Finite Group Databasebeta

← #13156505

Subgroups — #13156505

Order 1728.

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

normal subgroups only

This group has 580 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#11723667C211yesmaximal, characteristic
2#11723431C211yesmaximal, characteristic
3#11722042C211yesmaximal, characteristic, complemented
4#11722042C211yesmaximal, characteristic, complemented
5C3×#63575C211yesmaximal, characteristic, complemented
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×#63557C211yesmaximal, characteristic
8#10590709—31nomaximal
9#10590708—31nomaximal
10#10590450—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88814—21no—
19#88813—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21#88803—21no—
22#64106—31no—
23#63857—31no—
24#63857—31no—
25#63857—61no—
26#63849—31no—
27#63841—31no—
28#63841—31no—
29#63643—31no—
30#63643—31no—
31C3×(C2×D3)⋊C8—31no—
32C3×(C2×D3)⋊C8—31no—
33C3×#818—31no—
34C3×#817—31no—
35C3×#808—31no—
36C3×#807—31no—
37#63575D311yescharacteristic, complemented
38#63573D311yescharacteristic, complemented
39#63567—31no—
40#63567—31no—
41#63567—31no—
42#63567—31no—
43#63567—61no—
44#63567—61no—
45#63557D311yescharacteristic
46C22×C3×C3⋊D3D411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49#6308—21no—
50#6308—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3D411yescharacteristic
54C3×C4×C3⋊D3—21no—
55C3×C4×C3⋊D3—21no—
56C2×C32×Dic3—21no—
57C32×C3⋊C8—21no—
58C32×C3⋊C8—21no—
59#4888—91no—
60#4868—91no—
61#4723—91no—
62C2×C4×C3⋊D3C2×D311yescharacteristic
63C2×#634—31no—
64C2×#634—31no—
65C2×#634—31no—
66C2×#634—61no—
67C2×#634—61no—
68C2×#634—61no—
69C2×#634—121no—
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—31no—
72C2×C3×C4×D3—61no—
73C2×C32×C8—31no—
74C2×C32×C8—31no—
75C32×C4⋊C4—31no—
76#3752—61no—
77C4⋊(C3⋊Dic3)—61no—
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79C4⋊(C3⋊Dic3)C2×D311yescharacteristic
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3—31no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—61no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×C4⋊Dic3C2×D311yescharacteristic
89C2×C3×C3⋊C8—21no—
90C2×C3×C3⋊C8—21no—
91C2×C3×C3⋊C8C2×D311yescharacteristic
92C2×C3×C3⋊C8C2×D311yescharacteristic
93C2×C3×C3⋊C8C2×D311yescharacteristic
94C2×C3×C3⋊C8C2×D311yescharacteristic
95Dic3⋊Dic3—61no—
96Dic3⋊Dic3—61no—
97Dic3⋊Dic3—62no—
98#3722—61no—
99(C2×D3)⋊Dic3—61no—
100(C2×D3)⋊Dic3—61no—
101(C2×D3)⋊Dic3—62no—
102(C2×D3)⋊Dic3—61no—
103(C2×D3)⋊Dic3—61no—
104Dic32—61no—
105Dic32—61no—
106Dic32—61no—
107C22×C33C2×D411yescharacteristic
108C2×C3×C3⋊D3—21no—
109C2×C3×C3⋊D3—21no—
110C2×C3×C3⋊D3—21no—
111C33×C4#4111yescharacteristic
112C33×C4#4111yescharacteristic
113C3⋊(C3⋊Dic3)—41no—
114C3×C3⋊Dic3—21no—
115C3×C3⋊Dic3—41no—
116C3×C3⋊Dic3—41no—
117C32×Dic3—41no—
118#926—91no—
119#924—91no—
120#897—91no—
121#897—91no—
122#897—181no—
123#892—91no—
124#892—91no—
125C3×#107—91no—
126C3×#106—91no—
127C3×C22⋊C8—91no—
128(C2×D3)⋊C8—31no—
129(C2×D3)⋊C8—31no—
130#818—31no—
131#817—31no—
132#816—91no—
133#816—91no—
134#816—91no—
135#816—91no—
136#816—181no—
137#816—181no—
138#810—91no—
139#810—91no—
140#808—31no—
141#807—31no—
142C22×C3⋊D3D1211yescharacteristic
143C22×C3×D3—31no—
144C22×C3×D3—31no—
145C22×C3×D3—61no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—41no—
150C2×C32×C4—61no—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C3⋊Dic3—21no—
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
164C4×C3⋊D3—21no—
165C4×C3⋊D3—21no—
166#634—61no—
167#634—61no—
168#634—62no—
169#634—62no—
170#634—61no—
171#634—61no—
172#634—61no—
173#634—61no—
174#634—121no—
175#634—121no—
176#634—121no—
177#634—121no—
178#634—122no—
179#634—122no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—121no—
191C2×C3×Dic3—121no—
192C3×C4×D3—61no—
193C3×C4×D3—61no—
194C3×C4×D3—61no—
195C3×C4×D3—61no—
196C3×C4×D3—121no—
197C3×C4×D3—121no—
198C32×C8—61no—
199C32×C8—61no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8—21no—
203C3×C3⋊C8—21no—
204C3×C3⋊C8—41no—
205C3×C3⋊C8—41no—
206#485—271noSylow
207C2×C33#12311yescharacteristic
208C2×C33#13511yescharacteristic
209C2×C33#13711yescharacteristic
210C3×C3⋊D3—41no—
211C22×C3×C4—91no—
212C2×C4×D3—31no—
213C2×C4×D3—31no—
214C2×C4×D3—61no—
215C2×Dic6—91no—
216C2×Dic6—91no—
217C2×Dic6—91no—
218C2×Dic6—181no—
219C2×Dic6—181no—
220C2×Dic6—181no—
221C2×Dic6—361no—
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×C4⋊C4—91no—
232C3×C4⋊C4—91no—
233#217—181no—
234#217—181no—
235(C2×D3)⋊C4—181no—
236(C2×D3)⋊C4—181no—
237(C2×D3)⋊C4—181no—
238(C2×D3)⋊C4—181no—
239(C2×D3)⋊C4—182no—
240C4⋊Dic3—31no—
241C4⋊Dic3—31no—
242C4⋊Dic3—31no—
243C4⋊Dic3—31no—
244C4⋊Dic3—61no—
245C4⋊Dic3—61no—
246C4⋊Dic3—181no—
247C4⋊Dic3—181no—
248C4⋊Dic3—182no—
249C4⋊Dic3D3211yescharacteristic
250Dic3⋊C4—181no—
251C4×Dic3—181no—
252C4×Dic3—181no—
253C4×Dic3—181no—
254C2×C3⋊C8D3211yescharacteristic
255C2×C3⋊C8D3211yescharacteristic
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—41no—
260C22×C32C2×D1211yescharacteristic
261C22×C32D3×D411yescharacteristic
262C22×C32D3×D411yescharacteristic
263C2×C3⋊D3—21no—
264C2×C3⋊D3—21no—
265C2×C3⋊D3—21no—
266C2×C3×D3—62no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C2×C3×D3—61no—
271C2×C3×D3—61no—
272C2×C3×D3—61no—
273C2×C3×D3—121no—
274C2×C3×D3—121no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—41no—
282C32×C4—41no—
283C32×C4—121no—
284C32×C4#23511yescharacteristic
285C32×C4#23511yescharacteristic
286C32×C4#23711yescharacteristic
287C32×C4#23711yescharacteristic
288C32×C4#23911yescharacteristic
289C32×C4#23911yescharacteristic
290C3⋊Dic3—21no—
291C3⋊Dic3—41no—
292C3⋊Dic3—41no—
293C3⋊Dic3—122no—
294C3⋊Dic3—122no—
295C3⋊Dic3—122no—
296C3⋊Dic3—124no—
297C3⋊Dic3—121no—
298C3⋊Dic3—121no—
299C3⋊Dic3—121no—
300C3×Dic3—42no—
301C3×Dic3—41no—
302C3×Dic3—41no—
303C3×Dic3—61no—
304C3×Dic3—61no—
305C3×Dic3—121no—
306C3×Dic3—121no—
307C3×Dic3—121no—
308C3×Dic3—121no—
309C3×Dic3—121no—
310C3×Dic3—241no—
311C3×Dic3—241no—
312C22⋊Q8—271no—
313#117—271no—
314#107—91no—
315#106—91no—
316#103—271no—
317#103—271no—
318C22⋊C8—91no—
319C33#48511yescharacteristic, complemented, Sylow
320C23×C3—91no—
321C22×D3—31no—
322C22×D3—31no—
323C22×D3—61no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—41no—
328C2×C3×C4—62no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—91no—
332C2×C3×C4—181no—
333C2×C3×C4—181no—
334C2×C3×C4—181no—
335C2×C3×C4—181no—
336C2×C3×C4C2×D3211yescharacteristic
337C2×C3×C4C2×D3211yescharacteristic
338C2×C3×C4C2×D3211yescharacteristic
339C2×Dic3—21no—
340C2×Dic3—31no—
341C2×Dic3—31no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346C2×Dic3—61no—
347C2×Dic3—121no—
348C2×Dic3—121no—
349C2×Dic3—182no—
350C2×Dic3—181no—
351C2×Dic3—181no—
352C2×Dic3—181no—
353C2×Dic3—361no—
354C2×Dic3—361no—
355C2×Dic3—362no—
356C4×D3—61no—
357C4×D3—61no—
358C4×D3—61no—
359C4×D3—61no—
360C4×D3—121no—
361C4×D3—121no—
362Dic6—182no—
363Dic6—182no—
364Dic6—181no—
365Dic6—181no—
366Dic6—181no—
367Dic6—181no—
368Dic6—181no—
369Dic6—181no—
370Dic6—361no—
371Dic6—361no—
372Dic6—361no—
373Dic6—361no—
374Dic6—362no—
375Dic6—362no—
376C3×C8—61no—
377C3×C8—61no—
378C3×C8—61no—
379C3×C8—61no—
380C3×C8—121no—
381C3×C8—121no—
382C3⋊C8—21no—
383C3⋊C8—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—41no—
394C2×C32—41no—
395C2×C32—41no—
396C2×C32#88611yescharacteristic
397C2×C32#89411yescharacteristic
398C2×C32#89411yescharacteristic
399C2×C32#90411yescharacteristic
400C2×C32#90911yescharacteristic
401C2×C32#91211yescharacteristic
402C2×C32#91511yescharacteristic
403C2×C32#91611yescharacteristic
404C2×C32#91811yescharacteristic
405C3⋊D3—41no—
406C3×D3—122no—
407C3×D3—121no—
408C3×D3—121no—
409C2×Q8—271no—
410C22×C4—91no—
411C2×C8—31no—
412C2×C8—31no—
413C4⋊C4—31no—
414C4⋊C4—91no—
415C4⋊C4—91no—
416C4⋊C4—541no—
417C22⋊C4—541no—
418C22⋊C4—541no—
419C42—541no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—41no—
424C22×C3—181no—
425C22×C3—181no—
426C22×C3—181no—
427C22×C3D3×D1211yescharacteristic
428C22×C3D3×D1211yescharacteristic
429C22×C3#381111yescharacteristic
430C2×D3—62no—
431C2×D3—61no—
432C2×D3—61no—
433C2×D3—61no—
434C2×D3—61no—
435C2×D3—61no—
436C2×D3—61no—
437C2×D3—121no—
438C2×D3—121no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—122no—
448C3×C4—121no—
449C3×C4—121no—
450C3×C4—181no—
451C3×C4—361no—
452C3×C4—361no—
453C3×C4#379611yescharacteristic
454C3×C4#379611yescharacteristic
455C3×C4#379911yescharacteristic
456C3×C4#379911yescharacteristic
457C3×C4#380411yescharacteristic
458C3×C4#380411yescharacteristic
459Dic3—41no—
460Dic3—61no—
461Dic3—61no—
462Dic3—121no—
463Dic3—121no—
464Dic3—121no—
465Dic3—121no—
466Dic3—121no—
467Dic3—241no—
468Dic3—241no—
469Dic3—362no—
470Dic3—362no—
471Dic3—362no—
472Dic3—364no—
473Dic3—361no—
474Dic3—361no—
475Dic3—361no—
476C32—21no—
477C32—21no—
478C32—21no—
479C32—41no—
480C32#472311yescharacteristic, complemented
481C32#486811yescharacteristic, complemented
482C32#488811yescharacteristic, complemented
483C23—91no—
484Q8—541no—
485Q8—541no—
486C2×C4—61no—
487C2×C4—91no—
488C2×C4—181no—
489C2×C4—181no—
490C2×C4—181no—
491C2×C4—181no—
492C2×C4—541no—
493C2×C4D3311yescharacteristic
494C8—61no—
495C8—61no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—41no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—361no—
509C2×C3#6380011yescharacteristic
510C2×C3#6380111yescharacteristic
511C2×C3#6380611yescharacteristic
512C2×C3#6380711yescharacteristic
513C2×C3#6389011yescharacteristic
514C2×C3#6389011yescharacteristic
515C2×C3#6389711yescharacteristic
516C2×C3#6393611yescharacteristic
517C2×C3#6395111yescharacteristic
518D3—122no—
519D3—121no—
520D3—121no—
521C22—181no—
522C22—181no—
523C22—181no—
524C22#8895511yescharacteristic
525C4—121no—
526C4—181no—
527C4—361no—
528C4—361no—
529C4—1081no—
530C4#8896211yescharacteristic
531C4#8896211yescharacteristic
532C3—21no—
533C3—21no—
534C3—21no—
535C3—41no—
536C3#1059045011yescharacteristic, complemented
537C3#1059070811yescharacteristic, complemented
538C3#1059070911yescharacteristic, complemented
539C2—361no—
540C2#1172286411yescharacteristic
541C2#1172289611yescharacteristic
542C2#1172299911yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
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. #11722042C2semidirect
4. #11722042C2semidirect
5. C3×#63575C2semidirect
6. C3×#63573C2semidirect
37. #63575D3semidirect
38. #63573D3semidirect
319. C33#485semidirect
480. C32#4723semidirect
481. C32#4868semidirect
482. C32#4888semidirect
536. C3#10590450semidirect
537. C3#10590708semidirect
538. C3#10590709semidirect

Maximal subgroups

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

subgroupindex
#117236672
#117234312
#117220422
C3×#635752
C3×#635732
C3×#635572
#105907093
#105907083
#105904503