Finite Group Databasebeta

← #13147854

Subgroups — #13147854

Order 1728.

1074 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88651C212yesmaximal, complemented
2C4×C4⋊#272C211yesmaximal, characteristic
3#64078⋊C3C211yesmaximal, characteristic, complemented
4C4×#634⋊C3C211yesmaximal, characteristic, complemented
5#11721150C214yesmaximal, complemented
6#11721149C212yesmaximal
7#11721116C212yesmaximal, complemented
8#11721094C212yesmaximal, complemented
9#10593904—31nomaximal
10Dic6×D12—31nomaximal
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×#272C2212yes—
13C2×#634⋊C3C2212yes—
14C2×#634⋊C3C2211yescharacteristic
15C2×#6199C2214yes—
16C2×#6196C2214yes—
17#88651—28no—
18C42×H(3,𝔽3)C2211yescharacteristic
19#88542C2212yes—
20C4⋊#1084C2211yescharacteristic
21C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
22C4⋊((C3⋊Dic3)⋊C3)C2212yes—
23C4⋊((C3⋊Dic3)⋊C3)C2212yes—
24#3750⋊C3C2212yes—
25C4×(C3⋊Dic3)⋊C3C2212yes—
26#88451C2214yes—
27#88448C2214yes—
28C2×D3×Dic6—32no—
29C2×D3×Dic6—32no—
30#64078D311yescharacteristic, complemented
31C4×#634D311yescharacteristic, complemented
32C3×C4×D12—31no—
33C3×C4×D12—31no—
34C3×C4×D12—61no—
35C3×#869—31no—
36C3×C4×Dic6—31no—
37#63919—32no—
38#63918—32no—
39#63899—34no—
40#63898—32no—
41#63893—34no—
42Dic3×D12—32no—
43#63857—32no—
44#63856—32no—
45C22×#272C2312yes—
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49C2×#1084C2312yes—
50C4⋊#272Q814yescomplemented
51C4×#272—24no—
52C2×(C3⋊Dic3)⋊C3C2314yes—
53C2×(C3⋊Dic3)⋊C3C2312yes—
54C2×(C3⋊Dic3)⋊C3C2312yes—
55#634⋊C3—24no—
56#634⋊C3D414yescomplemented
57#6199—28no—
58#6196—28no—
59#5727—91no—
60Q8×D12—91no—
61D4×Dic6—91no—
62C2×#634C2×D312yes—
63C2×#634C2×D311yescharacteristic
64C2×C3×D12—31no—
65C2×C3×D12—31no—
66C2×C3×D12—61no—
67C2×C3×C4×D3—32no—
68C2×C3×C4×D3—32no—
69C2×C3×C4×D3—62no—
70C2×C3×Dic6—31no—
71C2×C3×Dic6—32no—
72C2×#627—34no—
73C2×#627—34no—
74C2×D3×Dic3—34no—
75C2×D3×Dic3—34no—
76D3×Dic6—68no—
77D3×Dic6—68no—
78C32×C42—21no—
79C32×C42C2×D311yescharacteristic
80C32×C42C2×D311yescharacteristic
81C4⋊(C3⋊Dic3)C2×D311yescharacteristic
82C4⋊(C3⋊Dic3)C2×D312yes—
83C4⋊(C3⋊Dic3)C2×D312yes—
84#3750C2×D312yes—
85C4×C3⋊Dic3C2×D312yes—
86C3×(C2×D3)⋊C4—32no—
87C3×(C2×D3)⋊C4—32no—
88C3×(C2×D3)⋊C4—62no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—32no—
91C3×C4⋊Dic3—32no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—61no—
95C3×Dic3⋊C4—32no—
96C3×C4×Dic3—32no—
97#3724—34no—
98Dic3⋊Dic3—34no—
99(C2×D3)⋊Dic3—34no—
100(C2×D3)⋊Dic3—34no—
101C22×H(3,𝔽3)C2411yescharacteristic
102C2×#272—22no—
103C2×#272—22no—
104C2×#272C2×Q814yescomplemented
105C4×H(3,𝔽3)—21no—
106C4×H(3,𝔽3)C2×D412yes—
107C4×H(3,𝔽3)C2×Q812yescomplemented
108#1084—22no—
109(C3⋊Dic3)⋊C3—24no—
110(C3⋊Dic3)⋊C3—22no—
111(C3⋊Dic3)⋊C3C2×D414yes—
112C2×D3×Q8—92no—
113C2×D3×Q8—92no—
114C22×Dic6—92no—
115C3×C4⋊Q8—91no—
116C3×C4×Q8—91no—
117C3×C4×D4—91no—
118#946—92no—
119D4×Dic3—92no—
120#924—92no—
121#897—94no—
122#897—92no—
123#896—94no—
124#893—92no—
125#888—92no—
126#888—92no—
127#878—94no—
128C4×D12—31no—
129C4×D12—31no—
130C4×D12—61no—
131C4×D12—92no—
132#869—31no—
133#869—91no—
134#869—92no—
135C4×Dic6—31no—
136C4×Dic6—91no—
137C22×C3×D3—32no—
138C22×C3×D3—32no—
139C22×C3×D3—62no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D311yescharacteristic
149C2×C3⋊Dic3C22×D312yes—
150C2×C3⋊Dic3C22×D314yes—
151C2×C3⋊Dic3C22×D312yes—
152#634—24no—
153#634D1214yescomplemented
154C2×C3×Dic3—32no—
155C2×C3×Dic3—32no—
156C2×C3×Dic3—34no—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—62no—
160C3×D12—34no—
161C3×D12—34no—
162C3×D12—64no—
163C3×C4×D3—64no—
164C3×C4×D3—64no—
165C3×C4×D3—124no—
166C3×Dic6—34no—
167C3×Dic6—64no—
168#627—68no—
169#627—68no—
170D3×Dic3—68no—
171D3×Dic3—68no—
172D4×Q8—271noSylow
173C2×H(3,𝔽3)C22×D411yescharacteristic
174C2×H(3,𝔽3)C22×Q811yescharacteristic
175C2×H(3,𝔽3)2−1+411yescharacteristic
176#272—24no—
177C2×C3×Q8—91no—
178C2×C3×Q8—92no—
179C2×C3×D4—91no—
180C22×C3×C4—92no—
181C22×Dic3—94no—
182D3×Q8—188no—
183D3×Q8—188no—
184C2×D12—31no—
185C2×D12—31no—
186C2×D12—61no—
187C2×C4×D3—32no—
188C2×C4×D3—32no—
189C2×C4×D3—62no—
190C2×C4×D3—94no—
191C2×C4×D3—94no—
192C2×Dic6—31no—
193C2×Dic6—32no—
194C2×Dic6—91no—
195C2×Dic6—92no—
196C2×Dic6—94no—
197C2×Dic6—94no—
198C2×Dic6—94no—
199C2×Dic6—188no—
200C3×C4⋊C4—92no—
201C3×C4⋊C4—92no—
202C3×C4⋊C4—92no—
203C3×C4⋊C4—91no—
204C3×C4⋊C4—91no—
205C3×C22⋊C4—92no—
206C3×C42—31no—
207C3×C42—31no—
208C3×C42—61no—
209C3×C42—92no—
210C3×C42D3211yescharacteristic
211#217—94no—
212(C2×D3)⋊C4—32no—
213(C2×D3)⋊C4—32no—
214(C2×D3)⋊C4—62no—
215(C2×D3)⋊C4—94no—
216(C2×D3)⋊C4—94no—
217C4⋊Dic3—31no—
218C4⋊Dic3—31no—
219C4⋊Dic3—31no—
220C4⋊Dic3—32no—
221C4⋊Dic3—32no—
222C4⋊Dic3—61no—
223C4⋊Dic3—91no—
224C4⋊Dic3—92no—
225C4⋊Dic3—92no—
226C4⋊Dic3—94no—
227Dic3⋊C4—32no—
228Dic3⋊C4—92no—
229Dic3⋊C4—94no—
230Dic3⋊C4—94no—
231C4×Dic3—32no—
232C4×Dic3—92no—
233C22×C32—21no—
234C22×C32C23×D311yescharacteristic
235C22×C32C23×D311yescharacteristic
236C2×C3×D3—34no—
237C2×C3×D3—34no—
238C2×C3×D3—64no—
239C2×C3×D3—62no—
240C2×C3×D3—62no—
241C2×C3×D3—62no—
242C2×C3×D3—62no—
243C2×C3×D3—122no—
244C2×C3×D3—122no—
245C32×C4—21no—
246C32×C4—21no—
247C32×C4—22no—
248C32×C4—22no—
249C32×C4—41no—
250C32×C4C2×D1212yes—
251C32×C4D3×D412yes—
252C32×C4D3×Q812yescomplemented
253C32×C4D3×Q812yescomplemented
254C3⋊Dic3—24no—
255C3⋊Dic3—22no—
256C3⋊Dic3C2×D1214yes—
257C3×Dic3—34no—
258C3×Dic3—64no—
259C3×Dic3—62no—
260C3×Dic3—62no—
261C3×Dic3—62no—
262C3×Dic3—122no—
263C22×Q8—272no—
264C4⋊Q8—91no—
265C4⋊Q8—272no—
266C22⋊Q8—274no—
267C22⋊Q8—272no—
268C4×Q8—91no—
269C4×D4—91no—
270C4×D4—272no—
271H(3,𝔽3)D4×Q811yescharacteristic, complemented, Sylow
272C23×C3—92no—
273C22×D3—32no—
274C22×D3—32no—
275C22×D3—62no—
276C3×Q8—94no—
277C3×Q8—184no—
278C3×D4—94no—
279C2×C3×C4—31no—
280C2×C3×C4—31no—
281C2×C3×C4—31no—
282C2×C3×C4—31no—
283C2×C3×C4—31no—
284C2×C3×C4—31no—
285C2×C3×C4—61no—
286C2×C3×C4—61no—
287C2×C3×C4—61no—
288C2×C3×C4—92no—
289C2×C3×C4—94no—
290C2×C3×C4—92no—
291C2×C3×C4—92no—
292C2×C3×C4—184no—
293C2×C3×C4C2×D3211yescharacteristic
294C2×C3×C4C2×D3211yescharacteristic
295C2×C3×C4C2×D3211yescharacteristic
296C2×Dic3—32no—
297C2×Dic3—32no—
298C2×Dic3—32no—
299C2×Dic3—32no—
300C2×Dic3—34no—
301C2×Dic3—62no—
302C2×Dic3—92no—
303C2×Dic3—92no—
304C2×Dic3—94no—
305C2×Dic3—188no—
306D12—34no—
307D12—34no—
308D12—64no—
309C4×D3—64no—
310C4×D3—64no—
311C4×D3—124no—
312C4×D3—188no—
313C4×D3—188no—
314Dic6—34no—
315Dic6—64no—
316Dic6—94no—
317Dic6—184no—
318Dic6—188no—
319Dic6—188no—
320Dic6—188no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32C22×D1211yescharacteristic
325C2×C32C2×D3×D411yescharacteristic
326C2×C32C2×D3×Q811yescharacteristic
327C2×C32C2×D3×Q811yescharacteristic
328C2×C32#100711yescharacteristic
329C2×C32#101011yescharacteristic
330C3×D3—64no—
331C3×D3—64no—
332C3×D3—124no—
333C2×Q8—91no—
334C2×Q8—92no—
335C2×Q8—274no—
336C2×Q8—548no—
337C2×D4—91no—
338C22×C4—92no—
339C22×C4—274no—
340C4⋊C4—91no—
341C4⋊C4—92no—
342C4⋊C4—92no—
343C4⋊C4—91no—
344C4⋊C4—92no—
345C4⋊C4—274no—
346C22⋊C4—92no—
347C22⋊C4—274no—
348C42—92no—
349C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
350C22×C3—31no—
351C22×C3—31no—
352C22×C3—61no—
353C22×C3—94no—
354C22×C3—182no—
355C22×C3—182no—
356C22×C3C22×D3211yescharacteristic
357C2×D3—34no—
358C2×D3—34no—
359C2×D3—64no—
360C2×D3—62no—
361C2×D3—62no—
362C2×D3—62no—
363C2×D3—62no—
364C2×D3—122no—
365C2×D3—122no—
366C3×C4—21no—
367C3×C4—32no—
368C3×C4—32no—
369C3×C4—32no—
370C3×C4—32no—
371C3×C4—61no—
372C3×C4—61no—
373C3×C4—62no—
374C3×C4—62no—
375C3×C4—94no—
376C3×C4—121no—
377C3×C4—184no—
378C3×C4—182no—
379C3×C4—182no—
380C3×C4#379712yes—
381C3×C4D3×D1212yes—
382Dic3—34no—
383Dic3—62no—
384Dic3—62no—
385Dic3—62no—
386Dic3—64no—
387Dic3—94no—
388Dic3—122no—
389Dic3—184no—
390Dic3—182no—
391C32—21no—
392C32Q8×D1211yescharacteristic, complemented
393C32#572711yescharacteristic, complemented
394C23—92no—
395Q8—94no—
396Q8—184no—
397Q8—548no—
398D4—94no—
399C2×C4—92no—
400C2×C4—92no—
401C2×C4—92no—
402C2×C4—94no—
403C2×C4—184no—
404C2×C4—548no—
405C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
406C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
407C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—31no—
414C2×C3—61no—
415C2×C3—61no—
416C2×C3—61no—
417C2×C3—184no—
418C2×C3#6429711yescharacteristic
419C2×C3C2×#379711yescharacteristic
420C2×C3C2×D3×D1211yescharacteristic
421D3—64no—
422D3—64no—
423D3—124no—
424C22—94no—
425C22—182no—
426C22—182no—
427C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
428C4—21no—
429C4—94no—
430C4—182no—
431C4—182no—
432C4—184no—
433C4#8865112yescomplemented
434C4#8865412yes—
435C3—31no—
436C3—31no—
437C3—61no—
438C3#1059390311yescharacteristic
439C2—184no—
440C2#1172248211yescharacteristic
441C2C2×#8865111yescharacteristic
442C2C2×#8865411yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×C4⋊#272C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)D4×Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#88651C2semidirect
3. #64078⋊C3C2semidirect
4. C4×#634⋊C3C2semidirect
5. #11721150C2semidirect
7. #11721116C2semidirect
8. #11721094C2semidirect
30. #64078D3semidirect
31. C4×#634D3semidirect
50. C4⋊#272Q8semidirect
56. #634⋊C3D4semidirect
104. C2×#272C2×Q8semidirect
107. C4×H(3,𝔽3)C2×Q8semidirect
153. #634D12semidirect
252. C32×C4D3×Q8semidirect
253. C32×C4D3×Q8semidirect
271. H(3,𝔽3)D4×Q8semidirect
392. C32Q8×D12semidirect
393. C32#5727semidirect
433. C4#88651semidirect

Maximal subgroups

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

subgroupindex
C2×#886512
C4×C4⋊#2722
#64078⋊C32
C4×#634⋊C32
#117211502
#117211492
#117211162
#117210942
#105939043
Dic6×D123