Finite Group Databasebeta

← #13138776

Subgroups — #13138776

Order 1728.

437 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#3743⋊C3C211yesmaximal, characteristic
2#11721105C211yesmaximal, characteristic
3C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic
4#11720547C211yesmaximal, characteristic, complemented
5#63641⋊C3C211yesmaximal, characteristic, complemented
6#11720311C211yesmaximal, characteristic, complemented
7#11720304C211yesmaximal, characteristic, complemented
8#10590702—31nomaximal
9#10590695—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C4×#1084C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17#3743⋊C3—22no—
18#88450C411yescharacteristic
19#88450C411yescharacteristic
20#88448C411yescharacteristic
21#88448C411yescharacteristic
22((C3⋊D3)⋊C3)⋊C8—22no—
23C2×#3743D311yescharacteristic
24C2×C3×ΓL(1,𝔽25)—31no—
25#63874—31no—
26#63855—31no—
27C2×#3710—31no—
28C2×D3×C3⋊C8—31no—
29#63641D311yescharacteristic, complemented
30C3×Dic3⋊C8—31no—
31C3×Dic3⋊C8—31no—
32C3×Dic3⋊C8—31no—
33C3×Dic3⋊C8—61no—
34#63572—31no—
35Dic3⋊(C3⋊C8)—31no—
36Dic3⋊(C3⋊C8)—31no—
37Dic3⋊(C3⋊C8)—31no—
38C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084C2×C411yescharacteristic
41C2×#1084C2×C411yescharacteristic
42C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
43C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
45C4×(C3⋊D3)⋊C3D412yes—
46C4×(C3⋊D3)⋊C3Q812yes—
47C8×H(3,𝔽3)—21no—
48#6180—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51#5119—91no—
52#4839—91no—
53#4839—91no—
54C2×C4×C3⋊D3C2×D311yescharacteristic
55C2×C3×C4×D3—31no—
56C2×C32×C8—21no—
57C2×C32×C8C2×D311yescharacteristic
58C2×C32×C8C2×D311yescharacteristic
59C4×C3⋊Dic3C2×D311yescharacteristic
60C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
62#3743—22no—
63C3×C4×Dic3—31no—
64C3×C4×Dic3—31no—
65C3×C4×Dic3—31no—
66C3×C4×Dic3—61no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—31no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—61no—
72C3×ΓL(1,𝔽25)—62no—
73#3724—31no—
74#3724—31no—
75Dic3⋊Dic3—31no—
76Dic3⋊Dic3—31no—
77#3722—31no—
78#3722—31no—
79(C2×D3)⋊Dic3—31no—
80(C2×D3)⋊Dic3—31no—
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)C2×D411yescharacteristic
88C4×H(3,𝔽3)C2×Q811yescharacteristic
89#1084—21no—
90#1084—21no—
91(C3⋊Dic3)⋊C3—21no—
92(C3⋊Dic3)⋊C3—21no—
93(C3⋊Dic3)⋊C3C4⋊C412yes—
94C2×C3×M16—91no—
95#926—91no—
96C22×C3⋊C8—91no—
97C2×ΓL(1,𝔽25)—31no—
98C2×ΓL(1,𝔽25)—91no—
99C2×D3×C8—91no—
100C2×D3×C8—91no—
101#872—91no—
102#872—91no—
103C3×C4⋊C8—91no—
104C3×C4⋊C8—91no—
105Dic3⋊C8—31no—
106Dic3⋊C8—31no—
107Dic3⋊C8—31no—
108Dic3⋊C8—61no—
109Dic3⋊C8—91no—
110Dic3⋊C8—91no—
111Dic3⋊C8—91no—
112Dic3⋊C8—91no—
113Dic3⋊C8—91no—
114C4⋊(C3⋊C8)—91no—
115C4⋊(C3⋊C8)—91no—
116C22×C3⋊D3C4×D311yescharacteristic
117C22×C3×D3—31no—
118C2×C32×C4—21no—
119C2×C32×C4C22×D311yescharacteristic
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3C4×D311yescharacteristic
122C2×C3⋊Dic3C4×D311yescharacteristic
123C2×C3⋊Dic3C4×D311yescharacteristic
124C4×C3⋊D3Dic612yes—
125C4×C3⋊D3#6712yes—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—31no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—31no—
132C2×C3×Dic3—31no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C3×C4×D3—32no—
136C3×C4×D3—32no—
137C32×C8—21no—
138C32×C8—21no—
139C32×C8—41no—
140C3⋊(C3⋊C8)—21no—
141C3⋊(C3⋊C8)—21no—
142C3×C3⋊C8—61no—
143C3×C3⋊C8—61no—
144C3×C3⋊C8—61no—
145C3×C3⋊C8—61no—
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—
155C3×M16—182no—
156C2×C3×C8—31no—
157C2×C3×C8—31no—
158C2×C3×C8—61no—
159C2×C3×C8—91no—
160C2×C3×C8—91no—
161C2×C3×C8—91no—
162C2×C3×C8D3211yescharacteristic
163C3×C42—91no—
164C3×C42—91no—
165#217—91no—
166#217—91no—
167(C2×D3)⋊C4—91no—
168(C2×D3)⋊C4—91no—
169(C2×D3)⋊C4—91no—
170(C2×D3)⋊C4—91no—
171C4⋊Dic3—91no—
172C4⋊Dic3—91no—
173Dic3⋊C4—91no—
174Dic3⋊C4—91no—
175Dic3⋊C4—91no—
176Dic3⋊C4—91no—
177C4×Dic3—31no—
178C4×Dic3—31no—
179C4×Dic3—31no—
180C4×Dic3—61no—
181C4×Dic3—91no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8—31no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—61no—
187C2×C3⋊C8—91no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—182no—
190ΓL(1,𝔽25)—62no—
191ΓL(1,𝔽25)—182no—
192D3×C8—182no—
193D3×C8—182no—
194C22×C32—21no—
195C22×C32C2×C4×D311yescharacteristic
196C22×C32C2×C4×D311yescharacteristic
197C2×C3⋊D3—21no—
198C2×C3⋊D3—21no—
199C2×C3⋊D3Dic3⋊C412yes—
200C2×C3×D3—32no—
201C2×C3×D3—61no—
202C2×C3×D3—61no—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4C2×Dic611yescharacteristic
206C32×C4D3×D411yescharacteristic
207C32×C4D3×Q811yescharacteristic
208C32×C4C2×#6711yescharacteristic
209C3⋊Dic3—21no—
210C3⋊Dic3—21no—
211C3⋊Dic3Dic3⋊C412yes—
212C3×Dic3—32no—
213C3×Dic3—62no—
214C3×Dic3—61no—
215C3×Dic3—61no—
216C3×Dic3—61no—
217C3×Dic3—61no—
218C3×Dic3—61no—
219C3×Dic3—61no—
220C3×Dic3—121no—
221C2×M16—91no—
222C22×C8—271no—
223#117—271no—
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—61no—
235C2×C3×C4—92no—
236C2×C3×C4—92no—
237C2×C3×C4—91no—
238C2×C3×C4—91no—
239C2×C3×C4—91no—
240C2×C3×C4—91no—
241C2×C3×C4—91no—
242C2×C3×C4C2×D3211yescharacteristic
243C2×Dic3—31no—
244C2×Dic3—31no—
245C2×Dic3—31no—
246C2×Dic3—31no—
247C2×Dic3—31no—
248C2×Dic3—31no—
249C2×Dic3—31no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—91no—
253C2×Dic3—91no—
254C2×Dic3—91no—
255C4×D3—32no—
256C4×D3—32no—
257C4×D3—92no—
258C4×D3—92no—
259C3×C8—21no—
260C3×C8—61no—
261C3×C8—61no—
262C3×C8—121no—
263C3×C8—181no—
264C3×C8—181no—
265C3×C8—181no—
266C3⋊C8—61no—
267C3⋊C8—61no—
268C3⋊C8—61no—
269C3⋊C8—61no—
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×C32#90711yescharacteristic
280C2×C32#90711yescharacteristic
281C2×C32C2×Dic3⋊C411yescharacteristic
282C3⋊D3—22no—
283C3×D3—62no—
284C22×C4—91no—
285M16—182no—
286C2×C8—91no—
287C2×C8—91no—
288C2×C8—91no—
289C2×C8—542no—
290C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
291C4⋊C4—271no—
292C4⋊C4—271no—
293C22⋊C4—271no—
294C22⋊C4—271no—
295C42—91no—
296C42—91no—
297C22×C3—31no—
298C22×C3—31no—
299C22×C3—61no—
300C22×C3—92no—
301C22×C3—181no—
302C22×C3—181no—
303C22×C3C4×D3211yescharacteristic
304C2×D3—32no—
305C2×D3—61no—
306C2×D3—61no—
307C2×D3—92no—
308C2×D3—181no—
309C2×D3—181no—
310C3×C4—31no—
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—61no—
315C3×C4—61no—
316C3×C4—92no—
317C3×C4—181no—
318C3×C4—181no—
319C3×C4—181no—
320C3×C4—181no—
321C3×C4D3×Dic611yescharacteristic
322C3×C4D3×#6711yescharacteristic
323Dic3—32no—
324Dic3—61no—
325Dic3—61no—
326Dic3—61no—
327Dic3—61no—
328Dic3—62no—
329Dic3—61no—
330Dic3—61no—
331Dic3—92no—
332Dic3—121no—
333Dic3—181no—
334Dic3—181no—
335C32—21no—
336C32#483911yescharacteristic, complemented
337C32#511911yescharacteristic, complemented
338C23—91no—
339C2×C4—91no—
340C2×C4—92no—
341C2×C4—92no—
342C2×C4—91no—
343C2×C4—91no—
344C2×C4—91no—
345C2×C4—91no—
346C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
347C8—21no—
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#6380411yescharacteristic
362C2×C3#6380511yescharacteristic
363C2×C3D3×Dic3⋊C411yescharacteristic
364D3—62no—
365D3—182no—
366C22—92no—
367C22—181no—
368C22—181no—
369C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
370C4—92no—
371C4—181no—
372C4—181no—
373C4—181no—
374C4—181no—
375C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
376C4#8867611yescharacteristic
377C3—31no—
378C3—31no—
379C3—61no—
380C3#1059069511yescharacteristic
381C2—182no—
382C2#1172084111yescharacteristic
383C2#1172084211yescharacteristic
384C2#1172111911yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#424
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. #11720547C2semidirect
5. #63641⋊C3C2semidirect
6. #11720311C2semidirect
7. #11720304C2semidirect
29. #63641D3semidirect
228. H(3,𝔽3)#424semidirect
336. C32#4839semidirect
337. C32#5119semidirect

Maximal subgroups

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

subgroupindex
C2×#3743⋊C32
#117211052
C2×((C3⋊D3)⋊C3)⋊C82
#117205472
#63641⋊C32
#117203112
#117203042
#105907023
#105906953