Finite Group Databasebeta

← #13137371

Subgroups — #13137371

Order 1728.

437 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64081⋊C3C211yesmaximal, characteristic
2C2×#88430C211yesmaximal, characteristic
3C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic
4C4⋊#6180C211yesmaximal, characteristic, complemented
5C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
6#11720311C211yesmaximal, characteristic, complemented
7#11720311C211yesmaximal, characteristic, complemented
8#10590332—31nomaximal
9#10590324—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C42×H(3,𝔽3)C2211yescharacteristic
12C2×#6180C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14#3752⋊C3C411yescharacteristic
15#3752⋊C3C411yescharacteristic
16#3750⋊C3C411yescharacteristic
17#3750⋊C3C411yescharacteristic
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88430—22no—
22((C3⋊D3)⋊C3)⋊C8—22no—
23#64081D311yescharacteristic
24C3×#872—31no—
25C2×#3712—31no—
26C2×#3711—31no—
27C2×#3710—31no—
28C2×D3×C3⋊C8—31no—
29C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
30C3×C4⋊(C3⋊C8)—31no—
31C3×C4⋊(C3⋊C8)—31no—
32C3×C4⋊(C3⋊C8)—31no—
33C3×C4⋊(C3⋊C8)—61no—
34#63572—31no—
35#63572—31no—
36Dic3⋊(C3⋊C8)—31no—
37Dic3⋊(C3⋊C8)—31no—
38C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
39C2×C4×H(3,𝔽3)C2×C411yescharacteristic
40C2×C4×H(3,𝔽3)C2×C411yescharacteristic
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
43C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
45C4×(C3⋊D3)⋊C3D412yes—
46C4×(C3⋊D3)⋊C3Q812yes—
47#6180—21no—
48#6180—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51#4999—91no—
52#4839—91no—
53#4839—91no—
54C2×C4×C3⋊D3C2×D311yescharacteristic
55C2×C3×C4×D3—31no—
56C32×C42—21no—
57C32×C42C2×D311yescharacteristic
58C32×C42C2×D311yescharacteristic
59#3752Dic311yescharacteristic
60#3752Dic311yescharacteristic
61#3750Dic311yescharacteristic
62#3750Dic311yescharacteristic
63C4×C3⋊Dic3C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
65C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
66C3×(C2×D3)⋊C4—31no—
67C3×(C2×D3)⋊C4—31no—
68C3×Dic3⋊C4—31no—
69C3×Dic3⋊C4—31no—
70C3×C4×Dic3—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—61no—
78C2×C3×C3⋊C8—61no—
79#3712—62no—
80#3711—62no—
81#3710—62no—
82D3×C3⋊C8—62no—
83C22×H(3,𝔽3)C22×C411yescharacteristic
84C2×(C3⋊D3)⋊C3—21no—
85C2×(C3⋊D3)⋊C3—21no—
86C2×(C3⋊D3)⋊C3C4⋊C412yes—
87C4×H(3,𝔽3)—21no—
88C4×H(3,𝔽3)—21no—
89C4×H(3,𝔽3)C2×D411yescharacteristic
90C4×H(3,𝔽3)C2×Q811yescharacteristic
91(C3⋊Dic3)⋊C3—21no—
92(C3⋊Dic3)⋊C3—21no—
93(C3⋊Dic3)⋊C3C4⋊C412yes—
94C3×#117—91no—
95C2×#208—91no—
96C22×C3⋊C8—91no—
97C2×ΓL(1,𝔽25)—91no—
98C2×ΓL(1,𝔽25)—91no—
99C2×D3×C8—91no—
100C2×D3×C8—91no—
101#872—31no—
102#872—91no—
103C3×C4⋊C8—91no—
104C3×C4⋊C8—91no—
105Dic3⋊C8—91no—
106Dic3⋊C8—91no—
107Dic3⋊C8—91no—
108Dic3⋊C8—91no—
109C4⋊(C3⋊C8)—31no—
110C4⋊(C3⋊C8)—31no—
111C4⋊(C3⋊C8)—31no—
112C4⋊(C3⋊C8)—61no—
113C4⋊(C3⋊C8)—91no—
114C4⋊(C3⋊C8)—91no—
115C4⋊(C3⋊C8)—91no—
116C22×C3⋊D3C2×Dic311yescharacteristic
117C22×C3×D3—31no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4C4×D311yescharacteristic
122C2×C32×C4C4×D311yescharacteristic
123C2×C32×C4C2×Dic311yescharacteristic
124C2×C32×C4C2×Dic311yescharacteristic
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3C2×Dic311yescharacteristic
128C2×C3⋊Dic3C2×Dic311yescharacteristic
129C2×C3⋊Dic3C2×Dic311yescharacteristic
130C4×C3⋊D3Dic612yes—
131C4×C3⋊D3D1212yes—
132C2×C3×Dic3—31no—
133C2×C3×Dic3—31no—
134C2×C3×Dic3—31no—
135C3×C4×D3—32no—
136C3×C4×D3—32no—
137C3⋊(C3⋊C8)—21no—
138C3⋊(C3⋊C8)—21no—
139C3×C3⋊C8—61no—
140C3×C3⋊C8—61no—
141C3×C3⋊C8—61no—
142C3×C3⋊C8—61no—
143C3×C3⋊C8—61no—
144C3×C3⋊C8—61no—
145C3×C3⋊C8—121no—
146C3×C3⋊C8—121no—
147#424—271noSylow
148C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
149C2×H(3,𝔽3)#13111yescharacteristic
150C2×H(3,𝔽3)#13111yescharacteristic
151(C3⋊D3)⋊C3—22no—
152C22×C3×C4—91no—
153C2×C4×D3—31no—
154C2×C4×D3—91no—
155C2×C3×C8—91no—
156C2×C3×C8—91no—
157C2×C3×C8—91no—
158C2×C3×C8—91no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4—91no—
161C3×C22⋊C4—91no—
162C3×C22⋊C4—91no—
163C3×C42—31no—
164C3×C42—31no—
165C3×C42—61no—
166C3×C42—91no—
167C3×C42D3211yescharacteristic
168(C2×D3)⋊C4—31no—
169(C2×D3)⋊C4—31no—
170(C2×D3)⋊C4—91no—
171(C2×D3)⋊C4—91no—
172Dic3⋊C4—31no—
173Dic3⋊C4—31no—
174Dic3⋊C4—91no—
175Dic3⋊C4—91no—
176C4×Dic3—31no—
177C4×Dic3—91no—
178#208—182no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8—31no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—61no—
186C2×C3⋊C8—61no—
187C2×C3⋊C8—91no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—182no—
190ΓL(1,𝔽25)—182no—
191ΓL(1,𝔽25)—182no—
192D3×C8—182no—
193D3×C8—182no—
194C22×C32—21no—
195C22×C32C2×C4×D311yescharacteristic
196C22×C32C22×Dic311yescharacteristic
197C2×C3⋊D3—21no—
198C2×C3⋊D3—21no—
199C2×C3⋊D3C4⋊Dic312yes—
200C2×C3×D3—32no—
201C2×C3×D3—61no—
202C2×C3×D3—61no—
203C32×C4—22no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—21no—
207C32×C4—21no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4—41no—
211C32×C4C2×Dic611yescharacteristic
212C32×C4C2×D1211yescharacteristic
213C32×C4D3×D411yescharacteristic
214C32×C4D3×Q811yescharacteristic
215C3⋊Dic3—21no—
216C3⋊Dic3—21no—
217C3⋊Dic3C4⋊Dic312yes—
218C3×Dic3—32no—
219C3×Dic3—61no—
220C3×Dic3—61no—
221C2×M16—271no—
222C22×C8—271no—
223#117—91no—
224C4⋊C8—91no—
225C4⋊C8—91no—
226C4⋊C8—271no—
227C4⋊C8—271no—
228H(3,𝔽3)#42411yescharacteristic, complemented, Sylow
229C23×C3—91no—
230C22×D3—31no—
231C22×D3—91no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—61no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—92no—
242C2×C3×C4—92no—
243C2×C3×C4—91no—
244C2×C3×C4—91no—
245C2×C3×C4—91no—
246C2×C3×C4D3×Dic311yescharacteristic
247C2×C3×C4D3×Dic311yescharacteristic
248C2×C3×C4C2×D3211yescharacteristic
249C2×Dic3—31no—
250C2×Dic3—31no—
251C2×Dic3—31no—
252C2×Dic3—91no—
253C2×Dic3—91no—
254C2×Dic3—91no—
255C4×D3—32no—
256C4×D3—32no—
257C4×D3—92no—
258C4×D3—92no—
259C3×C8—181no—
260C3×C8—181no—
261C3×C8—181no—
262C3×C8—181no—
263C3⋊C8—61no—
264C3⋊C8—61no—
265C3⋊C8—61no—
266C3⋊C8—61no—
267C3⋊C8—61no—
268C3⋊C8—61no—
269C3⋊C8—121no—
270C3⋊C8—121no—
271C3⋊C8—181no—
272C3⋊C8—181no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32D3×C4⋊C411yescharacteristic
277C2×C32#90111yescharacteristic
278C2×C32#90711yescharacteristic
279C2×C32C2×C4⋊Dic311yescharacteristic
280C2×C32#94811yescharacteristic
281C2×C32#94811yescharacteristic
282C3⋊D3—22no—
283C3×D3—62no—
284C22×C4—91no—
285M16—542no—
286C2×C8—91no—
287C2×C8—91no—
288C2×C8—91no—
289C2×C8—91no—
290C2×C8—542no—
291C4⋊C4—91no—
292C4⋊C4—91no—
293C22⋊C4—91no—
294C22⋊C4—91no—
295C42—91no—
296C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
297C22×C3—31no—
298C22×C3—31no—
299C22×C3—61no—
300C22×C3—92no—
301C22×C3—181no—
302C22×C3—181no—
303C22×C3C2×D3×Dic311yescharacteristic
304C2×D3—32no—
305C2×D3—61no—
306C2×D3—61no—
307C2×D3—92no—
308C2×D3—181no—
309C2×D3—181no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—31no—
315C3×C4—31no—
316C3×C4—61no—
317C3×C4—61no—
318C3×C4—61no—
319C3×C4—61no—
320C3×C4—62no—
321C3×C4—61no—
322C3×C4—61no—
323C3×C4—92no—
324C3×C4—121no—
325C3×C4—181no—
326C3×C4—181no—
327C3×C4D3×Dic611yescharacteristic
328C3×C4D3×D1211yescharacteristic
329Dic3—32no—
330Dic3—61no—
331Dic3—61no—
332Dic3—92no—
333Dic3—181no—
334Dic3—181no—
335C32—21no—
336C32#483911yescharacteristic, complemented
337C32#499911yescharacteristic, complemented
338C23—91no—
339C2×C4—91no—
340C2×C4—92no—
341C2×C4—92no—
342C2×C4—91no—
343C2×C4—91no—
344C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
345C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
346C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
347C8—181no—
348C8—181no—
349C8—181no—
350C8—181no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—61no—
358C2×C3—61no—
359C2×C3—61no—
360C2×C3—182no—
361C2×C3#6381411yescharacteristic
362C2×C3#6381511yescharacteristic
363C2×C3D3×C4⋊Dic311yescharacteristic
364D3—62no—
365D3—182no—
366C22—92no—
367C22—181no—
368C22—181no—
369C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
370C4—21no—
371C4—21no—
372C4—92no—
373C4—181no—
374C4—181no—
375C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
376C4#8865411yescharacteristic
377C3—31no—
378C3—31no—
379C3—61no—
380C3#1059032411yescharacteristic
381C2—182no—
382C2#1172105411yescharacteristic
383C2#1172105511yescharacteristic
384C2#1172112911yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×D3×Dic3
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#424
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
4. C4⋊#6180C2semidirect
5. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
6. #11720311C2semidirect
7. #11720311C2semidirect
29. C4⋊(C3⋊(C3⋊C8))D3semidirect
228. H(3,𝔽3)#424semidirect
336. C32#4839semidirect
337. C32#4999semidirect

Maximal subgroups

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

subgroupindex
#64081⋊C32
C2×#884302
C2×((C3⋊D3)⋊C3)⋊C82
C4⋊#61802
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#117203112
#105903323
#105903243