Finite Group Databasebeta

← #13156569

Subgroups — #13156569

Order 1728.

526 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723454C211yesmaximal, characteristic
2C3×#63866C211yesmaximal, characteristic
3#11722076C211yesmaximal, characteristic
4#11722040C211yesmaximal, characteristic
5#11722038C211yesmaximal, characteristic
6C3×#63566C211yesmaximal, characteristic
7C3×#63554C211yesmaximal, characteristic
8#10591613—31nomaximal
9#10590898—31nomaximal
10#10590289—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88807—21no—
19#88806—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22#64079—31no—
23C3×#944—31no—
24C3×#869—31no—
25#63869—31no—
26#63869—31no—
27#63869—61no—
28#63866D311yescharacteristic
29#63662—31no—
30#63643—31no—
31C3×#835—31no—
32C3×#816—31no—
33C3×#815—31no—
34C3×#803—31no—
35#63567—31no—
36#63566—31no—
37#63566—31no—
38#63566—31no—
39#63566—31no—
40#63566—31no—
41#63566—61no—
42#63566—61no—
43#63566D311yescharacteristic
44#63565—31no—
45#63554D311yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C3×#634—21no—
51C3×#634—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×Dic6—21no—
56C32×Dic6—21no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59#5082—91no—
60#4797—91no—
61#4703—91no—
62C2×C32×Q8—31no—
63C2×#634C2×D311yescharacteristic
64C2×C3×Dic6—21no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—61no—
68C2×C3×Dic6C2×D311yescharacteristic
69C2×C3×Dic6C2×D311yescharacteristic
70C2×C32×C8—31no—
71C32×C42—31no—
72C4⋊(C3⋊Dic3)—31no—
73C4⋊(C3⋊Dic3)—31no—
74C4⋊(C3⋊Dic3)—31no—
75C4⋊(C3⋊Dic3)—61no—
76C4⋊(C3⋊Dic3)—61no—
77C4⋊(C3⋊Dic3)—61no—
78C4⋊(C3⋊Dic3)—121no—
79C4⋊(C3⋊Dic3)C2×D311yescharacteristic
80#3750—61no—
81#3750—61no—
82C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—61no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3—61no—
88C3×Dic3⋊C4—61no—
89C3×Dic3⋊C4—61no—
90C3×C4×Dic3—21no—
91C3×C4×Dic3C2×D311yescharacteristic
92C3×C4×Dic3C2×D311yescharacteristic
93C2×C3×C3⋊C8—21no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—61no—
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99#3724—62no—
100#3724—61no—
101#3724—61no—
102Dic3⋊Dic3—21no—
103Dic3⋊Dic3—21no—
104Dic3⋊Dic3—62no—
105Dic3⋊Dic3—61no—
106Dic3⋊Dic3—61no—
107C22×C33C2×D411yescharacteristic
108C33×C4#4111yescharacteristic
109C33×C4#4111yescharacteristic
110C3⋊(C3⋊Dic3)—41no—
111C3×C3⋊Dic3—41no—
112C3×C3⋊Dic3—41no—
113C32×Dic3—21no—
114C32×Dic3—21no—
115C32×Dic3—41no—
116C3×C4⋊Q8—91no—
117#944—31no—
118#890—91no—
119#870—91no—
120#870—91no—
121#870—181no—
122#869—31no—
123C3×#103—91no—
124C3×#97—91no—
125#835—31no—
126#835—91no—
127#835—91no—
128#835—91no—
129#835—91no—
130#835—181no—
131#816—31no—
132#816—91no—
133#816—91no—
134#816—91no—
135#816—91no—
136#816—181no—
137#815—31no—
138#810—91no—
139#803—31no—
140C32×Q8—61no—
141C32×Q8—61no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—31no—
146C2×C32×C4—31no—
147C2×C32×C4—41no—
148C2×C32×C4—61no—
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C3⋊Dic3—21no—
153C2×C3⋊Dic3—21no—
154C2×C3⋊Dic3—61no—
155C2×C3⋊Dic3—62no—
156C2×C3⋊Dic3—61no—
157C2×C3⋊Dic3—61no—
158C2×C3⋊Dic3—121no—
159C2×C3⋊Dic3—121no—
160C2×C3⋊Dic3—122no—
161#634—21no—
162#634—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—41no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—121no—
174C2×C3×Dic3D1211yescharacteristic
175C2×C3×Dic3D1211yescharacteristic
176C2×C3×Dic3#6711yescharacteristic
177C2×C3×Dic3#6711yescharacteristic
178C3×Dic6—22no—
179C3×Dic6—21no—
180C3×Dic6—21no—
181C3×Dic6—21no—
182C3×Dic6—21no—
183C3×Dic6—41no—
184C3×Dic6—62no—
185C3×Dic6—61no—
186C3×Dic6—61no—
187C3×Dic6—61no—
188C3×Dic6—61no—
189C3×Dic6—121no—
190C32×C8—61no—
191C3⋊(C3⋊C8)—21no—
192C3×C3⋊C8—21no—
193C3×C3⋊C8—21no—
194C3×C3⋊C8—41no—
195C3×C3⋊C8—61no—
196C3×C3⋊C8—61no—
197C3×C3⋊C8—121no—
198#491—271noSylow
199C2×C33#12411yescharacteristic
200C2×C33#13711yescharacteristic
201C2×C33#13711yescharacteristic
202C2×C3×Q8—31no—
203C2×C3×Q8—31no—
204C2×C3×Q8—61no—
205C2×C3×Q8—91no—
206C2×Dic6—31no—
207C2×Dic6—31no—
208C2×Dic6—61no—
209C2×Dic6D3211yescharacteristic
210C2×C3×C8—31no—
211C2×C3×C8—31no—
212C2×C3×C8—61no—
213C2×C3×C8—91no—
214C3×C4⋊C4—91no—
215C3×C4⋊C4—181no—
216C3×C4⋊C4—181no—
217C3×C42—31no—
218C3×C42—31no—
219C3×C42—61no—
220C4⋊Dic3—31no—
221C4⋊Dic3—31no—
222C4⋊Dic3—61no—
223C4⋊Dic3—61no—
224C4⋊Dic3—61no—
225C4⋊Dic3—91no—
226C4⋊Dic3—91no—
227C4⋊Dic3—91no—
228C4⋊Dic3—181no—
229C4⋊Dic3—181no—
230C4⋊Dic3—181no—
231C4⋊Dic3—181no—
232C4⋊Dic3—361no—
233Dic3⋊C4—61no—
234Dic3⋊C4—61no—
235Dic3⋊C4—181no—
236Dic3⋊C4—181no—
237Dic3⋊C4—181no—
238Dic3⋊C4—181no—
239Dic3⋊C4—182no—
240Dic3⋊C4—182no—
241Dic3⋊C4—181no—
242C4×Dic3D3211yescharacteristic
243C2×C3⋊C8—31no—
244C2×C3⋊C8—31no—
245C2×C3⋊C8—61no—
246C2×C3⋊C8D3211yescharacteristic
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—41no—
251C22×C32C2×D1211yescharacteristic
252C22×C32D3×D411yescharacteristic
253C22×C32C2×#6711yescharacteristic
254C32×C4—21no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—41no—
261C32×C4—41no—
262C32×C4—61no—
263C32×C4—61no—
264C32×C4—121no—
265C32×C4#23511yescharacteristic
266C32×C4#23511yescharacteristic
267C32×C4#23511yescharacteristic
268C32×C4#23711yescharacteristic
269C32×C4#23711yescharacteristic
270C32×C4#23711yescharacteristic
271C3⋊Dic3—41no—
272C3⋊Dic3—41no—
273C3⋊Dic3—121no—
274C3⋊Dic3—122no—
275C3⋊Dic3—121no—
276C3⋊Dic3—121no—
277C3⋊Dic3—241no—
278C3⋊Dic3—241no—
279C3⋊Dic3—242no—
280C3×Dic3—21no—
281C3×Dic3—21no—
282C3×Dic3—21no—
283C3×Dic3—21no—
284C3×Dic3—41no—
285C3×Dic3—41no—
286C3×Dic3—42no—
287C3×Dic3—41no—
288C3×Dic3—41no—
289C3×Dic3—122no—
290C3×Dic3—122no—
291C3×Dic3—121no—
292C3×Dic3—121no—
293C3×Dic3—121no—
294C3×Dic3—121no—
295C4⋊Q8—91no—
296#125—271no—
297#103—91no—
298#103—271no—
299#103—271no—
300#103—271no—
301#97—91no—
302C33#49111yescharacteristic, complemented, Sylow
303C3×Q8—62no—
304C3×Q8—61no—
305C3×Q8—61no—
306C3×Q8—61no—
307C3×Q8—61no—
308C3×Q8—121no—
309C3×Q8—181no—
310C3×Q8—181no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—31no—
315C2×C3×C4—31no—
316C2×C3×C4—31no—
317C2×C3×C4—31no—
318C2×C3×C4—41no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—121no—
324C2×C3×C4—181no—
325C2×C3×C4—181no—
326C2×C3×C4C2×D3211yescharacteristic
327C2×C3×C4C2×D3211yescharacteristic
328C2×C3×C4C2×D3211yescharacteristic
329C2×Dic3—21no—
330C2×Dic3—62no—
331C2×Dic3—61no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—121no—
336C2×Dic3—182no—
337C2×Dic3—181no—
338C2×Dic3—181no—
339C2×Dic3—181no—
340C2×Dic3—361no—
341C2×Dic3—361no—
342C2×Dic3—362no—
343C2×Dic3#62611yescharacteristic
344C2×Dic3#62611yescharacteristic
345Dic6—21no—
346Dic6—21no—
347Dic6—62no—
348Dic6—61no—
349Dic6—61no—
350Dic6—61no—
351Dic6—61no—
352Dic6—121no—
353C3×C8—61no—
354C3×C8—61no—
355C3×C8—121no—
356C3×C8—181no—
357C3⋊C8—21no—
358C3⋊C8—61no—
359C3⋊C8—61no—
360C3⋊C8—121no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—41no—
371C2×C32—41no—
372C2×C32—41no—
373C2×C32#87511yescharacteristic
374C2×C32#88511yescharacteristic
375C2×C32#90911yescharacteristic
376C2×C32#90911yescharacteristic
377C2×C32#91511yescharacteristic
378C2×C32#91811yescharacteristic
379C2×C32#93611yescharacteristic
380C2×C32#94211yescharacteristic
381C2×C32#94211yescharacteristic
382C2×Q8—31no—
383C2×Q8—91no—
384C2×C8—31no—
385C2×C8—91no—
386C4⋊C4—91no—
387C4⋊C4—181no—
388C4⋊C4—181no—
389C4⋊C4—271no—
390C4⋊C4—541no—
391C4⋊C4—541no—
392C42—31no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—41no—
397C22×C3D3×D1211yescharacteristic
398C22×C3C2×#62611yescharacteristic
399C22×C3D3×#6711yescharacteristic
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—41no—
407C3×C4—41no—
408C3×C4—61no—
409C3×C4—61no—
410C3×C4—61no—
411C3×C4—61no—
412C3×C4—121no—
413C3×C4—121no—
414C3×C4—122no—
415C3×C4—121no—
416C3×C4—121no—
417C3×C4—361no—
418C3×C4—361no—
419C3×C4#379511yescharacteristic
420C3×C4#379511yescharacteristic
421C3×C4#379811yescharacteristic
422C3×C4#380411yescharacteristic
423C3×C4#380411yescharacteristic
424C3×C4#380511yescharacteristic
425Dic3—21no—
426Dic3—21no—
427Dic3—41no—
428Dic3—122no—
429Dic3—122no—
430Dic3—121no—
431Dic3—121no—
432Dic3—121no—
433Dic3—121no—
434Dic3—362no—
435Dic3—361no—
436Dic3—361no—
437Dic3—361no—
438Dic3—721no—
439Dic3—721no—
440Dic3—722no—
441C32—21no—
442C32—21no—
443C32—21no—
444C32—41no—
445C32#470311yescharacteristic, complemented
446C32#479711yescharacteristic, complemented
447C32#508211yescharacteristic, complemented
448Q8—61no—
449Q8—61no—
450Q8—181no—
451Q8—181no—
452C2×C4—31no—
453C2×C4—31no—
454C2×C4—61no—
455C2×C4—181no—
456C2×C4—181no—
457C2×C4—541no—
458C2×C4D3311yescharacteristic
459C8—61no—
460C8—181no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—41no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3#6380011yescharacteristic
474C2×C3#6380111yescharacteristic
475C2×C3#6383311yescharacteristic
476C2×C3#6383311yescharacteristic
477C2×C3#6385211yescharacteristic
478C2×C3#6386011yescharacteristic
479C2×C3#6394611yescharacteristic
480C2×C3#6394811yescharacteristic
481C2×C3#6395911yescharacteristic
482C22D3×#62611yescharacteristic
483C4—61no—
484C4—61no—
485C4—121no—
486C4—361no—
487C4—361no—
488C4—1081no—
489C4#8896011yescharacteristic
490C4#8896111yescharacteristic
491C3—21no—
492C3—21no—
493C3—21no—
494C3—41no—
495C3#1059028911yescharacteristic, complemented
496C3#1059089811yescharacteristic, complemented
497C3#1059161311yescharacteristic, complemented
498C2#1172288411yescharacteristic
499C2#1172288511yescharacteristic
500C2#1172300211yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#491
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
302. C33#491semidirect
445. C32#4703semidirect
446. C32#4797semidirect
447. C32#5082semidirect
495. C3#10590289semidirect
496. C3#10590898semidirect
497. C3#10591613semidirect

Maximal subgroups

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

subgroupindex
#117234542
C3×#638662
#117220762
#117220402
#117220382
C3×#635662
C3×#635542
#105916133
#105908983
#105902893