Finite Group Databasebeta

← #13156319

Subgroups — #13156319

Order 1728.

556 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723534C211yesmaximal, characteristic
2C3×#63844C211yesmaximal, characteristic
3#11722076C211yesmaximal, characteristic
4#11722057C211yesmaximal, characteristic
5#11722040C211yesmaximal, characteristic
6C3×#63575C211yesmaximal, characteristic
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic
8#10590805—31nomaximal
9#10590557—31nomaximal
10#10590283—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×#6293—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20C3×#3724—21no—
21C3×Dic32—21no—
22#64078—31no—
23C3×#887—31no—
24C3×C4×Dic6—31no—
25#63918—31no—
26#63918—31no—
27#63918—61no—
28#63844D311yescharacteristic
29#63643—31no—
30#63634—31no—
31C3×#818—31no—
32C3×Dic3⋊C8—31no—
33C3×#807—31no—
34C3×C4⋊(C3⋊C8)—31no—
35#63575—31no—
36#63575—31no—
37#63575—61no—
38#63575D311yescharacteristic
39Dic3⋊(C3⋊C8)D311yescharacteristic
40#63567—31no—
41#63566—31no—
42#63566—31no—
43#63566—31no—
44#63566—61no—
45#63565—31no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C3×#634—21no—
51C3×#634Q812yescomplemented
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55#6293—42no—
56C3×C3⋊(C3⋊C8)—21no—
57C32×C3⋊C8—21no—
58#5021—91no—
59#4875—91no—
60#4850—91no—
61C2×#634—61no—
62C2×#634C2×D311yescharacteristic
63C2×C3×Dic6—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—61no—
66C2×#627—62no—
67C2×#627—61no—
68C2×#627—61no—
69C2×C32×C8—31no—
70C32×C4⋊C4—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)—61no—
79C4⋊(C3⋊Dic3)—121no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—61no—
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×Dic3⋊C4—61no—
89C3×Dic3⋊C4—61no—
90C3×C4×Dic3—21no—
91C3×C4×Dic3—61no—
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—21no—
102Dic3⋊Dic3—62no—
103Dic3⋊Dic3—61no—
104Dic3⋊Dic3—61no—
105Dic32—21no—
106C22×C33C2×D411yescharacteristic
107C33×C4C2×Q811yescharacteristic
108C33×C4#4111yescharacteristic
109C3⋊(C3⋊Dic3)—41no—
110C3×C3⋊Dic3—22no—
111C3×C3⋊Dic3—41no—
112C32×Dic3—21no—
113C32×Dic3—41no—
114C32×Dic3Q1612yes—
115C3×C4×Q8—91no—
116#888—91no—
117#887—31no—
118#869—91no—
119#869—91no—
120#869—181no—
121C4×Dic6—31no—
122C3×#107—91no—
123C3×C4⋊C8—91no—
124#835—91no—
125#835—91no—
126#818—31no—
127#818—91no—
128#816—91no—
129#816—91no—
130#816—91no—
131#816—181no—
132Dic3⋊C8—31no—
133#810—91no—
134#807—31no—
135#807—91no—
136#807—91no—
137#807—181no—
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—21no—
159#634—122no—
160#634Dic612yescomplemented
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—41no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—121no—
172C2×C3×Dic3D1211yescharacteristic
173C2×C3×Dic3D1211yescharacteristic
174C2×C3×Dic3#6711yescharacteristic
175C2×C3×Dic3#6711yescharacteristic
176C3×Dic6—32no—
177C3×Dic6—32no—
178C3×Dic6—62no—
179C3×Dic6—61no—
180C3×Dic6—61no—
181C3×Dic6—121no—
182#627—124no—
183#627—122no—
184#627—122no—
185C32×C8—61no—
186C3⋊(C3⋊C8)—21no—
187C3×C3⋊C8—21no—
188C3×C3⋊C8—21no—
189C3×C3⋊C8—41no—
190C3×C3⋊C8—62no—
191C3×C3⋊C8—61no—
192C3×C3⋊C8—61no—
193#477—271noSylow
194C2×C33C22⋊Q811yescharacteristic
195C2×C33C2×Q1611yescharacteristic
196C2×C33AGL(1,ℤ/8)11yescharacteristic
197C2×C3×Q8—91no—
198C2×Dic6—31no—
199C2×Dic6—31no—
200C2×Dic6—61no—
201C2×Dic6—181no—
202C2×Dic6—181no—
203C2×Dic6—182no—
204C2×Dic6—181no—
205C2×C3×C8—31no—
206C2×C3×C8—31no—
207C2×C3×C8—61no—
208C2×C3×C8—91no—
209C3×C4⋊C4—31no—
210C3×C4⋊C4—31no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—91no—
213C3×C4⋊C4—181no—
214C3×C42—31no—
215C3×C42—31no—
216C3×C42—61no—
217C3×C42—181no—
218C4⋊Dic3—31no—
219C4⋊Dic3—31no—
220C4⋊Dic3—61no—
221C4⋊Dic3—91no—
222C4⋊Dic3—91no—
223C4⋊Dic3—91no—
224C4⋊Dic3—181no—
225C4⋊Dic3—181no—
226C4⋊Dic3—181no—
227C4⋊Dic3—181no—
228C4⋊Dic3—181no—
229C4⋊Dic3—182no—
230C4⋊Dic3—361no—
231C4⋊Dic3D3211yescharacteristic
232Dic3⋊C4—61no—
233Dic3⋊C4—61no—
234Dic3⋊C4—181no—
235C4×Dic3—61no—
236C4×Dic3—61no—
237C4×Dic3D3211yescharacteristic
238C2×C3⋊C8—31no—
239C2×C3⋊C8—31no—
240C2×C3⋊C8—61no—
241C2×C3⋊C8D3211yescharacteristic
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—41no—
246C22×C32C2×D1211yescharacteristic
247C22×C32D3×D411yescharacteristic
248C22×C32C2×#6711yescharacteristic
249C32×C4—21no—
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—21no—
255C32×C4—32no—
256C32×C4—41no—
257C32×C4—41no—
258C32×C4—61no—
259C32×C4—121no—
260C32×C4C2×Dic611yescharacteristic
261C32×C4#23511yescharacteristic
262C32×C4#23511yescharacteristic
263C32×C4#23711yescharacteristic
264C32×C4D3×Q811yescharacteristic
265C32×C4D3×Q811yescharacteristic
266C3⋊Dic3—22no—
267C3⋊Dic3—41no—
268C3⋊Dic3—121no—
269C3⋊Dic3—122no—
270C3⋊Dic3—121no—
271C3⋊Dic3—121no—
272C3⋊Dic3—241no—
273C3⋊Dic3—241no—
274C3⋊Dic3—242no—
275C3×Dic3—22no—
276C3×Dic3—21no—
277C3×Dic3—21no—
278C3×Dic3—22no—
279C3×Dic3—41no—
280C3×Dic3—41no—
281C3×Dic3—62no—
282C3×Dic3—62no—
283C3×Dic3—81no—
284C3×Dic3—122no—
285C3×Dic3—122no—
286C3×Dic3—121no—
287C3×Dic3—121no—
288C3×Dic3Dic1212yes—
289C3×Dic3#21612yes—
290C4⋊Q8—271no—
291C4×Q8—91no—
292#107—91no—
293#107—271no—
294C4⋊C8—91no—
295#103—271no—
296#103—271no—
297C33#47711yescharacteristic, complemented, Sylow
298C3×Q8—92no—
299C3×Q8—181no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—31no—
304C2×C3×C4—31no—
305C2×C3×C4—31no—
306C2×C3×C4—31no—
307C2×C3×C4—41no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—61no—
312C2×C3×C4—121no—
313C2×C3×C4—181no—
314C2×C3×C4—181no—
315C2×C3×C4C2×D3211yescharacteristic
316C2×C3×C4C2×D3211yescharacteristic
317C2×C3×C4C2×D3211yescharacteristic
318C2×Dic3—21no—
319C2×Dic3—61no—
320C2×Dic3—61no—
321C2×Dic3—62no—
322C2×Dic3—61no—
323C2×Dic3—61no—
324C2×Dic3—121no—
325C2×Dic3—182no—
326C2×Dic3—181no—
327C2×Dic3—181no—
328C2×Dic3—181no—
329C2×Dic3—361no—
330C2×Dic3—361no—
331C2×Dic3—362no—
332C2×Dic3#62611yescharacteristic
333C2×Dic3#62611yescharacteristic
334Dic6—32no—
335Dic6—32no—
336Dic6—62no—
337Dic6—61no—
338Dic6—61no—
339Dic6—121no—
340Dic6—362no—
341Dic6—364no—
342Dic6—362no—
343Dic6—362no—
344C3×C8—61no—
345C3×C8—61no—
346C3×C8—121no—
347C3×C8—181no—
348C3⋊C8—21no—
349C3⋊C8—62no—
350C3⋊C8—61no—
351C3⋊C8—61no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—41no—
362C2×C32—41no—
363C2×C32—41no—
364C2×C32#89611yescharacteristic
365C2×C32#89711yescharacteristic
366C2×C32C2×Dic1211yescharacteristic
367C2×C32#90811yescharacteristic
368C2×C32#91111yescharacteristic
369C2×C32D3×Q1611yescharacteristic
370C2×C32#92411yescharacteristic
371C2×C32C2×#21611yescharacteristic
372C2×C32#94911yescharacteristic
373C2×Q8—91no—
374C2×Q8—541no—
375C2×C8—31no—
376C2×C8—91no—
377C4⋊C4—31no—
378C4⋊C4—91no—
379C4⋊C4—181no—
380C4⋊C4—271no—
381C4⋊C4—541no—
382C42—31no—
383C42—181no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—41no—
388C22×C3D3×D1211yescharacteristic
389C22×C3C2×#62611yescharacteristic
390C22×C3D3×#6711yescharacteristic
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—32no—
398C3×C4—32no—
399C3×C4—41no—
400C3×C4—41no—
401C3×C4—62no—
402C3×C4—62no—
403C3×C4—61no—
404C3×C4—61no—
405C3×C4—121no—
406C3×C4—121no—
407C3×C4—182no—
408C3×C4—241no—
409C3×C4—361no—
410C3×C4D3×Dic611yescharacteristic
411C3×C4D3×Dic611yescharacteristic
412C3×C4#379511yescharacteristic
413C3×C4#379711yescharacteristic
414C3×C4#379811yescharacteristic
415C3×C4#380411yescharacteristic
416Dic3—21no—
417Dic3—41no—
418Dic3—62no—
419Dic3—62no—
420Dic3—122no—
421Dic3—122no—
422Dic3—121no—
423Dic3—121no—
424Dic3—362no—
425Dic3—361no—
426Dic3—361no—
427Dic3—361no—
428Dic3—721no—
429Dic3—721no—
430Dic3—722no—
431Dic3#371912yes—
432C32—21no—
433C32—21no—
434C32—21no—
435C32—41no—
436C32#485011yescharacteristic, complemented
437C32#487511yescharacteristic, complemented
438C32#502111yescharacteristic, complemented
439Q8—92no—
440Q8—181no—
441Q8—1082no—
442C2×C4—31no—
443C2×C4—31no—
444C2×C4—61no—
445C2×C4—181no—
446C2×C4—181no—
447C2×C4—541no—
448C2×C4D3311yescharacteristic
449C8—61no—
450C8—181no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3#6379511yescharacteristic
464C2×C3D3×Dic1211yescharacteristic
465C2×C3#6382511yescharacteristic
466C2×C3C2×#371911yescharacteristic
467C2×C3#6385111yescharacteristic
468C2×C3#6385711yescharacteristic
469C2×C3#6389511yescharacteristic
470C2×C3#6392511yescharacteristic
471C2×C3D3×#21611yescharacteristic
472C22D3×#62611yescharacteristic
473C4—32no—
474C4—61no—
475C4—121no—
476C4—182no—
477C4—361no—
478C4—1081no—
479C4#8895711yescharacteristic
480C4#8896011yescharacteristic
481C3—21no—
482C3—21no—
483C3—21no—
484C3—41no—
485C3#1059028311yescharacteristic, complemented
486C3#1059055711yescharacteristic, complemented
487C3#1059080511yescharacteristic, complemented
488C2D3×#371911yescharacteristic
489C2#1172287011yescharacteristic
490C2#1172297211yescharacteristic
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
51. C3×#634Q8semidirect
160. #634Dic6semidirect
297. C33#477semidirect
436. C32#4850semidirect
437. C32#4875semidirect
438. C32#5021semidirect
485. C3#10590283semidirect
486. C3#10590557semidirect
487. C3#10590805semidirect

Maximal subgroups

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

subgroupindex
#117235342
C3×#638442
#117220762
#117220572
#117220402
C3×#635752
C3×Dic3⋊(C3⋊C8)2
#105908053
#105905573
#105902833