Finite Group Databasebeta

← #13156556

Subgroups — #13156556

Order 1728.

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

normal subgroups only

This group has 614 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#11723536C211yesmaximal, characteristic, complemented
2#11723436C211yesmaximal, characteristic
3#11722075C211yesmaximal, characteristic
4#11722021C211yesmaximal, characteristic, complemented
5C3×#63566C211yesmaximal, characteristic, complemented
6C3×#63565C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic
8#10591632—31nomaximal
9#10590866—31nomaximal
10#10590541—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6293—21no—
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88806—21no—
20#88803—21no—
21#88803—21no—
22#64156—31no—
23#64097—31no—
24#63919—31no—
25#63919—31no—
26#63919—61no—
27#63863—31no—
28#63863—31no—
29#63863—61no—
30#63642—31no—
31C3×#835—31no—
32C3×#835—31no—
33C3×#821—31no—
34C3×#816—31no—
35C3×#810—31no—
36C3×#809—31no—
37#63566D311yescharacteristic, complemented
38#63565D311yescharacteristic, complemented
39#63564—31no—
40#63564D311yescharacteristic
41#63563—31no—
42#63562—31no—
43#63555—31no—
44#63555—31no—
45#63555—61no—
46C22×C3×C3⋊D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)D411yescharacteristic
49C2×C3⋊(C3⋊Dic3)D411yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C32×Dic6—21no—
56C32×Dic6—21no—
57#6293—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60#4794—91no—
61#4794—91no—
62#4765—91no—
63C2×C32×Q8—31no—
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×#634—61no—
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×D12—61no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6C2×D311yescharacteristic
71C2×C3×Dic6C2×D311yescharacteristic
72C2×#627—61no—
73C2×#627—61no—
74C2×#627—62no—
75C2×C32×C8—31no—
76C32×C4⋊C4—31no—
77#3752—61no—
78#3752—61no—
79C4⋊(C3⋊Dic3)C2×D311yescharacteristic
80#3750—61no—
81C4×C3⋊Dic3—31no—
82C4×C3⋊Dic3—31no—
83C4×C3⋊Dic3—31no—
84C4×C3⋊Dic3—61no—
85C4×C3⋊Dic3—61no—
86C4×C3⋊Dic3—61no—
87C4×C3⋊Dic3—121no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
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#3724—61no—
102#3724—61no—
103#3724—62no—
104(C2×D3)⋊Dic3—61no—
105(C2×D3)⋊Dic3—61no—
106(C2×D3)⋊Dic3—61no—
107(C2×D3)⋊Dic3—61no—
108(C2×D3)⋊Dic3—62no—
109(C2×D3)⋊Dic3—62no—
110C22×C33C2×D411yescharacteristic
111C2×C3×C3⋊D3—41no—
112C2×C3×C3⋊D3—41no—
113C2×C3×C3⋊D3—41no—
114C33×C4#4111yescharacteristic
115C33×C4#4111yescharacteristic
116C3⋊(C3⋊Dic3)—21no—
117C3⋊(C3⋊Dic3)—21no—
118C3×C3⋊Dic3—41no—
119C32×Dic3—41no—
120C32×Dic3—41no—
121#947—91no—
122#947—91no—
123#947—181no—
124#936—91no—
125#888—91no—
126#888—91no—
127#888—91no—
128#888—181no—
129C3×#103—91no—
130C3×#103—91no—
131C3×#102—91no—
132#835—31no—
133#835—31no—
134#832—91no—
135#821—31no—
136#821—91no—
137#816—31no—
138#815—91no—
139#815—91no—
140#815—91no—
141#815—181no—
142#810—31no—
143#809—31no—
144#809—91no—
145C22×C3⋊D3—21no—
146C22×C3×D3—62no—
147C22×C3×D3—61no—
148C22×C3×D3—61no—
149C32×Q8—61no—
150C32×Q8—61no—
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—31no—
162C2×C3⋊Dic3—31no—
163C2×C3⋊Dic3—31no—
164C2×C3⋊Dic3—31no—
165C2×C3⋊Dic3—31no—
166C2×C3⋊Dic3—31no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—61no—
173C2×C3⋊Dic3—121no—
174C2×C3⋊Dic3—121no—
175C4⋊(C3⋊D3)—21no—
176C4⋊(C3⋊D3)—21no—
177#634—122no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—62no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C3×D12—61no—
188C3×D12—61no—
189C3×D12—61no—
190C3×D12—61no—
191C3×D12—121no—
192C3×D12—121no—
193C3×Dic6—21no—
194C3×Dic6—21no—
195C3×Dic6—21no—
196C3×Dic6—21no—
197C3×Dic6—41no—
198C3×Dic6—41no—
199#627—124no—
200#627—122no—
201#627—122no—
202C32×C8—61no—
203C3⋊(C3⋊C8)—21no—
204C3×C3⋊C8—22no—
205C3×C3⋊C8—21no—
206C3×C3⋊C8—21no—
207C3×C3⋊C8—62no—
208C3×C3⋊C8—61no—
209C3×C3⋊C8—61no—
210#489—271noSylow
211C2×C33#12411yescharacteristic
212C2×C33AGL(1,ℤ/8)11yescharacteristic
213C2×C33#13711yescharacteristic
214C3×C3⋊D3—81no—
215C2×C3×Q8—31no—
216C2×C3×Q8—31no—
217C2×C3×Q8—61no—
218C2×C3×D4—91no—
219C2×D12—31no—
220C2×D12—31no—
221C2×D12—61no—
222C2×Dic6—181no—
223C2×Dic6—181no—
224C2×Dic6—181no—
225C2×Dic6—182no—
226C2×Dic6D3211yescharacteristic
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—61no—
234C3×C4⋊C4—91no—
235#217—181no—
236#217—181no—
237(C2×D3)⋊C4—181no—
238(C2×D3)⋊C4—181no—
239(C2×D3)⋊C4—181no—
240(C2×D3)⋊C4—181no—
241(C2×D3)⋊C4—182no—
242(C2×D3)⋊C4—182no—
243C4⋊Dic3—31no—
244C4⋊Dic3—31no—
245C4⋊Dic3—61no—
246C4⋊Dic3D3211yescharacteristic
247Dic3⋊C4—181no—
248Dic3⋊C4—181no—
249Dic3⋊C4—181no—
250Dic3⋊C4—182no—
251C4×Dic3—91no—
252C4×Dic3—91no—
253C4×Dic3—91no—
254C4×Dic3—181no—
255C4×Dic3—181no—
256C4×Dic3—181no—
257C4×Dic3—361no—
258C2×C3⋊C8—31no—
259C2×C3⋊C8—31no—
260C2×C3⋊C8—61no—
261C2×C3⋊C8D3211yescharacteristic
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—41no—
266C22×C32D3×D411yescharacteristic
267C22×C32D3×D411yescharacteristic
268C22×C32D3×D411yescharacteristic
269C2×C3⋊D3—41no—
270C2×C3⋊D3—41no—
271C2×C3⋊D3—41no—
272C2×C3×D3—122no—
273C2×C3×D3—122no—
274C2×C3×D3—122no—
275C2×C3×D3—121no—
276C2×C3×D3—121no—
277C2×C3×D3—121no—
278C2×C3×D3—121no—
279C2×C3×D3—121no—
280C2×C3×D3—121no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—41no—
288C32×C4—41no—
289C32×C4—121no—
290C32×C4—121no—
291C32×C4#23511yescharacteristic
292C32×C4#23511yescharacteristic
293C32×C4#23511yescharacteristic
294C32×C4#23711yescharacteristic
295C32×C4#23711yescharacteristic
296C32×C4#23711yescharacteristic
297C3⋊Dic3—41no—
298C3⋊Dic3—61no—
299C3⋊Dic3—61no—
300C3⋊Dic3—62no—
301C3⋊Dic3—62no—
302C3⋊Dic3—61no—
303C3⋊Dic3—61no—
304C3⋊Dic3—61no—
305C3⋊Dic3—61no—
306C3⋊Dic3—121no—
307C3⋊Dic3—121no—
308C3⋊Dic3—121no—
309C3⋊Dic3—121no—
310C3⋊Dic3—122no—
311C3⋊Dic3—122no—
312C3×Dic3—42no—
313C3×Dic3—42no—
314C3×Dic3—41no—
315C3×Dic3—41no—
316C3×Dic3—41no—
317C3×Dic3—41no—
318C3×Dic3—122no—
319C3×Dic3—121no—
320C3×Dic3—121no—
321C4⋊Q8—271no—
322#124—271no—
323#103—91no—
324#103—91no—
325#102—91no—
326#102—271no—
327#97—271no—
328C33#48911yescharacteristic, complemented, Sylow
329C23×C3—181no—
330C22×D3—62no—
331C22×D3—61no—
332C22×D3—61no—
333C3×Q8—61no—
334C3×Q8—61no—
335C3×Q8—61no—
336C3×Q8—61no—
337C3×Q8—121no—
338C3×Q8—121no—
339C3×D4—181no—
340C3×D4—181no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—41no—
345C2×C3×C4—62no—
346C2×C3×C4—62no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—181no—
352C2×C3×C4C2×D3211yescharacteristic
353C2×C3×C4C2×D3211yescharacteristic
354C2×C3×C4C2×D3211yescharacteristic
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—62no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360C2×Dic3—91no—
361C2×Dic3—91no—
362C2×Dic3—91no—
363C2×Dic3—91no—
364C2×Dic3—91no—
365C2×Dic3—91no—
366C2×Dic3—181no—
367C2×Dic3—181no—
368C2×Dic3—181no—
369C2×Dic3—181no—
370C2×Dic3—181no—
371C2×Dic3—181no—
372C2×Dic3—361no—
373C2×Dic3—361no—
374D12—61no—
375D12—61no—
376D12—61no—
377D12—61no—
378D12—121no—
379D12—121no—
380Dic6—21no—
381Dic6—21no—
382Dic6—362no—
383Dic6—364no—
384Dic6—362no—
385Dic6—362no—
386C3×C8—62no—
387C3×C8—61no—
388C3×C8—61no—
389C3×C8—181no—
390C3⋊C8—21no—
391C3⋊C8—62no—
392C3⋊C8—61no—
393C3⋊C8—61no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—41no—
404C2×C32—41no—
405C2×C32—41no—
406C2×C32#88511yescharacteristic
407C2×C32#88511yescharacteristic
408C2×C32#88511yescharacteristic
409C2×C32#91111yescharacteristic
410C2×C32#91411yescharacteristic
411C2×C32#91411yescharacteristic
412C2×C32#91511yescharacteristic
413C2×C32#91811yescharacteristic
414C2×C32#91811yescharacteristic
415C3⋊D3—81no—
416C3×D3—242no—
417C3×D3—241no—
418C3×D3—241no—
419C2×Q8—31no—
420C2×Q8—541no—
421C2×D4—91no—
422C2×C8—31no—
423C2×C8—91no—
424C4⋊C4—31no—
425C4⋊C4—91no—
426C4⋊C4—541no—
427C22⋊C4—541no—
428C22⋊C4—541no—
429C42—271no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—41no—
434C22×C3—361no—
435C22×C3—361no—
436C22×C3—361no—
437C22×C3#380211yescharacteristic
438C22×C3#381111yescharacteristic
439C22×C3#381111yescharacteristic
440C2×D3—122no—
441C2×D3—122no—
442C2×D3—122no—
443C2×D3—121no—
444C2×D3—121no—
445C2×D3—121no—
446C2×D3—121no—
447C2×D3—121no—
448C2×D3—121no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—21no—
455C3×C4—41no—
456C3×C4—41no—
457C3×C4—122no—
458C3×C4—122no—
459C3×C4—121no—
460C3×C4—121no—
461C3×C4—121no—
462C3×C4—121no—
463C3×C4—361no—
464C3×C4#379511yescharacteristic
465C3×C4#379511yescharacteristic
466C3×C4#379811yescharacteristic
467C3×C4#380411yescharacteristic
468C3×C4#380411yescharacteristic
469C3×C4#380511yescharacteristic
470Dic3—41no—
471Dic3—41no—
472Dic3—122no—
473Dic3—121no—
474Dic3—121no—
475Dic3—182no—
476Dic3—182no—
477Dic3—181no—
478Dic3—181no—
479Dic3—181no—
480Dic3—181no—
481Dic3—181no—
482Dic3—181no—
483Dic3—361no—
484Dic3—361no—
485Dic3—361no—
486Dic3—361no—
487Dic3—362no—
488Dic3—362no—
489C32—21no—
490C32—21no—
491C32—21no—
492C32—41no—
493C32#476511yescharacteristic, complemented
494C32#479411yescharacteristic, complemented
495C32#479411yescharacteristic, complemented
496C23—181no—
497Q8—61no—
498Q8—61no—
499Q8—1082no—
500D4—181no—
501D4—181no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—181no—
505C2×C4—271no—
506C2×C4—271no—
507C2×C4D3311yescharacteristic
508C8—61no—
509C8—181no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—41no—
520C2×C3—41no—
521C2×C3—41no—
522C2×C3—721no—
523C2×C3#6379811yescharacteristic
524C2×C3#6381111yescharacteristic
525C2×C3#6391311yescharacteristic
526C2×C3#6391511yescharacteristic
527C2×C3#6393711yescharacteristic
528C2×C3#6394111yescharacteristic
529C2×C3#6394511yescharacteristic
530C2×C3#6395111yescharacteristic
531C2×C3#6395311yescharacteristic
532D3—242no—
533D3—241no—
534D3—241no—
535C22—361no—
536C22—361no—
537C22—361no—
538C22#8895411yescharacteristic
539C4—121no—
540C4—121no—
541C4—361no—
542C4—541no—
543C4—541no—
544C4#8896011yescharacteristic
545C4#8896111yescharacteristic
546C3—21no—
547C3—21no—
548C3—21no—
549C3—41no—
550C3#1059054111yescharacteristic, complemented
551C3#1059086611yescharacteristic, complemented
552C3#1059163211yescharacteristic, complemented
553C2—721no—
554C2#1172287811yescharacteristic
555C2#1172289411yescharacteristic
556C2#1172300911yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#489
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
1. #11723536C2semidirect
4. #11722021C2semidirect
5. C3×#63566C2semidirect
6. C3×#63565C2semidirect
37. #63566D3semidirect
38. #63565D3semidirect
328. C33#489semidirect
493. C32#4765semidirect
494. C32#4794semidirect
495. C32#4794semidirect
550. C3#10590541semidirect
551. C3#10590866semidirect
552. C3#10591632semidirect

Maximal subgroups

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

subgroupindex
#117235362
#117234362
#117220752
#117220212
C3×#635662
C3×#635652
C3×#635642
#105916323
#105908663
#105905413