Finite Group Databasebeta

← #13156658

Subgroups — #13156658

Order 1728.

498 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723447C211yesmaximal, characteristic
2C3×#63872C211yesmaximal, characteristic
3#11722070C211yesmaximal, characteristic
4#11722057C211yesmaximal, characteristic
5#11722054C211yesmaximal, characteristic
6C3×#63576C211yesmaximal, characteristic
7C3×#63574C211yesmaximal, characteristic
8#10590417—31nomaximal
9#10590414—31nomaximal
10#10590413—31nomaximal
11C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18#88806—21no—
19Dic3⋊(C3⋊Dic3)—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22#64099—31no—
23C3×#890—31no—
24C3×#890—31no—
25#63872D311yescharacteristic
26#63869—31no—
27#63869—31no—
28#63869—61no—
29#63635—31no—
30#63634—31no—
31C3×#808—31no—
32C3×#808—31no—
33C3×#807—31no—
34C3×#807—31no—
35#63576D311yescharacteristic
36#63575—31no—
37#63575—31no—
38#63575—61no—
39#63574D311yescharacteristic
40#63573—31no—
41#63573—31no—
42#63573—61no—
43#63553—31no—
44Dic3×C3⋊C8—31no—
45Dic3×C3⋊C8—31no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3×C3⋊Dic3D411yescharacteristic
49C2×C3×C3⋊Dic3D411yescharacteristic
50C2×C32×Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C3×C3⋊(C3⋊C8)Q812yes—
54C3×C3⋊(C3⋊C8)Q812yes—
55#5065—91no—
56#4878—91no—
57#4858—91no—
58C32×C4⋊C4—31no—
59C32×C4⋊C4—31no—
60C32×C4⋊C4—31no—
61C4⋊(C3⋊Dic3)—31no—
62C4⋊(C3⋊Dic3)—31no—
63C4⋊(C3⋊Dic3)—31no—
64C4⋊(C3⋊Dic3)—61no—
65C4⋊(C3⋊Dic3)—61no—
66C4⋊(C3⋊Dic3)—61no—
67C4⋊(C3⋊Dic3)—61no—
68C4⋊(C3⋊Dic3)—121no—
69#3750—61no—
70C4×C3⋊Dic3C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×C4⋊Dic3—21no—
74C3×C4⋊Dic3—21no—
75C3×C4⋊Dic3—21no—
76C3×C4⋊Dic3—61no—
77C3×C4⋊Dic3—61no—
78C3×C4⋊Dic3C2×D311yescharacteristic
79C3×C4⋊Dic3C2×D311yescharacteristic
80C3×C4⋊Dic3C2×D311yescharacteristic
81C3×C4⋊Dic3C2×D311yescharacteristic
82C3×C4⋊Dic3C2×D311yescharacteristic
83C3×C4⋊Dic3C2×D311yescharacteristic
84C3×Dic3⋊C4—61no—
85C3×Dic3⋊C4—61no—
86C3×C4×Dic3—31no—
87C3×C4×Dic3—31no—
88C3×C4×Dic3—61no—
89C2×C3×C3⋊C8—31no—
90C2×C3×C3⋊C8—31no—
91C2×C3×C3⋊C8—31no—
92C2×C3×C3⋊C8—31no—
93C2×C3×C3⋊C8—61no—
94C2×C3×C3⋊C8—61no—
95#3724—61no—
96#3724—61no—
97#3724—62no—
98Dic3⋊Dic3—21no—
99Dic3⋊Dic3—21no—
100Dic3⋊Dic3—61no—
101Dic3⋊Dic3—61no—
102Dic3⋊Dic3—62no—
103C22×C33C2×D411yescharacteristic
104C33×C4C2×Q811yescharacteristic
105C33×C4C2×Q811yescharacteristic
106C3⋊(C3⋊Dic3)—41no—
107C3×C3⋊Dic3—21no—
108C3×C3⋊Dic3—21no—
109C32×Dic3—41no—
110C32×Dic3—41no—
111C32×Dic3—41no—
112C3×#125—91no—
113#890—31no—
114#890—31no—
115#890—91no—
116#890—91no—
117#890—181no—
118#870—91no—
119C3×#107—91no—
120C3×#106—91no—
121#818—91no—
122#817—91no—
123C8×Dic3—91no—
124C8×Dic3—91no—
125#808—31no—
126#808—31no—
127#808—91no—
128#808—91no—
129#808—181no—
130#807—31no—
131#807—31no—
132#807—91no—
133#807—91no—
134#807—181no—
135C4×C3⋊C8—91no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—41no—
140C2×C32×C4—61no—
141C2×C32×C4—61no—
142C2×C32×C4—61no—
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146C2×C3⋊Dic3—61no—
147C2×C3⋊Dic3—62no—
148C2×C3⋊Dic3—61no—
149C2×C3⋊Dic3—61no—
150C2×C3⋊Dic3—121no—
151C2×C3⋊Dic3—121no—
152C2×C3⋊Dic3—122no—
153C2×C3⋊Dic3#6711yescharacteristic
154C2×C3⋊Dic3#6711yescharacteristic
155C2×C3×Dic3—22no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
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—41no—
167C2×C3×Dic3—41no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C3⋊(C3⋊C8)Dic612yes—
171C3⋊(C3⋊C8)Dic612yes—
172C3×C3⋊C8—32no—
173C3×C3⋊C8—32no—
174C3×C3⋊C8—32no—
175C3×C3⋊C8—32no—
176C3×C3⋊C8—62no—
177C3×C3⋊C8—62no—
178#499—271noSylow
179C2×C33C4⋊Q811yescharacteristic
180C2×C33#13511yescharacteristic
181C2×C33#13511yescharacteristic
182C2×C3×C8—91no—
183C2×C3×C8—91no—
184C3×C4⋊C4—31no—
185C3×C4⋊C4—31no—
186C3×C4⋊C4—31no—
187C3×C4⋊C4—31no—
188C3×C4⋊C4—31no—
189C3×C4⋊C4—31no—
190C3×C4⋊C4—61no—
191C3×C4⋊C4—61no—
192C3×C4⋊C4—61no—
193C3×C4⋊C4—181no—
194C3×C4⋊C4—181no—
195C3×C42—91no—
196C4⋊Dic3—61no—
197C4⋊Dic3—61no—
198C4⋊Dic3—91no—
199C4⋊Dic3—91no—
200C4⋊Dic3—91no—
201C4⋊Dic3—181no—
202C4⋊Dic3—181no—
203C4⋊Dic3—181no—
204C4⋊Dic3—181no—
205C4⋊Dic3—181no—
206C4⋊Dic3—182no—
207C4⋊Dic3—361no—
208C4⋊Dic3D3211yescharacteristic
209C4⋊Dic3D3211yescharacteristic
210C4⋊Dic3D3211yescharacteristic
211Dic3⋊C4—61no—
212Dic3⋊C4—61no—
213Dic3⋊C4—181no—
214Dic3⋊C4—181no—
215Dic3⋊C4—181no—
216Dic3⋊C4—181no—
217Dic3⋊C4—182no—
218C4×Dic3—31no—
219C4×Dic3—31no—
220C4×Dic3—61no—
221C2×C3⋊C8—31no—
222C2×C3⋊C8—31no—
223C2×C3⋊C8—31no—
224C2×C3⋊C8—31no—
225C2×C3⋊C8—61no—
226C2×C3⋊C8—61no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—21no—
230C22×C32—41no—
231C22×C32D3×D411yescharacteristic
232C22×C32D3×D411yescharacteristic
233C22×C32C2×#6711yescharacteristic
234C32×C4—21no—
235C32×C4—21no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4—21no—
240C32×C4—41no—
241C32×C4—41no—
242C32×C4—121no—
243C32×C4—121no—
244C32×C4—121no—
245C32×C4C2×Dic611yescharacteristic
246C32×C4C2×Dic611yescharacteristic
247C32×C4D3×Q811yescharacteristic
248C32×C4D3×Q811yescharacteristic
249C32×C4D3×Q811yescharacteristic
250C32×C4D3×Q811yescharacteristic
251C3⋊Dic3—21no—
252C3⋊Dic3—21no—
253C3⋊Dic3—121no—
254C3⋊Dic3—122no—
255C3⋊Dic3—121no—
256C3⋊Dic3—121no—
257C3⋊Dic3—241no—
258C3⋊Dic3—241no—
259C3⋊Dic3—242no—
260C3×Dic3—42no—
261C3×Dic3—41no—
262C3×Dic3—41no—
263C3×Dic3—41no—
264C3×Dic3—41no—
265C3×Dic3—41no—
266C3×Dic3—41no—
267C3×Dic3—61no—
268C3×Dic3—61no—
269C3×Dic3—61no—
270C3×Dic3—61no—
271C3×Dic3—81no—
272C3×Dic3—81no—
273C3×Dic3—121no—
274C3×Dic3—121no—
275#125—91no—
276#125—271no—
277#107—91no—
278#107—271no—
279#106—91no—
280#106—271no—
281C4×C8—271no—
282C33#49911yescharacteristic, complemented, Sylow
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—41no—
287C2×C3×C4—62no—
288C2×C3×C4—61no—
289C2×C3×C4—61no—
290C2×C3×C4—61no—
291C2×C3×C4—61no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—91no—
295C2×C3×C4—91no—
296C2×C3×C4—121no—
297C2×C3×C4—121no—
298C2×C3×C4C2×D3211yescharacteristic
299C2×C3×C4C2×D3211yescharacteristic
300C2×C3×C4C2×D3211yescharacteristic
301C2×Dic3—21no—
302C2×Dic3—21no—
303C2×Dic3—21no—
304C2×Dic3—31no—
305C2×Dic3—31no—
306C2×Dic3—31no—
307C2×Dic3—31no—
308C2×Dic3—61no—
309C2×Dic3—61no—
310C2×Dic3—182no—
311C2×Dic3—181no—
312C2×Dic3—181no—
313C2×Dic3—181no—
314C2×Dic3—361no—
315C2×Dic3—361no—
316C2×Dic3—362no—
317C3×C8—92no—
318C3×C8—92no—
319C3⋊C8—32no—
320C3⋊C8—32no—
321C3⋊C8—32no—
322C3⋊C8—32no—
323C3⋊C8—62no—
324C3⋊C8—62no—
325C2×C32—21no—
326C2×C32—21no—
327C2×C32—21no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—41no—
335C2×C32—41no—
336C2×C32—41no—
337C2×C32#88811yescharacteristic
338C2×C32#88811yescharacteristic
339C2×C32#91211yescharacteristic
340C2×C32#91611yescharacteristic
341C2×C32#91611yescharacteristic
342C2×C32#91911yescharacteristic
343C2×C32#94411yescharacteristic
344C2×C32#95011yescharacteristic
345C2×C32#95011yescharacteristic
346C2×C8—91no—
347C2×C8—91no—
348C4⋊C4—31no—
349C4⋊C4—31no—
350C4⋊C4—31no—
351C4⋊C4—181no—
352C4⋊C4—181no—
353C4⋊C4—271no—
354C4⋊C4—541no—
355C4⋊C4—541no—
356C42—91no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—41no—
361C22×C3D3×#6711yescharacteristic
362C22×C3D3×#6711yescharacteristic
363C22×C3#381111yescharacteristic
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—41no—
371C3×C4—41no—
372C3×C4—122no—
373C3×C4—121no—
374C3×C4—121no—
375C3×C4—121no—
376C3×C4—121no—
377C3×C4—121no—
378C3×C4—121no—
379C3×C4—181no—
380C3×C4—181no—
381C3×C4—241no—
382C3×C4—241no—
383C3×C4D3×Dic611yescharacteristic
384C3×C4D3×Dic611yescharacteristic
385C3×C4D3×Dic611yescharacteristic
386C3×C4D3×Dic611yescharacteristic
387C3×C4#379711yescharacteristic
388C3×C4#379711yescharacteristic
389Dic3—41no—
390Dic3—41no—
391Dic3—41no—
392Dic3—61no—
393Dic3—61no—
394Dic3—61no—
395Dic3—61no—
396Dic3—121no—
397Dic3—121no—
398Dic3—362no—
399Dic3—361no—
400Dic3—361no—
401Dic3—361no—
402Dic3—721no—
403Dic3—721no—
404Dic3—722no—
405C32—21no—
406C32—21no—
407C32—21no—
408C32—41no—
409C32#485811yescharacteristic, complemented
410C32#487811yescharacteristic, complemented
411C32#506511yescharacteristic, complemented
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—61no—
415C2×C4—91no—
416C2×C4—91no—
417C2×C4—541no—
418C2×C4D3311yescharacteristic
419C8—92no—
420C8—92no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—41no—
431C2×C3—41no—
432C2×C3—41no—
433C2×C3#6383911yescharacteristic
434C2×C3#6383911yescharacteristic
435C2×C3#6384011yescharacteristic
436C2×C3#6393311yescharacteristic
437C2×C3#6393511yescharacteristic
438C2×C3#6393611yescharacteristic
439C2×C3#6394711yescharacteristic
440C2×C3#6394911yescharacteristic
441C2×C3#6395011yescharacteristic
442C22#8895211yescharacteristic
443C4—121no—
444C4—121no—
445C4—121no—
446C4—181no—
447C4—181no—
448C4—1081no—
449C4#8895711yescharacteristic
450C4#8895711yescharacteristic
451C3—21no—
452C3—21no—
453C3—21no—
454C3—41no—
455C3#1059041311yescharacteristic, complemented
456C3#1059041411yescharacteristic, complemented
457C3#1059041711yescharacteristic, complemented
458C2#1172285711yescharacteristic
459C2#1172285811yescharacteristic
460C2#1172303411yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#499
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
282. C33#499semidirect
409. C32#4858semidirect
410. C32#4878semidirect
411. C32#5065semidirect
455. C3#10590413semidirect
456. C3#10590414semidirect
457. C3#10590417semidirect

Maximal subgroups

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

subgroupindex
#117234472
C3×#638722
#117220702
#117220572
#117220542
C3×#635762
C3×#635742
#105904173
#105904143
#105904133