Finite Group Databasebeta

← #13156285

Subgroups — #13156285

Order 1728.

552 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×#634C211yesmaximal, characteristic
2C3×#63918C211yesmaximal, characteristic
3#11722076C211yesmaximal, characteristic
4#11722054C211yesmaximal, characteristic
5#11722052C211yesmaximal, characteristic
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic
7C3×#63567C211yesmaximal, characteristic
8#10590803—31nomaximal
9#10590486—31nomaximal
10#10590307—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C3×#627—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
19#88807—21no—
20Dic3×C3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22C4×#634—31no—
23C3×#888—31no—
24C3×#869—31no—
25#63918D311yescharacteristic
26Dic3×Dic6—31no—
27Dic3×Dic6—31no—
28Dic3×Dic6—61no—
29#63644—31no—
30#63635—31no—
31C3×#816—31no—
32C3×Dic3⋊C8—31no—
33C3×#810—31no—
34C3×C4⋊(C3⋊C8)—31no—
35#63573—31no—
36#63573—31no—
37#63573—31no—
38#63573—31no—
39#63573—61no—
40#63573—61no—
41Dic3⋊(C3⋊C8)D311yescharacteristic
42#63567—31no—
43#63567D311yescharacteristic
44#63566—31no—
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
55C3×#627—42no—
56C3×C3⋊(C3⋊C8)—21no—
57C32×C3⋊C8—21no—
58#5094—91no—
59#4851—91no—
60#4791—91no—
61C2×#634C2×D311yescharacteristic
62C2×C3×Dic6—31no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6—61no—
65C2×C3×Dic6—61no—
66C2×C3×Dic6—61no—
67C2×#627—21no—
68C2×C32×C8—31no—
69C32×C4⋊C4—31no—
70C32×C42—31no—
71C4⋊(C3⋊Dic3)—31no—
72C4⋊(C3⋊Dic3)—31no—
73C4⋊(C3⋊Dic3)—31no—
74C4⋊(C3⋊Dic3)—61no—
75C4⋊(C3⋊Dic3)—61no—
76C4⋊(C3⋊Dic3)—61no—
77C4⋊(C3⋊Dic3)—121no—
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79#3750—61no—
80C4×C3⋊Dic3—61no—
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
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
99Dic3⋊Dic3—21no—
100Dic3⋊Dic3—62no—
101Dic3⋊Dic3—61no—
102Dic3⋊Dic3—61no—
103Dic32—62no—
104Dic32—61no—
105Dic32—61no—
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×Dic3SD1612yes—
115C3×C4⋊Q8—91no—
116Q8×Dic3—91no—
117#888—31no—
118#869—31no—
119C4×Dic6—91no—
120C4×Dic6—91no—
121C4×Dic6—181no—
122C3×C4⋊C8—91no—
123C3×#103—91no—
124#835—91no—
125#817—91no—
126#817—91no—
127#817—91no—
128#817—181no—
129#816—31no—
130#816—91no—
131Dic3⋊C8—31no—
132#810—31no—
133#810—91no—
134#808—91no—
135#808—91no—
136#808—91no—
137#808—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#634Dic612yescomplemented
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—41no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—121no—
171C2×C3×Dic3D1211yescharacteristic
172C2×C3×Dic3D1211yescharacteristic
173C2×C3×Dic3#6711yescharacteristic
174C2×C3×Dic3#6711yescharacteristic
175C3×Dic6—32no—
176C3×Dic6—32no—
177C3×Dic6—62no—
178C3×Dic6—62no—
179C3×Dic6—61no—
180C3×Dic6—61no—
181C3×Dic6—122no—
182C3×Dic6—122no—
183#627—42no—
184C32×C8—61no—
185C3⋊(C3⋊C8)—21no—
186C3×C3⋊C8—22no—
187C3×C3⋊C8—21no—
188C3×C3⋊C8—21no—
189C3×C3⋊C8—62no—
190C3×C3⋊C8—61no—
191C3×C3⋊C8—61no—
192#475—271noSylow
193C2×C33C22⋊Q811yescharacteristic
194C2×C33C2×SD1611yescharacteristic
195C2×C33#13711yescharacteristic
196C2×C3×Q8—91no—
197C2×C3×Q8—181no—
198C2×Dic6—31no—
199C2×Dic6—31no—
200C2×Dic6—61no—
201C2×Dic6—61no—
202C2×Dic6—61no—
203C2×C3×C8—31no—
204C2×C3×C8—31no—
205C2×C3×C8—61no—
206C2×C3×C8—91no—
207C3×C4⋊C4—31no—
208C3×C4⋊C4—31no—
209C3×C4⋊C4—61no—
210C3×C4⋊C4—91no—
211C3×C4⋊C4—181no—
212C3×C42—31no—
213C3×C42—31no—
214C3×C42—61no—
215C4⋊Dic3—31no—
216C4⋊Dic3—31no—
217C4⋊Dic3—61no—
218C4⋊Dic3—61no—
219C4⋊Dic3—91no—
220C4⋊Dic3—91no—
221C4⋊Dic3—91no—
222C4⋊Dic3—181no—
223C4⋊Dic3—181no—
224C4⋊Dic3—181no—
225C4⋊Dic3—181no—
226C4⋊Dic3—361no—
227C4⋊Dic3D3211yescharacteristic
228Dic3⋊C4—61no—
229Dic3⋊C4—181no—
230Dic3⋊C4—181no—
231Dic3⋊C4—182no—
232C4×Dic3—181no—
233C4×Dic3—181no—
234C4×Dic3—182no—
235C4×Dic3—181no—
236C4×Dic3D3211yescharacteristic
237C2×C3⋊C8—31no—
238C2×C3⋊C8—31no—
239C2×C3⋊C8—61no—
240C2×C3⋊C8D3211yescharacteristic
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—41no—
245C22×C32C2×D1211yescharacteristic
246C22×C32D3×D411yescharacteristic
247C22×C32C2×#6711yescharacteristic
248C32×C4—21no—
249C32×C4—21no—
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—32no—
255C32×C4—41no—
256C32×C4—41no—
257C32×C4—61no—
258C32×C4—121no—
259C32×C4C2×Dic611yescharacteristic
260C32×C4#23711yescharacteristic
261C32×C4#23711yescharacteristic
262C32×C4D3×Q811yescharacteristic
263C32×C4D3×Q811yescharacteristic
264C32×C4#23911yescharacteristic
265C3⋊Dic3—22no—
266C3⋊Dic3—41no—
267C3⋊Dic3—121no—
268C3⋊Dic3—122no—
269C3⋊Dic3—121no—
270C3⋊Dic3—121no—
271C3⋊Dic3—241no—
272C3⋊Dic3—241no—
273C3⋊Dic3—242no—
274C3×Dic3—21no—
275C3×Dic3—21no—
276C3×Dic3—22no—
277C3×Dic3—41no—
278C3×Dic3—41no—
279C3×Dic3—41no—
280C3×Dic3—64no—
281C3×Dic3—62no—
282C3×Dic3—62no—
283C3×Dic3—81no—
284C3×Dic3—121no—
285C3×Dic3—121no—
286C3×Dic3—241no—
287C3×Dic3#20412yes—
288C3×Dic3#21412yes—
289C4⋊Q8—91no—
290C4×Q8—271no—
291#106—271no—
292#106—271no—
293C4⋊C8—91no—
294#103—91no—
295#103—271no—
296C33#47511yescharacteristic, complemented, Sylow
297C3×Q8—92no—
298C3×Q8—181no—
299C3×Q8—362no—
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—62no—
320C2×Dic3—61no—
321C2×Dic3—61no—
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—62no—
338Dic6—61no—
339Dic6—61no—
340Dic6—122no—
341Dic6—122no—
342C3×C8—62no—
343C3×C8—61no—
344C3×C8—61no—
345C3×C8—181no—
346C3⋊C8—21no—
347C3⋊C8—62no—
348C3⋊C8—61no—
349C3⋊C8—61no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—41no—
360C2×C32—41no—
361C2×C32—41no—
362C2×C32#87811yescharacteristic
363C2×C32#89711yescharacteristic
364C2×C32C2×#20411yescharacteristic
365C2×C32#90911yescharacteristic
366C2×C32D3×SD1611yescharacteristic
367C2×C32#91811yescharacteristic
368C2×C32C2×#21411yescharacteristic
369C2×C32#94211yescharacteristic
370C2×C32#94611yescharacteristic
371C2×Q8—91no—
372C2×Q8—181no—
373C2×C8—31no—
374C2×C8—91no—
375C4⋊C4—31no—
376C4⋊C4—91no—
377C4⋊C4—181no—
378C4⋊C4—271no—
379C4⋊C4—541no—
380C42—31no—
381C42—541no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—41no—
386C22×C3D3×D1211yescharacteristic
387C22×C3C2×#62611yescharacteristic
388C22×C3D3×#6711yescharacteristic
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—32no—
396C3×C4—32no—
397C3×C4—41no—
398C3×C4—41no—
399C3×C4—62no—
400C3×C4—61no—
401C3×C4—61no—
402C3×C4—121no—
403C3×C4—121no—
404C3×C4—121no—
405C3×C4—182no—
406C3×C4—241no—
407C3×C4—361no—
408C3×C4D3×Dic611yescharacteristic
409C3×C4D3×Dic611yescharacteristic
410C3×C4#379611yescharacteristic
411C3×C4#379611yescharacteristic
412C3×C4#379711yescharacteristic
413C3×C4#380511yescharacteristic
414Dic3—21no—
415Dic3—41no—
416Dic3—64no—
417Dic3—62no—
418Dic3—62no—
419Dic3—121no—
420Dic3—121no—
421Dic3—241no—
422Dic3—362no—
423Dic3—361no—
424Dic3—361no—
425Dic3—361no—
426Dic3—721no—
427Dic3—721no—
428Dic3—722no—
429Dic3#371612yes—
430C32—21no—
431C32—21no—
432C32—21no—
433C32—41no—
434C32#479111yescharacteristic, complemented
435C32#485111yescharacteristic, complemented
436C32#509411yescharacteristic, complemented
437Q8—92no—
438Q8—181no—
439Q8—362no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—61no—
443C2×C4—181no—
444C2×C4—181no—
445C2×C4—541no—
446C2×C4D3311yescharacteristic
447C8—61no—
448C8—181no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—41no—
459C2×C3—41no—
460C2×C3—41no—
461C2×C3D3×#20411yescharacteristic
462C2×C3#6380111yescharacteristic
463C2×C3C2×#371611yescharacteristic
464C2×C3#6383311yescharacteristic
465C2×C3#6387111yescharacteristic
466C2×C3#6388711yescharacteristic
467C2×C3#6389311yescharacteristic
468C2×C3D3×#21411yescharacteristic
469C2×C3#6394811yescharacteristic
470C22D3×#62611yescharacteristic
471C4—32no—
472C4—61no—
473C4—121no—
474C4—182no—
475C4—361no—
476C4—1081no—
477C4#8895711yescharacteristic
478C4#8896411yescharacteristic
479C3—21no—
480C3—21no—
481C3—21no—
482C3—41no—
483C3#1059030711yescharacteristic, complemented
484C3#1059048611yescharacteristic, complemented
485C3#1059080311yescharacteristic, complemented
486C2D3×#371611yescharacteristic
487C2#1172288511yescharacteristic
488C2#1172297411yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#475
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
159. #634Dic6semidirect
296. C33#475semidirect
434. C32#4791semidirect
435. C32#4851semidirect
436. C32#5094semidirect
483. C3#10590307semidirect
484. C3#10590486semidirect
485. C3#10590803semidirect

Maximal subgroups

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

subgroupindex
Dic3×#6342
C3×#639182
#117220762
#117220542
#117220522
C3×Dic3⋊(C3⋊C8)2
C3×#635672
#105908033
#105904863
#105903073