156C22×C3×D3—62no—
157C22×C3×D3—61no—
158C22×C3×D3—61no—
159C32×Q8—61no—
160C32×Q8—61no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—41no—
165C2×C32×C4—61no—
166C2×C32×C4—61no—
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3—31no—
172C2×C3⋊Dic3—31no—
173C2×C3⋊Dic3—31no—
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—31no—
177C2×C3⋊Dic3—61no—
178C2×C3⋊Dic3—61no—
179C2×C3⋊Dic3—61no—
180C2×C3⋊Dic3—61no—
181C2×C3⋊Dic3—61no—
182C2×C3⋊Dic3—61no—
183C2×C3⋊Dic3—121no—
184C2×C3⋊Dic3—121no—
185C4⋊(C3⋊D3)—21no—
186C4⋊(C3⋊D3)#6712yes—
187C4×C3⋊D3—122no—
188#634—21no—
189#634—21no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3—22no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—62no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—61no—
199C3×D12—32no—
200C3×D12—32no—
201C3×D12—62no—
202C3×D12—62no—
203C3×D12—61no—
204C3×D12—61no—
205C3×Dic6—21no—
206C3×Dic6—21no—
207C3×Dic6—21no—
208C3×Dic6—21no—
209C3×Dic6—41no—
210C3×Dic6—41no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—61no—
214C3×Dic6—61no—
215C3×Dic6—121no—
216C3×Dic6—121no—
217D3×Dic3—122no—
218D3×Dic3—122no—
219D3×Dic3—124no—
220C32×C8—61no—
221C3⋊(C3⋊C8)—21no—
222C3×C3⋊C8—21no—
223C3×C3⋊C8—21no—
224C3×C3⋊C8—41no—
225C3×C3⋊C8—61no—
226C3×C3⋊C8—61no—
227C3×C3⋊C8—121no—
228#461—271noSylow
229C2×C33#12111yescharacteristic
230C2×C33C2×SD1611yescharacteristic
231C2×C33#13711yescharacteristic
232C3×C3⋊D3—42no—
233C2×C3×Q8—31no—
234C2×C3×Q8—31no—
235C2×C3×Q8—61no—
236C2×C3×Q8—91no—
237C2×C3×D4—91no—
238C22×Dic3—181no—
239C2×D12—31no—
240C2×D12—31no—
241C2×D12—61no—
242C2×C4×D3—181no—
243C2×C4×D3—181no—
244C2×C4×D3—182no—
245C2×Dic6—31no—
246C2×Dic6—31no—
247C2×Dic6—61no—
248C2×Dic6D3211yescharacteristic
249C3×SD16—182no—
250C2×C3×C8—31no—
251C2×C3×C8—31no—
252C2×C3×C8—61no—
253C2×C3×C8—91no—
254C3×C4⋊C4—31no—
255C3×C4⋊C4—31no—
256C3×C4⋊C4—61no—
257#217—181no—
258#215—62no—
259#215—182no—
260#214—182no—
261(C2×D3)⋊C4—181no—
262(C2×D3)⋊C4—181no—
263(C2×D3)⋊C4—182no—
264C4⋊Dic3D3211yescharacteristic
265Dic3⋊C4—181no—
266Dic3⋊C4—181no—
267Dic3⋊C4—181no—
268Dic3⋊C4—181no—
269Dic3⋊C4—181no—
270Dic3⋊C4—181no—
271Dic3⋊C4—182no—
272Dic3⋊C4—182no—
273C4×Dic3—91no—
274C4×Dic3—91no—
275C4×Dic3—91no—
276C4×Dic3—181no—
277C4×Dic3—181no—
278C4×Dic3—181no—
279C4×Dic3—361no—
280C2×C3⋊C8—31no—
281C2×C3⋊C8—31no—
282C2×C3⋊C8—61no—
283C2×C3⋊C8D3211yescharacteristic
284#204—62no—
285#204—182no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—41no—
290C22×C32D3×D411yescharacteristic
291C22×C32D3×D411yescharacteristic
292C22×C32D3×D411yescharacteristic
293C2×C3⋊D3—22no—
294C2×C3⋊D3—41no—
295C2×C3⋊D3—41no—
296C2×C3×D3—64no—
297C2×C3×D3—62no—
298C2×C3×D3—62no—
299C2×C3×D3—122no—
300C2×C3×D3—122no—
301C2×C3×D3—121no—
302C2×C3×D3—121no—
303C2×C3×D3—121no—
304C2×C3×D3—121no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—41no—
312C32×C4—41no—
313C32×C4—121no—
314C32×C4—121no—
315C32×C4#23511yescharacteristic
316C32×C4#23511yescharacteristic
317C32×C4D3×D411yescharacteristic
318C32×C4D3×D411yescharacteristic
319C32×C4#23711yescharacteristic
320C32×C4C2×#6711yescharacteristic
321C3⋊Dic3—32no—
322C3⋊Dic3—32no—
323C3⋊Dic3—32no—
324C3⋊Dic3—41no—
325C3⋊Dic3—62no—
326C3⋊Dic3—62no—
327C3⋊Dic3—62no—
328C3⋊Dic3—61no—
329C3⋊Dic3—62no—
330C3⋊Dic3—61no—
331C3⋊Dic3—61no—
332C3⋊Dic3—122no—
333C3⋊Dic3—121no—
334C3⋊Dic3—121no—
335C3⋊Dic3—122no—
336C3×Dic3—42no—
337C3×Dic3—42no—
338C3×Dic3—41no—
339C3×Dic3—41no—
340C3×Dic3—41no—
341C3×Dic3—41no—
342C3×Dic3—122no—
343C3×Dic3—121no—
344C3×Dic3—121no—
345C2×SD16—91no—
346C2×SD16—271no—
347C4⋊Q8—271no—
348C4×D4—271no—
349C4⋊C8—271no—
350#103—91no—
351#103—91no—
352C33#46111yescharacteristic, complemented, Sylow
353C23×C3—181no—
354C22×D3—62no—
355C22×D3—61no—
356C22×D3—61no—
357C3×Q8—61no—
358C3×Q8—61no—
359C3×Q8—61no—
360C3×Q8—61no—
361C3×Q8—121no—
362C3×Q8—121no—
363C3×Q8—181no—
364C3×Q8—181no—
365C3×D4—92no—
366C3×D4—181no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—41no—
371C2×C3×C4—62no—
372C2×C3×C4—62no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—181no—
378C2×C3×C4C2×D3211yescharacteristic
379C2×C3×C4C2×D3211yescharacteristic
380C2×C3×C4C2×D3211yescharacteristic
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—62no—
384C2×Dic3—61no—
385C2×Dic3—61no—
386C2×Dic3—91no—
387C2×Dic3—91no—
388C2×Dic3—91no—
389C2×Dic3—91no—
390C2×Dic3—91no—
391C2×Dic3—91no—
392C2×Dic3—181no—
393C2×Dic3—181no—
394C2×Dic3—181no—
395C2×Dic3—181no—
396C2×Dic3—181no—
397C2×Dic3—181no—
398C2×Dic3—361no—
399C2×Dic3—361no—
400C2×Dic3—362no—
401D12—32no—
402D12—32no—
403D12—62no—
404D12—62no—
405D12—61no—
406D12—61no—
407C4×D3—362no—
408C4×D3—362no—
409C4×D3—364no—
410Dic6—21no—
411Dic6—21no—
412Dic6—61no—
413Dic6—61no—
414Dic6—61no—
415Dic6—61no—
416Dic6—121no—
417Dic6—121no—
418C3×C8—61no—
419C3×C8—61no—
420C3×C8—121no—
421C3×C8—181no—
422C3⋊C8—21no—
423C3⋊C8—61no—
424C3⋊C8—61no—
425C3⋊C8—121no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—41no—
436C2×C32—41no—
437C2×C32—41no—
438C2×C32#88411yescharacteristic
439C2×C32#88411yescharacteristic
440C2×C32D3×SD1611yescharacteristic
441C2×C32D3×SD1611yescharacteristic
442C2×C32D3×SD1611yescharacteristic
443C2×C32#91511yescharacteristic
444C2×C32#91811yescharacteristic
445C2×C32#91811yescharacteristic
446C2×C32#93911yescharacteristic
447C3⋊D3—42no—
448C3×D3—124no—
449C3×D3—122no—
450C3×D3—122no—
451C2×Q8—31no—
452C2×Q8—91no—
453C2×D4—91no—
454C22×C4—541no—
455SD16—182no—
456SD16—542no—
457C2×C8—31no—
458C2×C8—91no—
459C4⋊C4—31no—
460C4⋊C4—541no—
461C4⋊C4—541no—
462C22⋊C4—541no—
463C42—271no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—41no—
468C22×C3—182no—
469C22×C3—361no—
470C22×C3—361no—
471C22×C3#380211yescharacteristic
472C22×C3#381111yescharacteristic
473C22×C3#381111yescharacteristic
474C2×D3—64no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—122no—
478C2×D3—122no—
479C2×D3—121no—
480C2×D3—121no—
481C2×D3—121no—
482C2×D3—121no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—122no—
492C3×C4—122no—
493C3×C4—121no—
494C3×C4—121no—
495C3×C4—121no—
496C3×C4—121no—
497C3×C4—361no—
498C3×C4#379511yescharacteristic
499C3×C4#379811yescharacteristic
500C3×C4#380211yescharacteristic
501C3×C4#380411yescharacteristic
502C3×C4D3×#6711yescharacteristic
503C3×C4D3×#6711yescharacteristic
504Dic3—41no—
505Dic3—41no—
506Dic3—92no—
507Dic3—92no—
508Dic3—92no—
509Dic3—122no—
510Dic3—121no—
511Dic3—121no—
512Dic3—182no—
513Dic3—182no—
514Dic3—182no—
515Dic3—182no—
516Dic3—181no—
517Dic3—181no—
518Dic3—181no—
519Dic3—362no—
520Dic3—361no—
521Dic3—361no—
522Dic3—362no—
523C32—21no—
524C32—21no—
525C32—21no—
526C32—41no—
527C32#481811yescharacteristic, complemented
528C32#481811yescharacteristic, complemented
529C32#516211yescharacteristic, complemented
530C23—181no—
531Q8—61no—
532Q8—61no—
533Q8—181no—
534Q8—181no—
535D4—92no—
536D4—181no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—181no—
540C2×C4—271no—
541C2×C4—271no—
542C2×C4—1082no—
543C2×C4D3311yescharacteristic
544C8—61no—
545C8—181no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—41no—
556C2×C3—41no—
557C2×C3—41no—
558C2×C3—362no—
559C2×C3#6379611yescharacteristic
560C2×C3#6381111yescharacteristic
561C2×C3#6388511yescharacteristic
562C2×C3#6391211yescharacteristic
563C2×C3#6393011yescharacteristic
564C2×C3#6394011yescharacteristic
565C2×C3#6394511yescharacteristic
566C2×C3#6395111yescharacteristic
567C2×C3#6397111yescharacteristic
568D3—124no—
569D3—122no—
570D3—122no—
571C22—182no—
572C22—361no—
573C22—361no—
574C22#8895411yescharacteristic
575C4—121no—
576C4—121no—
577C4—272no—
578C4—361no—
579C4—541no—
580C4#8895311yescharacteristic
581C4#8896011yescharacteristic
582C3—21no—
583C3—21no—
584C3—21no—
585C3—41no—
586C3#1059056511yescharacteristic, complemented
587C3#1059074811yescharacteristic, complemented
588C3#1059156911yescharacteristic, complemented
589C2—362no—
590C2#1172284111yescharacteristic
591C2#1172289411yescharacteristic
592C2#1172296711yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#461
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
3. #11723451C2semidirect
5. #11722050C2semidirect
6. C3×#63567C2semidirect
7. C3×#63565C2semidirect
46. #63567D3semidirect
47. #63565D3semidirect
127. C3⋊(C3⋊Dic3)SD16semidirect
352. C33#461semidirect
527. C32#4818semidirect
528. C32#4818semidirect
529. C32#5162semidirect
586. C3#10590565semidirect
587. C3#10590748semidirect
588. C3#10591569semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
#117235012
#117234512
C2×C3×#37172
#117220502
C3×#635672
C3×#635652
#105915693
#105907483
#105905653