Finite Group Databasebeta

← #13156384

Subgroups — #13156384

Order 1728.

520 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723668C211yesmaximal, characteristic
2#11723447C211yesmaximal, characteristic
3#11722052C211yesmaximal, characteristic
4#11722040C211yesmaximal, characteristic
5C3×#63575C211yesmaximal, characteristic
6C3×#63572C211yesmaximal, characteristic
7C3×#63567C211yesmaximal, characteristic
8#10590815—31nomaximal
9#10590814—31nomaximal
10#10590494—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88815—21no—
19#88813—21no—
20#88806—21no—
21Dic3⋊(C3⋊Dic3)—21no—
22#64099—31no—
23#63869—31no—
24#63869—31no—
25#63869—61no—
26#63844—31no—
27#63843—31no—
28Dic3×Dic6—31no—
29#63644—31no—
30#63643—31no—
31C3×#818—31no—
32C3×#816—31no—
33C3×Dic3⋊C8—31no—
34C3×Dic3⋊C8—31no—
35C3×#810—31no—
36C3×#807—31no—
37#63575D311yescharacteristic
38#63573—31no—
39#63573—31no—
40#63573—61no—
41#63572D311yescharacteristic
42#63567D311yescharacteristic
43#63566—31no—
44#63566—31no—
45#63566—61no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×#634—21no—
53C3×#634Q812yes—
54C2×C32×Dic3—21no—
55C32×C3⋊C8—21no—
56C32×C3⋊C8—21no—
57#4877—91no—
58#4829—91no—
59#4799—91no—
60C2×#634C2×D311yescharacteristic
61C2×C3×Dic6—31no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—61no—
64C2×C32×C8—31no—
65C2×C32×C8—31no—
66C32×C4⋊C4—31no—
67C4⋊(C3⋊Dic3)—31no—
68C4⋊(C3⋊Dic3)—31no—
69C4⋊(C3⋊Dic3)—31no—
70C4⋊(C3⋊Dic3)—61no—
71C4⋊(C3⋊Dic3)—61no—
72C4⋊(C3⋊Dic3)—61no—
73C4⋊(C3⋊Dic3)—61no—
74C4⋊(C3⋊Dic3)—121no—
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic
76#3750—61no—
77C4×C3⋊Dic3C2×D311yescharacteristic
78C3×C4⋊Dic3—21no—
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—61no—
82C3×C4⋊Dic3C2×D311yescharacteristic
83C3×C4⋊Dic3C2×D311yescharacteristic
84C3×C4×Dic3—31no—
85C3×C4×Dic3—31no—
86C3×C4×Dic3—61no—
87C2×C3×C3⋊C8—21no—
88C2×C3×C3⋊C8—21no—
89C2×C3×C3⋊C8C2×D311yescharacteristic
90C2×C3×C3⋊C8C2×D311yescharacteristic
91C2×C3×C3⋊C8C2×D311yescharacteristic
92C2×C3×C3⋊C8C2×D311yescharacteristic
93#3724—61no—
94#3724—61no—
95#3724—62no—
96#3724—61no—
97Dic3⋊Dic3—61no—
98Dic3⋊Dic3—61no—
99Dic3⋊Dic3—62no—
100Dic3⋊Dic3—61no—
101Dic3⋊Dic3—61no—
102Dic32—61no—
103Dic32—61no—
104Dic32—61no—
105C22×C33C2×D411yescharacteristic
106C33×C4C2×Q811yescharacteristic
107C33×C4#4111yescharacteristic
108C3⋊(C3⋊Dic3)—41no—
109C3×C3⋊Dic3—22no—
110C3×C3⋊Dic3—21no—
111C3×C3⋊Dic3—21no—
112C3×C3⋊Dic3—41no—
113C32×Dic3—41no—
114Q8×Dic3—91no—
115#890—91no—
116#890—91no—
117#890—181no—
118#887—91no—
119#870—91no—
120C4×Dic6—91no—
121C3×#107—91no—
122C3×C4⋊C8—91no—
123C3×#103—91no—
124#835—91no—
125#818—31no—
126#817—91no—
127#817—91no—
128#817—181no—
129#816—31no—
130#816—91no—
131#816—91no—
132#816—181no—
133Dic3⋊C8—31no—
134Dic3⋊C8—31no—
135#810—31no—
136#808—91no—
137#807—31no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—41no—
142C2×C32×C4—61no—
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146C2×C3⋊Dic3—21no—
147C2×C3⋊Dic3—21no—
148C2×C3⋊Dic3—61no—
149C2×C3⋊Dic3—62no—
150C2×C3⋊Dic3—61no—
151C2×C3⋊Dic3—61no—
152C2×C3⋊Dic3—121no—
153C2×C3⋊Dic3—121no—
154C2×C3⋊Dic3—122no—
155C2×C3⋊Dic3D1211yescharacteristic
156C2×C3⋊Dic3D1211yescharacteristic
157#634—21no—
158#634Dic612yes—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—31no—
163C2×C3×Dic3—31no—
164C2×C3×Dic3—31no—
165C2×C3×Dic3—31no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—121no—
173C2×C3×Dic3—121no—
174C3×Dic6—32no—
175C3×Dic6—32no—
176C3×Dic6—62no—
177C3×Dic6—61no—
178C3×Dic6—61no—
179C3×Dic6—121no—
180C32×C8—61no—
181C32×C8—61no—
182C3×C3⋊C8—22no—
183C3×C3⋊C8—21no—
184C3×C3⋊C8—21no—
185C3×C3⋊C8—21no—
186C3×C3⋊C8—21no—
187C3×C3⋊C8—41no—
188#479—271noSylow
189C2×C33C22⋊Q811yescharacteristic
190C2×C33#13511yescharacteristic
191C2×C33#13711yescharacteristic
192C2×C3×Q8—91no—
193C2×Dic6—31no—
194C2×Dic6—31no—
195C2×Dic6—61no—
196C2×C3×C8—31no—
197C2×C3×C8—31no—
198C2×C3×C8—31no—
199C2×C3×C8—31no—
200C2×C3×C8—61no—
201C2×C3×C8—61no—
202C3×C4⋊C4—31no—
203C3×C4⋊C4—31no—
204C3×C4⋊C4—61no—
205C3×C4⋊C4—91no—
206C3×C42—91no—
207C4⋊Dic3—31no—
208C4⋊Dic3—31no—
209C4⋊Dic3—61no—
210C4⋊Dic3—91no—
211C4⋊Dic3—91no—
212C4⋊Dic3—91no—
213C4⋊Dic3—181no—
214C4⋊Dic3—181no—
215C4⋊Dic3—181no—
216C4⋊Dic3—181no—
217C4⋊Dic3—181no—
218C4⋊Dic3—181no—
219C4⋊Dic3—182no—
220C4⋊Dic3—361no—
221C4⋊Dic3D3211yescharacteristic
222Dic3⋊C4—181no—
223Dic3⋊C4—181no—
224Dic3⋊C4—181no—
225Dic3⋊C4—181no—
226Dic3⋊C4—181no—
227Dic3⋊C4—181no—
228Dic3⋊C4—182no—
229C4×Dic3—31no—
230C4×Dic3—31no—
231C4×Dic3—61no—
232C4×Dic3—181no—
233C4×Dic3—181no—
234C4×Dic3—181no—
235C2×C3⋊C8D3211yescharacteristic
236C2×C3⋊C8D3211yescharacteristic
237C22×C32—21no—
238C22×C32—21no—
239C22×C32—21no—
240C22×C32—41no—
241C22×C32C2×D1211yescharacteristic
242C22×C32D3×D411yescharacteristic
243C22×C32D3×D411yescharacteristic
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—21no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4—21no—
250C32×C4—41no—
251C32×C4—41no—
252C32×C4—121no—
253C32×C4C2×Dic611yescharacteristic
254C32×C4#23511yescharacteristic
255C32×C4#23711yescharacteristic
256C32×C4#23711yescharacteristic
257C32×C4D3×Q811yescharacteristic
258C32×C4D3×Q811yescharacteristic
259C3⋊Dic3—22no—
260C3⋊Dic3—21no—
261C3⋊Dic3—21no—
262C3⋊Dic3—41no—
263C3⋊Dic3—121no—
264C3⋊Dic3—122no—
265C3⋊Dic3—121no—
266C3⋊Dic3—121no—
267C3⋊Dic3—241no—
268C3⋊Dic3—241no—
269C3⋊Dic3—242no—
270C3×Dic3—42no—
271C3×Dic3—41no—
272C3×Dic3—41no—
273C3×Dic3—62no—
274C3×Dic3—62no—
275C3×Dic3—61no—
276C3×Dic3—61no—
277C3×Dic3—61no—
278C3×Dic3—61no—
279C3×Dic3—122no—
280C3×Dic3—121no—
281C3×Dic3—121no—
282C3×Dic3—121no—
283C3×Dic3—121no—
284C3×Dic3—241no—
285#125—271no—
286C4×Q8—271no—
287#107—91no—
288#106—271no—
289C4⋊C8—91no—
290#103—91no—
291#103—271no—
292C33#47911yescharacteristic, complemented, Sylow
293C3×Q8—92no—
294C3×Q8—181no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—41no—
299C2×C3×C4—62no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—91no—
303C2×C3×C4—91no—
304C2×C3×C4—181no—
305C2×C3×C4—181no—
306C2×C3×C4C2×D3211yescharacteristic
307C2×C3×C4C2×D3211yescharacteristic
308C2×C3×C4C2×D3211yescharacteristic
309C2×Dic3—21no—
310C2×Dic3—31no—
311C2×Dic3—31no—
312C2×Dic3—31no—
313C2×Dic3—31no—
314C2×Dic3—61no—
315C2×Dic3—61no—
316C2×Dic3—61no—
317C2×Dic3—61no—
318C2×Dic3—61no—
319C2×Dic3—61no—
320C2×Dic3—121no—
321C2×Dic3—121no—
322C2×Dic3—182no—
323C2×Dic3—181no—
324C2×Dic3—181no—
325C2×Dic3—181no—
326C2×Dic3—361no—
327C2×Dic3—361no—
328C2×Dic3—362no—
329Dic6—32no—
330Dic6—32no—
331Dic6—62no—
332Dic6—61no—
333Dic6—61no—
334Dic6—121no—
335C3×C8—62no—
336C3×C8—61no—
337C3×C8—61no—
338C3×C8—61no—
339C3×C8—61no—
340C3×C8—121no—
341C3⋊C8—21no—
342C3⋊C8—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—41no—
353C2×C32—41no—
354C2×C32—41no—
355C2×C32#87811yescharacteristic
356C2×C32#89611yescharacteristic
357C2×C32#89711yescharacteristic
358C2×C32#90411yescharacteristic
359C2×C32#90911yescharacteristic
360C2×C32#91211yescharacteristic
361C2×C32#91611yescharacteristic
362C2×C32#91811yescharacteristic
363C2×C32#91811yescharacteristic
364C2×Q8—91no—
365C2×C8—31no—
366C2×C8—31no—
367C4⋊C4—31no—
368C4⋊C4—91no—
369C4⋊C4—271no—
370C4⋊C4—541no—
371C4⋊C4—541no—
372C4⋊C4—541no—
373C42—91no—
374C42—541no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—41no—
379C22×C3D3×D1211yescharacteristic
380C22×C3D3×D1211yescharacteristic
381C22×C3#381111yescharacteristic
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—41no—
389C3×C4—41no—
390C3×C4—122no—
391C3×C4—121no—
392C3×C4—121no—
393C3×C4—182no—
394C3×C4—181no—
395C3×C4—181no—
396C3×C4—361no—
397C3×C4D3×Dic611yescharacteristic
398C3×C4D3×Dic611yescharacteristic
399C3×C4#379511yescharacteristic
400C3×C4#379711yescharacteristic
401C3×C4#380411yescharacteristic
402C3×C4#380511yescharacteristic
403Dic3—41no—
404Dic3—61no—
405Dic3—61no—
406Dic3—61no—
407Dic3—61no—
408Dic3—62no—
409Dic3—62no—
410Dic3—122no—
411Dic3—121no—
412Dic3—121no—
413Dic3—121no—
414Dic3—121no—
415Dic3—241no—
416Dic3—362no—
417Dic3—361no—
418Dic3—361no—
419Dic3—361no—
420Dic3—721no—
421Dic3—721no—
422Dic3—722no—
423C32—21no—
424C32—21no—
425C32—21no—
426C32—41no—
427C32#479911yescharacteristic, complemented
428C32#482911yescharacteristic, complemented
429C32#487711yescharacteristic, complemented
430Q8—92no—
431Q8—181no—
432C2×C4—61no—
433C2×C4—91no—
434C2×C4—91no—
435C2×C4—181no—
436C2×C4—181no—
437C2×C4—541no—
438C2×C4D3311yescharacteristic
439C8—61no—
440C8—61no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—41no—
451C2×C3—41no—
452C2×C3—41no—
453C2×C3#6380111yescharacteristic
454C2×C3#6380111yescharacteristic
455C2×C3#6380611yescharacteristic
456C2×C3#6380711yescharacteristic
457C2×C3#6389311yescharacteristic
458C2×C3#6389511yescharacteristic
459C2×C3#6389611yescharacteristic
460C2×C3#6393611yescharacteristic
461C2×C3#6394511yescharacteristic
462C22#8895511yescharacteristic
463C4—121no—
464C4—181no—
465C4—181no—
466C4—182no—
467C4—361no—
468C4—1081no—
469C4#8895711yescharacteristic
470C4#8896111yescharacteristic
471C3—21no—
472C3—21no—
473C3—21no—
474C3—41no—
475C3#1059049411yescharacteristic, complemented
476C3#1059081411yescharacteristic, complemented
477C3#1059081511yescharacteristic, complemented
478C2#1172286411yescharacteristic
479C2#1172289711yescharacteristic
480C2#1172298411yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#479
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
292. C33#479semidirect
427. C32#4799semidirect
428. C32#4829semidirect
429. C32#4877semidirect
475. C3#10590494semidirect
476. C3#10590814semidirect
477. C3#10590815semidirect

Maximal subgroups

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

subgroupindex
#117236682
#117234472
#117220522
#117220402
C3×#635752
C3×#635722
C3×#635672
#105908153
#105908143
#105904943