Finite Group Databasebeta

← #13156317

Subgroups — #13156317

Order 1728.

576 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#11723399C211yesmaximal, characteristic
2C3×#63918C211yesmaximal, characteristic
3#11722080C211yesmaximal, characteristic
4#11722042C211yesmaximal, characteristic
5#11722041C211yesmaximal, characteristic
6C3×#63575C211yesmaximal, characteristic
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic
8#10590805—31nomaximal
9#10590488—31nomaximal
10#10590284—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×#627—21no—
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88806—21no—
20Dic3×C3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22C4×#634—31no—
23C3×#888—31no—
24C3×#869—31no—
25#63918D311yescharacteristic
26#63844—31no—
27#63844—31no—
28#63844—61no—
29#63643—31no—
30#63637—31no—
31C3×#818—31no—
32C3×Dic3⋊C8—31no—
33C3×#807—31no—
34C3×C4⋊(C3⋊C8)—31no—
35#63576—31no—
36#63575—31no—
37#63575—31no—
38#63575D311yescharacteristic
39Dic3⋊(C3⋊C8)D311yescharacteristic
40#63567—31no—
41#63567—31no—
42#63567—31no—
43#63567—31no—
44#63567—61no—
45#63567—61no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48#6308—21no—
49#6308Q812yescomplemented
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C3×#627—42no—
56C3×C3⋊(C3⋊C8)—21no—
57C32×C3⋊C8—21no—
58#5091—91no—
59#4875—91no—
60#4850—91no—
61C2×#634—31no—
62C2×#634—31no—
63C2×#634—31no—
64C2×#634—61no—
65C2×#634—61no—
66C2×#634—61no—
67C2×#634—121no—
68C2×C3×Dic6—61no—
69C2×C3×Dic6—61no—
70C2×#627—21no—
71C2×C32×C8—31no—
72C32×C4⋊C4—31no—
73C32×C42—31no—
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic
76#3750—61no—
77C4×C3⋊Dic3—61no—
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C3×C4⋊Dic3—21no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—31no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—61no—
85C3×C4⋊Dic3—61no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×C4⋊Dic3C2×D311yescharacteristic
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—
103Dic32—62no—
104Dic32—61no—
105Dic32—61no—
106C22×C33C2×D411yescharacteristic
107C33×C4C2×Q811yescharacteristic
108C33×C4#4111yescharacteristic
109C3⋊(C3⋊Dic3)—22no—
110C3×C3⋊Dic3—41no—
111C3×C3⋊Dic3—41no—
112C32×Dic3—21no—
113C32×Dic3—41no—
114C32×Dic3Q1612yes—
115C3×C4⋊Q8—91no—
116#888—31no—
117#887—91no—
118#869—31no—
119C4×Dic6—91no—
120C4×Dic6—91no—
121C4×Dic6—181no—
122C3×#107—91no—
123C3×C4⋊C8—91no—
124#818—31no—
125#818—91no—
126#816—91no—
127#816—91no—
128#816—91no—
129#816—181no—
130Dic3⋊C8—31no—
131#810—91no—
132#810—91no—
133#810—91no—
134#810—181no—
135#807—31no—
136#807—91no—
137#807—91no—
138C4⋊(C3⋊C8)—31no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—31no—
143C2×C32×C4—31no—
144C2×C32×C4—41no—
145C2×C32×C4—61no—
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D311yescharacteristic
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3—21no—
151C2×C3⋊Dic3—61no—
152C2×C3⋊Dic3—62no—
153C2×C3⋊Dic3—61no—
154C2×C3⋊Dic3—61no—
155C2×C3⋊Dic3—121no—
156C2×C3⋊Dic3—121no—
157C2×C3⋊Dic3—122no—
158#634—32no—
159#634—32no—
160#634—32no—
161#634—62no—
162#634—62no—
163#634—62no—
164#634—61no—
165#634—62no—
166#634—61no—
167#634—61no—
168#634—122no—
169#634—121no—
170#634—121no—
171#634—122no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—41no—
177C2×C3×Dic3—62no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—121no—
183C2×C3×Dic3D1211yescharacteristic
184C2×C3×Dic3D1211yescharacteristic
185C2×C3×Dic3#6711yescharacteristic
186C2×C3×Dic3#6711yescharacteristic
187C3×Dic6—122no—
188C3×Dic6—122no—
189#627—42no—
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#477—271noSylow
199C2×C33C22⋊Q811yescharacteristic
200C2×C33C2×Q1611yescharacteristic
201C2×C33AGL(1,ℤ/8)11yescharacteristic
202C2×C3×Q8—181no—
203C2×Dic6—61no—
204C2×Dic6—61no—
205C2×Dic6—91no—
206C2×Dic6—91no—
207C2×Dic6—91no—
208C2×Dic6—181no—
209C2×Dic6—181no—
210C2×Dic6—181no—
211C2×Dic6—361no—
212C2×C3×C8—31no—
213C2×C3×C8—31no—
214C2×C3×C8—61no—
215C2×C3×C8—91no—
216C3×C4⋊C4—31no—
217C3×C4⋊C4—31no—
218C3×C4⋊C4—61no—
219C3×C4⋊C4—91no—
220C3×C4⋊C4—91no—
221C3×C4⋊C4—181no—
222C3×C42—31no—
223C3×C42—31no—
224C3×C42—61no—
225C4⋊Dic3—31no—
226C4⋊Dic3—31no—
227C4⋊Dic3—31no—
228C4⋊Dic3—31no—
229C4⋊Dic3—61no—
230C4⋊Dic3—61no—
231C4⋊Dic3—61no—
232C4⋊Dic3D3211yescharacteristic
233Dic3⋊C4—61no—
234Dic3⋊C4—181no—
235Dic3⋊C4—181no—
236Dic3⋊C4—182no—
237Dic3⋊C4—181no—
238C4×Dic3—181no—
239C4×Dic3—181no—
240C4×Dic3—182no—
241C4×Dic3—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—32no—
261C32×C4—41no—
262C32×C4—41no—
263C32×C4—61no—
264C32×C4—121no—
265C32×C4#23511yescharacteristic
266C32×C4#23711yescharacteristic
267C32×C4D3×Q811yescharacteristic
268C32×C4D3×Q811yescharacteristic
269C32×C4D3×Q811yescharacteristic
270C32×C4#23911yescharacteristic
271C3⋊Dic3—41no—
272C3⋊Dic3—41no—
273C3⋊Dic3—62no—
274C3⋊Dic3—64no—
275C3⋊Dic3—62no—
276C3⋊Dic3—62no—
277C3⋊Dic3—122no—
278C3⋊Dic3—122no—
279C3⋊Dic3—124no—
280C3×Dic3—21no—
281C3×Dic3—21no—
282C3×Dic3—22no—
283C3×Dic3—41no—
284C3×Dic3—41no—
285C3×Dic3—41no—
286C3×Dic3—81no—
287C3×Dic3—122no—
288C3×Dic3—121no—
289C3×Dic3—121no—
290C3×Dic3—121no—
291C3×Dic3—121no—
292C3×Dic3—241no—
293C3×Dic3Dic1212yes—
294C3×Dic3#21612yes—
295C4⋊Q8—91no—
296C4×Q8—271no—
297#107—91no—
298#107—271no—
299C4⋊C8—91no—
300#103—271no—
301#103—271no—
302C33#47711yescharacteristic, complemented, Sylow
303C3×Q8—362no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—31no—
310C2×C3×C4—31no—
311C2×C3×C4—41no—
312C2×C3×C4—61no—
313C2×C3×C4—61no—
314C2×C3×C4—61no—
315C2×C3×C4—61no—
316C2×C3×C4—121no—
317C2×C3×C4—181no—
318C2×C3×C4—181no—
319C2×C3×C4C2×D3211yescharacteristic
320C2×C3×C4C2×D3211yescharacteristic
321C2×C3×C4C2×D3211yescharacteristic
322C2×Dic3—21no—
323C2×Dic3—62no—
324C2×Dic3—61no—
325C2×Dic3—61no—
326C2×Dic3—61no—
327C2×Dic3—61no—
328C2×Dic3—121no—
329C2×Dic3—182no—
330C2×Dic3—181no—
331C2×Dic3—181no—
332C2×Dic3—181no—
333C2×Dic3—361no—
334C2×Dic3—361no—
335C2×Dic3—362no—
336C2×Dic3#62611yescharacteristic
337C2×Dic3#62611yescharacteristic
338Dic6—92no—
339Dic6—92no—
340Dic6—92no—
341Dic6—122no—
342Dic6—122no—
343Dic6—182no—
344Dic6—182no—
345Dic6—182no—
346Dic6—182no—
347Dic6—181no—
348Dic6—181no—
349Dic6—181no—
350Dic6—362no—
351Dic6—361no—
352Dic6—361no—
353Dic6—362no—
354C3×C8—61no—
355C3×C8—61no—
356C3×C8—121no—
357C3×C8—181no—
358C3⋊C8—21no—
359C3⋊C8—61no—
360C3⋊C8—61no—
361C3⋊C8—121no—
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—21no—
371C2×C32—41no—
372C2×C32—41no—
373C2×C32—41no—
374C2×C32#89611yescharacteristic
375C2×C32#89711yescharacteristic
376C2×C32C2×Dic1211yescharacteristic
377C2×C32#90811yescharacteristic
378C2×C32#91111yescharacteristic
379C2×C32D3×Q1611yescharacteristic
380C2×C32#93211yescharacteristic
381C2×C32C2×#21611yescharacteristic
382C2×C32#94611yescharacteristic
383C2×Q8—181no—
384C2×Q8—271no—
385C2×C8—31no—
386C2×C8—91no—
387C4⋊C4—31no—
388C4⋊C4—91no—
389C4⋊C4—91no—
390C4⋊C4—181no—
391C4⋊C4—541no—
392C42—31no—
393C42—541no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—41no—
398C22×C3D3×D1211yescharacteristic
399C22×C3C2×#62611yescharacteristic
400C22×C3D3×#6711yescharacteristic
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—32no—
408C3×C4—32no—
409C3×C4—41no—
410C3×C4—41no—
411C3×C4—62no—
412C3×C4—61no—
413C3×C4—61no—
414C3×C4—121no—
415C3×C4—121no—
416C3×C4—121no—
417C3×C4—241no—
418C3×C4—361no—
419C3×C4—361no—
420C3×C4#379611yescharacteristic
421C3×C4#379711yescharacteristic
422C3×C4#379711yescharacteristic
423C3×C4#379711yescharacteristic
424C3×C4#379911yescharacteristic
425C3×C4#380411yescharacteristic
426Dic3—21no—
427Dic3—41no—
428Dic3—122no—
429Dic3—121no—
430Dic3—121no—
431Dic3—121no—
432Dic3—121no—
433Dic3—184no—
434Dic3—182no—
435Dic3—182no—
436Dic3—182no—
437Dic3—241no—
438Dic3—362no—
439Dic3—362no—
440Dic3—364no—
441Dic3#371912yes—
442C32—21no—
443C32—21no—
444C32—21no—
445C32—41no—
446C32#485011yescharacteristic, complemented
447C32#487511yescharacteristic, complemented
448C32#509111yescharacteristic, complemented
449Q8—272no—
450Q8—362no—
451Q8—541no—
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#6379511yescharacteristic
474C2×C3D3×Dic1211yescharacteristic
475C2×C3#6383011yescharacteristic
476C2×C3C2×#371911yescharacteristic
477C2×C3#6387111yescharacteristic
478C2×C3#6388811yescharacteristic
479C2×C3#6389511yescharacteristic
480C2×C3#6393211yescharacteristic
481C2×C3D3×#21611yescharacteristic
482C22D3×#62611yescharacteristic
483C4—32no—
484C4—61no—
485C4—121no—
486C4—361no—
487C4—361no—
488C4—542no—
489C4#8895811yescharacteristic
490C4#8896211yescharacteristic
491C3—21no—
492C3—21no—
493C3—21no—
494C3—41no—
495C3#1059028411yescharacteristic, complemented
496C3#1059048811yescharacteristic, complemented
497C3#1059080511yescharacteristic, complemented
498C2D3×#371911yescharacteristic
499C2#1172286911yescharacteristic
500C2#1172297311yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#477
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
49. #6308Q8semidirect
302. C33#477semidirect
446. C32#4850semidirect
447. C32#4875semidirect
448. C32#5091semidirect
495. C3#10590284semidirect
496. C3#10590488semidirect
497. C3#10590805semidirect

Maximal subgroups

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

subgroupindex
#117233992
C3×#639182
#117220802
#117220422
#117220412
C3×#635752
C3×Dic3⋊(C3⋊C8)2
#105908053
#105904883
#105902843