Finite Group Databasebeta

← #13156557

Subgroups — #13156557

Order 1728.

586 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 98 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#11723677C211yesmaximal, characteristic
2#11723535C211yesmaximal, characteristic, complemented
3#11722040C211yesmaximal, characteristic, complemented
4#11722034C211yesmaximal, characteristic
5C3×#63567C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic
7C3×#63555C211yesmaximal, characteristic, complemented
8#10590895—31nomaximal
9#10590893—31nomaximal
10#10590541—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×#6293—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88814—21no—
20#88814—21no—
21#88807—21no—
22#64097—31no—
23#63918—31no—
24#63918—31no—
25#63918—61no—
26#63864—31no—
27#63863—31no—
28#63860—31no—
29#63648—31no—
30#63643—31no—
31C3×#821—31no—
32C3×#816—31no—
33C3×#815—31no—
34C3×#815—31no—
35C3×#810—31no—
36C3×#809—31no—
37#63567D311yescharacteristic, complemented
38#63566—31no—
39#63566—31no—
40#63566—61no—
41#63564D311yescharacteristic
42#63563—31no—
43#63563—31no—
44#63563—61no—
45#63555D311yescharacteristic, complemented
46C22×C3×C3⋊D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×#634—21no—
55C3×#634—21no—
56C2×C32×Dic3—21no—
57#6293—42no—
58C32×C3⋊C8—21no—
59C32×C3⋊C8—21no—
60#4794—91no—
61#4765—91no—
62#4714—91no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×#634—61no—
65C2×#634C2×D311yescharacteristic
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×D12—61no—
69C2×C3×Dic6—31no—
70C2×C3×Dic6—31no—
71C2×C3×Dic6—61no—
72C2×#627—61no—
73C2×#627—61no—
74C2×#627—62no—
75C2×C32×C8—31no—
76C2×C32×C8—31no—
77C32×C4⋊C4—31no—
78C4⋊(C3⋊Dic3)—31no—
79C4⋊(C3⋊Dic3)—31no—
80C4⋊(C3⋊Dic3)—31no—
81C4⋊(C3⋊Dic3)—61no—
82C4⋊(C3⋊Dic3)—61no—
83C4⋊(C3⋊Dic3)—61no—
84C4⋊(C3⋊Dic3)—121no—
85#3750—61no—
86C4×C3⋊Dic3C2×D311yescharacteristic
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4×Dic3—31no—
91C3×C4×Dic3—31no—
92C3×C4×Dic3—61no—
93C2×C3×C3⋊C8—21no—
94C2×C3×C3⋊C8—21no—
95C2×C3×C3⋊C8C2×D311yescharacteristic
96C2×C3×C3⋊C8C2×D311yescharacteristic
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99Dic3⋊Dic3—61no—
100Dic3⋊Dic3—61no—
101Dic3⋊Dic3—62no—
102#3722—61no—
103#3722—61no—
104(C2×D3)⋊Dic3—61no—
105(C2×D3)⋊Dic3—61no—
106(C2×D3)⋊Dic3—61no—
107(C2×D3)⋊Dic3—61no—
108C22×C33C2×D411yescharacteristic
109C2×C3×C3⋊D3—41no—
110C2×C3×C3⋊D3—41no—
111C2×C3×C3⋊D3—41no—
112C33×C4#4111yescharacteristic
113C33×C4#4111yescharacteristic
114C3⋊(C3⋊Dic3)—41no—
115C3×C3⋊Dic3—21no—
116C3×C3⋊Dic3—21no—
117C3×C3⋊Dic3—41no—
118C32×Dic3—41no—
119#947—91no—
120#936—91no—
121#888—91no—
122#888—91no—
123#888—181no—
124#875—91no—
125#869—91no—
126C3×#103—91no—
127C3×#102—91no—
128C3×#97—91no—
129#835—91no—
130#832—91no—
131#821—31no—
132#821—91no—
133#821—91no—
134#821—181no—
135#816—31no—
136#816—91no—
137#816—91no—
138#816—181no—
139#815—31no—
140#815—31no—
141#810—31no—
142#809—31no—
143C22×C3⋊D3—21no—
144C22×C3×D3—61no—
145C22×C3×D3—61no—
146C22×C3×D3—121no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—41no—
151C2×C32×C4—61no—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C3⋊Dic3—21no—
156C2×C3⋊Dic3—61no—
157C2×C3⋊Dic3—62no—
158C2×C3⋊Dic3—61no—
159C2×C3⋊Dic3—61no—
160C2×C3⋊Dic3—121no—
161C2×C3⋊Dic3—121no—
162C2×C3⋊Dic3—122no—
163C2×C3⋊Dic3D1211yescharacteristic
164C2×C3⋊Dic3D1211yescharacteristic
165C4⋊(C3⋊D3)—21no—
166C4⋊(C3⋊D3)—21no—
167#634—21no—
168#634—21no—
169#634—122no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
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—121no—
182C3×D12—62no—
183C3×D12—61no—
184C3×D12—61no—
185C3×D12—61no—
186C3×D12—61no—
187C3×D12—121no—
188C3×Dic6—62no—
189C3×Dic6—61no—
190C3×Dic6—61no—
191C3×Dic6—61no—
192C3×Dic6—61no—
193C3×Dic6—121no—
194#627—124no—
195#627—122no—
196#627—122no—
197C32×C8—61no—
198C32×C8—61no—
199C3×C3⋊C8—21no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8—21no—
203C3×C3⋊C8—41no—
204C3×C3⋊C8—41no—
205#489—271noSylow
206C2×C33#12411yescharacteristic
207C2×C33AGL(1,ℤ/8)11yescharacteristic
208C2×C33#13711yescharacteristic
209C3×C3⋊D3—81no—
210C2×C3×Q8—91no—
211C2×C3×D4—91no—
212C2×D12—31no—
213C2×D12—31no—
214C2×D12—61no—
215C2×Dic6—31no—
216C2×Dic6—31no—
217C2×Dic6—61no—
218C2×Dic6—181no—
219C2×Dic6—181no—
220C2×Dic6—181no—
221C2×Dic6—182no—
222C2×C3×C8—31no—
223C2×C3×C8—31no—
224C2×C3×C8—31no—
225C2×C3×C8—31no—
226C2×C3×C8—61no—
227C2×C3×C8—61no—
228C3×C4⋊C4—31no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—61no—
231C3×C42—91no—
232#217—181no—
233#217—181no—
234(C2×D3)⋊C4—181no—
235(C2×D3)⋊C4—181no—
236(C2×D3)⋊C4—181no—
237(C2×D3)⋊C4—181no—
238C4⋊Dic3—91no—
239C4⋊Dic3—91no—
240C4⋊Dic3—91no—
241C4⋊Dic3—181no—
242C4⋊Dic3—181no—
243C4⋊Dic3—181no—
244C4⋊Dic3—181no—
245C4⋊Dic3—361no—
246C4⋊Dic3D3211yescharacteristic
247Dic3⋊C4—181no—
248Dic3⋊C4—181no—
249Dic3⋊C4—182no—
250C4×Dic3—31no—
251C4×Dic3—31no—
252C4×Dic3—61no—
253C2×C3⋊C8D3211yescharacteristic
254C2×C3⋊C8D3211yescharacteristic
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—41no—
259C22×C32C2×D1211yescharacteristic
260C22×C32D3×D411yescharacteristic
261C22×C32D3×D411yescharacteristic
262C2×C3⋊D3—41no—
263C2×C3⋊D3—41no—
264C2×C3⋊D3—41no—
265C2×C3×D3—122no—
266C2×C3×D3—121no—
267C2×C3×D3—121no—
268C2×C3×D3—121no—
269C2×C3×D3—121no—
270C2×C3×D3—121no—
271C2×C3×D3—121no—
272C2×C3×D3—241no—
273C2×C3×D3—241no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—41no—
281C32×C4—41no—
282C32×C4—121no—
283C32×C4#23511yescharacteristic
284C32×C4#23511yescharacteristic
285C32×C4#23511yescharacteristic
286C32×C4#23511yescharacteristic
287C32×C4#23711yescharacteristic
288C32×C4#23711yescharacteristic
289C3⋊Dic3—21no—
290C3⋊Dic3—21no—
291C3⋊Dic3—41no—
292C3⋊Dic3—121no—
293C3⋊Dic3—122no—
294C3⋊Dic3—121no—
295C3⋊Dic3—121no—
296C3⋊Dic3—241no—
297C3⋊Dic3—241no—
298C3⋊Dic3—242no—
299C3×Dic3—42no—
300C3×Dic3—41no—
301C3×Dic3—41no—
302C3×Dic3—62no—
303C3×Dic3—62no—
304C3×Dic3—61no—
305C3×Dic3—61no—
306C3×Dic3—61no—
307C3×Dic3—61no—
308C3×Dic3—122no—
309C3×Dic3—121no—
310C3×Dic3—121no—
311C4⋊Q8—271no—
312#124—271no—
313#103—91no—
314#103—271no—
315#102—91no—
316#102—271no—
317#97—91no—
318C33#48911yescharacteristic, complemented, Sylow
319C23×C3—181no—
320C22×D3—61no—
321C22×D3—61no—
322C22×D3—121no—
323C3×Q8—181no—
324C3×Q8—181no—
325C3×D4—181no—
326C3×D4—181no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—41no—
331C2×C3×C4—62no—
332C2×C3×C4—61no—
333C2×C3×C4—61no—
334C2×C3×C4—91no—
335C2×C3×C4—91no—
336C2×C3×C4—181no—
337C2×C3×C4C2×D3211yescharacteristic
338C2×C3×C4C2×D3211yescharacteristic
339C2×C3×C4C2×D3211yescharacteristic
340C2×Dic3—21no—
341C2×Dic3—31no—
342C2×Dic3—31no—
343C2×Dic3—31no—
344C2×Dic3—31no—
345C2×Dic3—61no—
346C2×Dic3—61no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—121no—
350C2×Dic3—182no—
351C2×Dic3—181no—
352C2×Dic3—181no—
353C2×Dic3—181no—
354C2×Dic3—361no—
355C2×Dic3—361no—
356C2×Dic3—362no—
357D12—62no—
358D12—61no—
359D12—61no—
360D12—61no—
361D12—61no—
362D12—121no—
363Dic6—62no—
364Dic6—61no—
365Dic6—61no—
366Dic6—61no—
367Dic6—61no—
368Dic6—121no—
369Dic6—362no—
370Dic6—364no—
371Dic6—362no—
372Dic6—362no—
373C3×C8—61no—
374C3×C8—61no—
375C3×C8—61no—
376C3×C8—61no—
377C3×C8—121no—
378C3×C8—121no—
379C3⋊C8—21no—
380C3⋊C8—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—41no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32#87511yescharacteristic
394C2×C32#88511yescharacteristic
395C2×C32#88511yescharacteristic
396C2×C32#90811yescharacteristic
397C2×C32#90911yescharacteristic
398C2×C32#91111yescharacteristic
399C2×C32#91411yescharacteristic
400C2×C32#91511yescharacteristic
401C2×C32#91811yescharacteristic
402C3⋊D3—81no—
403C3×D3—242no—
404C3×D3—241no—
405C3×D3—241no—
406C2×Q8—91no—
407C2×Q8—541no—
408C2×D4—91no—
409C2×C8—31no—
410C2×C8—31no—
411C4⋊C4—31no—
412C4⋊C4—271no—
413C4⋊C4—541no—
414C22⋊C4—541no—
415C22⋊C4—541no—
416C42—91no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—41no—
421C22×C3—361no—
422C22×C3—361no—
423C22×C3—361no—
424C22×C3D3×D1211yescharacteristic
425C22×C3D3×D1211yescharacteristic
426C22×C3#381111yescharacteristic
427C2×D3—122no—
428C2×D3—121no—
429C2×D3—121no—
430C2×D3—121no—
431C2×D3—121no—
432C2×D3—121no—
433C2×D3—121no—
434C2×D3—241no—
435C2×D3—241no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—41no—
443C3×C4—41no—
444C3×C4—122no—
445C3×C4—121no—
446C3×C4—121no—
447C3×C4—181no—
448C3×C4—181no—
449C3×C4—361no—
450C3×C4#379511yescharacteristic
451C3×C4#379511yescharacteristic
452C3×C4#379811yescharacteristic
453C3×C4#379811yescharacteristic
454C3×C4#380411yescharacteristic
455C3×C4#380411yescharacteristic
456Dic3—41no—
457Dic3—62no—
458Dic3—62no—
459Dic3—61no—
460Dic3—61no—
461Dic3—61no—
462Dic3—61no—
463Dic3—122no—
464Dic3—121no—
465Dic3—121no—
466Dic3—362no—
467Dic3—361no—
468Dic3—361no—
469Dic3—361no—
470Dic3—721no—
471Dic3—721no—
472Dic3—722no—
473C32—21no—
474C32—21no—
475C32—21no—
476C32—41no—
477C32#471411yescharacteristic, complemented
478C32#476511yescharacteristic, complemented
479C32#479411yescharacteristic, complemented
480C23—181no—
481Q8—181no—
482Q8—181no—
483Q8—1082no—
484D4—181no—
485D4—181no—
486C2×C4—61no—
487C2×C4—91no—
488C2×C4—91no—
489C2×C4—181no—
490C2×C4—541no—
491C2×C4D3311yescharacteristic
492C8—61no—
493C8—61no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—41no—
504C2×C3—41no—
505C2×C3—41no—
506C2×C3—721no—
507C2×C3#6379411yescharacteristic
508C2×C3#6379511yescharacteristic
509C2×C3#6380011yescharacteristic
510C2×C3#6380111yescharacteristic
511C2×C3#6385211yescharacteristic
512C2×C3#6385211yescharacteristic
513C2×C3#6391311yescharacteristic
514C2×C3#6393711yescharacteristic
515C2×C3#6395111yescharacteristic
516D3—242no—
517D3—241no—
518D3—241no—
519C22—361no—
520C22—361no—
521C22—361no—
522C22#8895511yescharacteristic
523C4—121no—
524C4—181no—
525C4—181no—
526C4—361no—
527C4—1081no—
528C4#8896011yescharacteristic
529C4#8896011yescharacteristic
530C3—21no—
531C3—21no—
532C3—21no—
533C3—41no—
534C3#1059054111yescharacteristic, complemented
535C3#1059089311yescharacteristic, complemented
536C3#1059089511yescharacteristic, complemented
537C2—721no—
538C2#1172287911yescharacteristic
539C2#1172289611yescharacteristic
540C2#1172301011yescharacteristic
subgroupG/H
centerC22#88955
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
2. #11723535C2semidirect
3. #11722040C2semidirect
5. C3×#63567C2semidirect
7. C3×#63555C2semidirect
37. #63567D3semidirect
45. #63555D3semidirect
318. C33#489semidirect
477. C32#4714semidirect
478. C32#4765semidirect
479. C32#4794semidirect
534. C3#10590541semidirect
535. C3#10590893semidirect
536. C3#10590895semidirect

Maximal subgroups

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

subgroupindex
#117236772
#117235352
#117220402
#117220342
C3×#635672
C3×#635642
C3×#635552
#105908953
#105908933
#105905413