Finite Group Databasebeta

← #13156504

Subgroups — #13156504

Order 1728.

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

normal subgroups only

This group has 638 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#11723449C211yesmaximal, characteristic
2#11723413C211yesmaximal, characteristic
3#11722076C211yesmaximal, characteristic, complemented
4#11722024C211yesmaximal, characteristic
5C3×#63576C211yesmaximal, characteristic, complemented
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×#63567C211yesmaximal, characteristic, complemented
8#10590713—31nomaximal
9#10590526—31nomaximal
10#10590450—31nomaximal
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18Dic3⋊(C3⋊Dic3)—21no—
19#88804—21no—
20#88804—21no—
21Dic3×C3⋊Dic3—21no—
22#64106—31no—
23#64101—31no—
24#63871—31no—
25#63871—31no—
26#63871—61no—
27#63849—31no—
28#63849—31no—
29#63849—61no—
30#63647—31no—
31C3×#817—31no—
32C3×#816—31no—
33C3×#810—31no—
34C3×#808—31no—
35C3×#807—31no—
36C3×#807—31no—
37#63576D311yescharacteristic, complemented
38#63573D311yescharacteristic, complemented
39#63567—31no—
40#63567D311yescharacteristic, complemented
41#63566—31no—
42#63565—31no—
43#63557—31no—
44#63557—31no—
45#63557—61no—
46C22×C3⋊(C3⋊D3)D411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)D411yescharacteristic
49C4×C3⋊(C3⋊D3)—21no—
50C4×C3⋊(C3⋊D3)—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C3×#634—21no—
54C3×#634—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59#4888—91no—
60#4868—91no—
61#4815—91no—
62C2×C4×C3⋊D3—31no—
63C2×C4×C3⋊D3—31no—
64C2×C4×C3⋊D3—31no—
65C2×C4×C3⋊D3—61no—
66C2×C4×C3⋊D3—61no—
67C2×C4×C3⋊D3—61no—
68C2×C4×C3⋊D3—121no—
69C2×#634C2×D311yescharacteristic
70C2×C3×Dic6—31no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—61no—
73C2×C32×C8—31no—
74C32×C4⋊C4—31no—
75C32×C4⋊C4—31no—
76#3752—61no—
77#3752—61no—
78C4⋊(C3⋊Dic3)—61no—
79C4⋊(C3⋊Dic3)C2×D311yescharacteristic
80C4×C3⋊Dic3—61no—
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D311yescharacteristic
91C2×C3×C3⋊C8—21no—
92C2×C3×C3⋊C8—31no—
93C2×C3×C3⋊C8—31no—
94C2×C3×C3⋊C8—61no—
95C2×C3×C3⋊C8C2×D311yescharacteristic
96C2×C3×C3⋊C8C2×D311yescharacteristic
97Dic3⋊Dic3—61no—
98Dic3⋊Dic3—61no—
99Dic3⋊Dic3—62no—
100#3722—61no—
101#3722—61no—
102#3722—61no—
103#3722—61no—
104#3722—62no—
105#3722—62no—
106Dic32—61no—
107Dic32—61no—
108Dic32—62no—
109C22×C33C2×D411yescharacteristic
110C2×C3⋊(C3⋊D3)—21no—
111C2×C3⋊(C3⋊D3)—21no—
112C2×C3⋊(C3⋊D3)—21no—
113C33×C4#4111yescharacteristic
114C33×C4#4111yescharacteristic
115C3⋊(C3⋊Dic3)—21no—
116C3×C3⋊Dic3—41no—
117C3×C3⋊Dic3—41no—
118C32×Dic3—41no—
119C32×Dic3—41no—
120#946—91no—
121#897—91no—
122#897—91no—
123#897—181no—
124#892—91no—
125#892—91no—
126#892—91no—
127#892—181no—
128C3×#107—91no—
129C3×#106—91no—
130C3×#103—91no—
131#835—91no—
132(C2×D3)⋊C8—91no—
133(C2×D3)⋊C8—91no—
134(C2×D3)⋊C8—91no—
135(C2×D3)⋊C8—181no—
136#817—31no—
137#816—31no—
138#816—91no—
139#810—31no—
140#810—91no—
141#808—31no—
142#807—31no—
143#807—31no—
144C22×C3⋊D3—31no—
145C22×C3⋊D3—31no—
146C22×C3⋊D3—31no—
147C22×C3⋊D3—61no—
148C22×C3⋊D3—61no—
149C22×C3⋊D3—61no—
150C22×C3⋊D3—121no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—41no—
155C2×C32×C4—61no—
156C2×C32×C4—61no—
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3—31no—
163C2×C3⋊Dic3—31no—
164C2×C3⋊Dic3—31no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—121no—
169C4×C3⋊D3—61no—
170C4×C3⋊D3—61no—
171C4×C3⋊D3—62no—
172C4×C3⋊D3—62no—
173C4×C3⋊D3—61no—
174C4×C3⋊D3—61no—
175C4×C3⋊D3—61no—
176C4×C3⋊D3—61no—
177C4×C3⋊D3—121no—
178C4×C3⋊D3—121no—
179C4×C3⋊D3—121no—
180C4×C3⋊D3—121no—
181C4×C3⋊D3—122no—
182C4×C3⋊D3—122no—
183#634—21no—
184#634—21no—
185C2×C3×Dic3—22no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—62no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C3×Dic6—61no—
198C3×Dic6—61no—
199C3×Dic6—61no—
200C3×Dic6—61no—
201C3×Dic6—121no—
202C3×Dic6—121no—
203C32×C8—61no—
204C3⋊(C3⋊C8)—21no—
205C3×C3⋊C8—21no—
206C3×C3⋊C8—21no—
207C3×C3⋊C8—41no—
208C3×C3⋊C8—61no—
209C3×C3⋊C8—61no—
210C3×C3⋊C8—121no—
211#485—271noSylow
212C2×C33#12311yescharacteristic
213C2×C33#13511yescharacteristic
214C2×C33#13711yescharacteristic
215C3⋊(C3⋊D3)—41no—
216C2×C3×Q8—91no—
217C2×C4×D3—91no—
218C2×C4×D3—91no—
219C2×C4×D3—91no—
220C2×C4×D3—181no—
221C2×C4×D3—181no—
222C2×C4×D3—181no—
223C2×C4×D3—361no—
224C2×Dic6—31no—
225C2×Dic6—31no—
226C2×Dic6—61no—
227C2×C3×C8—31no—
228C2×C3×C8—31no—
229C2×C3×C8—61no—
230C2×C3×C8—91no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4—61no—
237C3×C4⋊C4—91no—
238(C2×D3)⋊C4—181no—
239(C2×D3)⋊C4—181no—
240(C2×D3)⋊C4—181no—
241(C2×D3)⋊C4—181no—
242(C2×D3)⋊C4—181no—
243(C2×D3)⋊C4—181no—
244(C2×D3)⋊C4—182no—
245(C2×D3)⋊C4—182no—
246C4⋊Dic3—31no—
247C4⋊Dic3—31no—
248C4⋊Dic3—61no—
249C4⋊Dic3—181no—
250C4⋊Dic3—181no—
251C4⋊Dic3—182no—
252C4⋊Dic3D3211yescharacteristic
253C4⋊Dic3D3211yescharacteristic
254Dic3⋊C4—181no—
255C4×Dic3—181no—
256C4×Dic3—181no—
257C4×Dic3—181no—
258C4×Dic3—182no—
259C2×C3⋊C8—31no—
260C2×C3⋊C8—31no—
261C2×C3⋊C8—61no—
262C2×C3⋊C8D3211yescharacteristic
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32—41no—
267C22×C32D3×D411yescharacteristic
268C22×C32D3×D411yescharacteristic
269C22×C32D3×D411yescharacteristic
270C2×C3⋊D3—61no—
271C2×C3⋊D3—61no—
272C2×C3⋊D3—61no—
273C2×C3⋊D3—62no—
274C2×C3⋊D3—62no—
275C2×C3⋊D3—62no—
276C2×C3⋊D3—61no—
277C2×C3⋊D3—61no—
278C2×C3⋊D3—61no—
279C2×C3⋊D3—61no—
280C2×C3⋊D3—61no—
281C2×C3⋊D3—61no—
282C2×C3⋊D3—121no—
283C2×C3⋊D3—121no—
284C2×C3⋊D3—121no—
285C2×C3⋊D3—121no—
286C2×C3⋊D3—121no—
287C2×C3⋊D3—121no—
288C2×C3⋊D3—122no—
289C2×C3⋊D3—122no—
290C2×C3⋊D3—122no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—41no—
298C32×C4—41no—
299C32×C4—121no—
300C32×C4—121no—
301C32×C4#23511yescharacteristic
302C32×C4#23511yescharacteristic
303C32×C4#23511yescharacteristic
304C32×C4#23711yescharacteristic
305C32×C4#23911yescharacteristic
306C32×C4#23911yescharacteristic
307C3⋊Dic3—41no—
308C3⋊Dic3—41no—
309C3⋊Dic3—61no—
310C3⋊Dic3—62no—
311C3⋊Dic3—61no—
312C3⋊Dic3—61no—
313C3⋊Dic3—121no—
314C3⋊Dic3—121no—
315C3⋊Dic3—122no—
316C3×Dic3—42no—
317C3×Dic3—42no—
318C3×Dic3—41no—
319C3×Dic3—41no—
320C3×Dic3—41no—
321C3×Dic3—41no—
322C3×Dic3—122no—
323C3×Dic3—122no—
324C3×Dic3—121no—
325C3×Dic3—121no—
326C3×Dic3—121no—
327C3×Dic3—121no—
328C22⋊Q8—271no—
329#117—271no—
330#107—91no—
331#106—91no—
332#103—91no—
333#103—271no—
334C22⋊C8—271no—
335C33#48511yescharacteristic, complemented, Sylow
336C22×D3—91no—
337C22×D3—91no—
338C22×D3—91no—
339C22×D3—181no—
340C22×D3—181no—
341C22×D3—181no—
342C22×D3—361no—
343C3×Q8—181no—
344C3×Q8—181no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—41no—
349C2×C3×C4—62no—
350C2×C3×C4—62no—
351C2×C3×C4—61no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—181no—
356C2×C3×C4—181no—
357C2×C3×C4C2×D3211yescharacteristic
358C2×C3×C4C2×D3211yescharacteristic
359C2×C3×C4C2×D3211yescharacteristic
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—62no—
363C2×Dic3—62no—
364C2×Dic3—61no—
365C2×Dic3—61no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—91no—
369C2×Dic3—91no—
370C2×Dic3—91no—
371C2×Dic3—181no—
372C2×Dic3—181no—
373C2×Dic3—181no—
374C2×Dic3—361no—
375C4×D3—182no—
376C4×D3—182no—
377C4×D3—181no—
378C4×D3—181no—
379C4×D3—181no—
380C4×D3—181no—
381C4×D3—181no—
382C4×D3—181no—
383C4×D3—361no—
384C4×D3—361no—
385C4×D3—361no—
386C4×D3—361no—
387C4×D3—362no—
388C4×D3—362no—
389Dic6—61no—
390Dic6—61no—
391Dic6—61no—
392Dic6—61no—
393Dic6—121no—
394Dic6—121no—
395C3×C8—61no—
396C3×C8—61no—
397C3×C8—121no—
398C3×C8—181no—
399C3⋊C8—21no—
400C3⋊C8—61no—
401C3⋊C8—61no—
402C3⋊C8—121no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—41no—
413C2×C32—41no—
414C2×C32—41no—
415C2×C32#88311yescharacteristic
416C2×C32#89411yescharacteristic
417C2×C32#89411yescharacteristic
418C2×C32#91211yescharacteristic
419C2×C32#91511yescharacteristic
420C2×C32#91611yescharacteristic
421C2×C32#91611yescharacteristic
422C2×C32#91811yescharacteristic
423C2×C32#91811yescharacteristic
424C3⋊D3—122no—
425C3⋊D3—122no—
426C3⋊D3—122no—
427C3⋊D3—124no—
428C3⋊D3—121no—
429C3⋊D3—121no—
430C3⋊D3—121no—
431C2×Q8—91no—
432C22×C4—271no—
433C2×C8—31no—
434C2×C8—91no—
435C4⋊C4—31no—
436C4⋊C4—31no—
437C4⋊C4—91no—
438C4⋊C4—541no—
439C22⋊C4—541no—
440C22⋊C4—541no—
441C42—541no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—41no—
446C22×C3#380211yescharacteristic
447C22×C3#381111yescharacteristic
448C22×C3#381111yescharacteristic
449C2×D3—182no—
450C2×D3—182no—
451C2×D3—182no—
452C2×D3—181no—
453C2×D3—181no—
454C2×D3—181no—
455C2×D3—181no—
456C2×D3—181no—
457C2×D3—181no—
458C2×D3—181no—
459C2×D3—181no—
460C2×D3—181no—
461C2×D3—361no—
462C2×D3—361no—
463C2×D3—361no—
464C2×D3—361no—
465C2×D3—361no—
466C2×D3—361no—
467C2×D3—362no—
468C2×D3—362no—
469C2×D3—362no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—122no—
479C3×C4—122no—
480C3×C4—121no—
481C3×C4—121no—
482C3×C4—121no—
483C3×C4—121no—
484C3×C4—361no—
485C3×C4—361no—
486C3×C4#379611yescharacteristic
487C3×C4#379811yescharacteristic
488C3×C4#379911yescharacteristic
489C3×C4#379911yescharacteristic
490C3×C4#379911yescharacteristic
491C3×C4#380411yescharacteristic
492Dic3—41no—
493Dic3—41no—
494Dic3—122no—
495Dic3—122no—
496Dic3—121no—
497Dic3—121no—
498Dic3—121no—
499Dic3—121no—
500Dic3—182no—
501Dic3—181no—
502Dic3—181no—
503Dic3—181no—
504Dic3—361no—
505Dic3—361no—
506Dic3—362no—
507C32—21no—
508C32—21no—
509C32—21no—
510C32—41no—
511C32#481511yescharacteristic, complemented
512C32#486811yescharacteristic, complemented
513C32#488811yescharacteristic, complemented
514C23—271no—
515Q8—181no—
516Q8—181no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—181no—
520C2×C4—181no—
521C2×C4—271no—
522C2×C4—541no—
523C2×C4—541no—
524C2×C4D3311yescharacteristic
525C8—61no—
526C8—181no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—41no—
537C2×C3—41no—
538C2×C3—41no—
539C2×C3#6380911yescharacteristic
540C2×C3#6381111yescharacteristic
541C2×C3#6385311yescharacteristic
542C2×C3#6385411yescharacteristic
543C2×C3#6389711yescharacteristic
544C2×C3#6393611yescharacteristic
545C2×C3#6393611yescharacteristic
546C2×C3#6394511yescharacteristic
547C2×C3#6395111yescharacteristic
548D3—362no—
549D3—362no—
550D3—362no—
551D3—364no—
552D3—361no—
553D3—361no—
554D3—361no—
555C22—541no—
556C22—541no—
557C22—541no—
558C22#8895411yescharacteristic
559C4—121no—
560C4—121no—
561C4—361no—
562C4—361no—
563C4—541no—
564C4#8896211yescharacteristic
565C4#8896511yescharacteristic
566C3—21no—
567C3—21no—
568C3—21no—
569C3—41no—
570C3#1059045011yescharacteristic, complemented
571C3#1059052611yescharacteristic, complemented
572C3#1059071311yescharacteristic, complemented
573C2—1081no—
574C2#1172286111yescharacteristic
575C2#1172289411yescharacteristic
576C2#1172300011yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypocenterC33#485
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. #11722076C2semidirect
5. C3×#63576C2semidirect
6. C3×#63573C2semidirect
7. C3×#63567C2semidirect
37. #63576D3semidirect
38. #63573D3semidirect
40. #63567D3semidirect
335. C33#485semidirect
511. C32#4815semidirect
512. C32#4868semidirect
513. C32#4888semidirect
570. C3#10590450semidirect
571. C3#10590526semidirect
572. C3#10590713semidirect

Maximal subgroups

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

subgroupindex
#117234492
#117234132
#117220762
#117220242
C3×#635762
C3×#635732
C3×#635672
#105907133
#105905263
#105904503