Finite Group Databasebeta

← #13138786

Subgroups — #13138786

Order 1728.

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

normal subgroups only

This group has 471 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×#3744⋊C3C211yesmaximal, characteristic
2#11721136C211yesmaximal, characteristic
3#11721084C211yesmaximal, characteristic, complemented
4C8×#1084C211yesmaximal, characteristic, complemented
5#63644⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic
7#11720313C211yesmaximal, characteristic, complemented
8#10590758—31nomaximal
9Dic3×#204—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×((C3⋊D3)⋊C3)⋊C4—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4×#1084C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3744⋊C3C414yescomplemented
19#88450—21no—
20#88448—21no—
21#88447—21no—
22C2×#3744D311yescharacteristic
23C2×C3×#204—31no—
24#63894—31no—
25Dic3×D12—31no—
26#63844—31no—
27Dic3×Dic6—31no—
28#63644D311yescharacteristic, complemented
29C3×#817—31no—
30C3×C8×Dic3—31no—
31C3×C8×Dic3—31no—
32C3×C8×Dic3—61no—
33#63567—31no—
34#63566—31no—
35#63564—31no—
36#63563—31no—
37C22×(C3⋊D3)⋊C3—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084D411yescharacteristic
40C2×#1084D411yescharacteristic
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C4⋊((C3⋊D3)⋊C3)—21no—
45C4⋊((C3⋊D3)⋊C3)C2×C412yes—
46#634⋊C3—21no—
47#634⋊C3C2×C412yescomplemented
48((C3⋊D3)⋊C3)⋊C4—42no—
49C8×H(3,𝔽3)C2×C412yescomplemented
50#6180—21no—
51Dic3×SD16—91no—
52#4852—91no—
53C4×#204—91no—
54C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
55C2×#634C2×D311yescharacteristic
56C2×C3×D12—31no—
57C2×C3×Dic6—31no—
58C2×#624—61no—
59C2×D3×Dic3—61no—
60C2×C32×C8—21no—
61C2×C32×C8C2×D311yescharacteristic
62C2×C32×C8C2×D311yescharacteristic
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65#3744Dic314yescomplemented
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4×Dic3—31no—
69C3×C4×Dic3—31no—
70C3×C4×Dic3—61no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—61no—
74C3×#204—34no—
75#3724—61no—
76Dic3⋊Dic3—61no—
77#3722—61no—
78(C2×D3)⋊Dic3—61no—
79Dic32—61no—
80Dic32—61no—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C2×(C3⋊D3)⋊C3—22no—
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C4×H(3,𝔽3)C22×C411yescharacteristic
86C4×H(3,𝔽3)#4111yescharacteristic
87#1084—21no—
88#1084SD1612yescomplemented
89(C3⋊Dic3)⋊C3—22no—
90(C3⋊Dic3)⋊C3—41no—
91(C3⋊Dic3)⋊C3—41no—
92C2×C3×SD16—91no—
93Q8×Dic3—91no—
94D4×Dic3—91no—
95C2×#204—31no—
96C2×#204—91no—
97#893—91no—
98#887—91no—
99C4×D12—91no—
100C4×Dic6—91no—
101C3×#106—91no—
102C3×C4×C8—91no—
103#835—91no—
104#832—91no—
105#821—91no—
106#817—31no—
107#817—91no—
108#816—91no—
109C8×Dic3—31no—
110C8×Dic3—31no—
111C8×Dic3—61no—
112#810—91no—
113#809—91no—
114C22×C3⋊D3—21no—
115C22×C3×D3—61no—
116C2×C32×C4—21no—
117C2×C32×C4C22×D311yescharacteristic
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—21no—
122C4⋊(C3⋊D3)—21no—
123C4⋊(C3⋊D3)C2×Dic312yes—
124#634—21no—
125#634C2×Dic312yescomplemented
126C2×C3×Dic3—31no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—31no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C3×D12—32no—
136C3×D12—61no—
137C3×Dic6—32no—
138C3×Dic6—61no—
139#624—122no—
140D3×Dic3—122no—
141C32×C8—22no—
142C32×C8C4×D312yescomplemented
143C32×C8C2×Dic312yescomplemented
144C3×C3⋊C8—62no—
145C3×C3⋊C8—61no—
146C3×C3⋊C8—61no—
147C4×SD16—271noSylow
148C2×H(3,𝔽3)C4×D411yescharacteristic
149C2×H(3,𝔽3)C2×SD1611yescharacteristic
150C2×H(3,𝔽3)#13511yescharacteristic
151(C3⋊D3)⋊C3—42no—
152C2×C3×Q8—91no—
153C2×C3×D4—91no—
154C22×Dic3—181no—
155C2×D12—31no—
156C2×D12—91no—
157C2×C4×D3—181no—
158C2×C4×D3—181no—
159C2×Dic6—31no—
160C2×Dic6—91no—
161C3×SD16—94no—
162C2×C3×C8—31no—
163C2×C3×C8—31no—
164C2×C3×C8—61no—
165C2×C3×C8—91no—
166C2×C3×C8D3211yescharacteristic
167C3×C4⋊C4—91no—
168C3×C4⋊C4—91no—
169C3×C42—91no—
170#217—181no—
171(C2×D3)⋊C4—181no—
172(C2×D3)⋊C4—181no—
173C4⋊Dic3—31no—
174C4⋊Dic3—31no—
175C4⋊Dic3—91no—
176C4⋊Dic3—91no—
177C4⋊Dic3—181no—
178Dic3⋊C4—181no—
179Dic3⋊C4—181no—
180C4×Dic3—31no—
181C4×Dic3—31no—
182C4×Dic3—61no—
183C4×Dic3—181no—
184C4×Dic3—181no—
185C4×Dic3—181no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—61no—
189#204—34no—
190#204—94no—
191C22×C32—21no—
192C22×C32D3×D411yescharacteristic
193C22×C32D3×D411yescharacteristic
194C2×C3⋊D3—22no—
195C2×C3⋊D3—41no—
196C2×C3⋊D3—41no—
197C2×C3×D3—62no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C32×C4—21no—
201C32×C4—21no—
202C32×C4C2×C4×D311yescharacteristic
203C32×C4#23711yescharacteristic
204C32×C4#23911yescharacteristic
205C32×C4C22×Dic311yescharacteristic
206C3⋊Dic3—22no—
207C3⋊Dic3—41no—
208C3⋊Dic3—41no—
209C3×Dic3—32no—
210C3×Dic3—32no—
211C3×Dic3—62no—
212C3×Dic3—62no—
213C3×Dic3—62no—
214C3×Dic3—61no—
215C3×Dic3—61no—
216C3×Dic3—121no—
217C3×Dic3—121no—
218C2×SD16—91no—
219C4×Q8—271no—
220C4×D4—271no—
221#106—91no—
222#103—271no—
223#102—271no—
224C4×C8—91no—
225H(3,𝔽3)C4×SD1611yescharacteristic, complemented, Sylow
226C23×C3—181no—
227C22×D3—61no—
228C22×D3—181no—
229C3×Q8—92no—
230C3×Q8—181no—
231C3×D4—92no—
232C3×D4—181no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—61no—
236C2×C3×C4—91no—
237C2×C3×C4—91no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4—181no—
241C2×C3×C4C2×D3211yescharacteristic
242C2×Dic3—31no—
243C2×Dic3—31no—
244C2×Dic3—31no—
245C2×Dic3—31no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—181no—
252C2×Dic3—181no—
253C2×Dic3—181no—
254C2×Dic3—362no—
255D12—32no—
256D12—61no—
257D12—92no—
258D12—181no—
259C4×D3—362no—
260C4×D3—362no—
261Dic6—32no—
262Dic6—61no—
263Dic6—92no—
264Dic6—181no—
265C3×C8—32no—
266C3×C8—32no—
267C3×C8—62no—
268C3×C8—181no—
269C3×C8D3×Dic312yes—
270C3⋊C8—61no—
271C3⋊C8—61no—
272C3⋊C8—62no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32#89311yescharacteristic
277C2×C32D3×SD1611yescharacteristic
278C2×C32D3×SD1611yescharacteristic
279C2×C32#91611yescharacteristic
280C2×C32#91611yescharacteristic
281C2×C32D4×Dic311yescharacteristic
282C3⋊D3—42no—
283C3×D3—122no—
284C2×Q8—91no—
285C2×D4—91no—
286C22×C4—541no—
287SD16—94no—
288C2×C8—91no—
289C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
290C4⋊C4—91no—
291C4⋊C4—91no—
292C4⋊C4—541no—
293C22⋊C4—541no—
294C42—91no—
295C42—541no—
296C22×C3—31no—
297C22×C3—31no—
298C22×C3—61no—
299C22×C3—182no—
300C22×C3—361no—
301C22×C3—361no—
302C22×C3#380211yescharacteristic
303C2×D3—62no—
304C2×D3—121no—
305C2×D3—121no—
306C2×D3—182no—
307C2×D3—361no—
308C2×D3—361no—
309C3×C4—31no—
310C3×C4—31no—
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—61no—
314C3×C4—61no—
315C3×C4—92no—
316C3×C4—182no—
317C3×C4—181no—
318C3×C4—361no—
319C3×C4—361no—
320C3×C4#379611yescharacteristic
321C3×C4C2×D3×Dic311yescharacteristic
322Dic3—32no—
323Dic3—32no—
324Dic3—62no—
325Dic3—61no—
326Dic3—61no—
327Dic3—62no—
328Dic3—62no—
329Dic3—121no—
330Dic3—121no—
331Dic3—182no—
332Dic3—361no—
333Dic3—361no—
334C32—21no—
335C32#485211yescharacteristic, complemented
336C32Dic3×SD1611yescharacteristic, complemented
337C23—181no—
338Q8—92no—
339Q8—181no—
340D4—92no—
341D4—181no—
342C2×C4—91no—
343C2×C4—91no—
344C2×C4—181no—
345C2×C4—181no—
346C2×C4—181no—
347C2×C4—1082no—
348C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
349C8—181no—
350C8((C3⋊D3)⋊C3)⋊C412yescomplemented
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—362no—
361C2×C3#6379611yescharacteristic
362C2×C3#6380911yescharacteristic
363C2×C3#6389811yescharacteristic
364D3—122no—
365D3—362no—
366C22—182no—
367C22—361no—
368C22—361no—
369C22#8865511yescharacteristic
370C4—92no—
371C4—181no—
372C4—182no—
373C4—361no—
374C4—361no—
375C4#8864911yescharacteristic
376C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
377C3—31no—
378C3—31no—
379C3—61no—
380C3#1059077511yescharacteristic
381C2—362no—
382C2#1172083711yescharacteristic
383C2#1172084311yescharacteristic
384C2#1172113611yescharacteristic
subgroupG/H
centerC22#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×SD16
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. #11721084C2semidirect
4. C8×#1084C2semidirect
5. #63644⋊C3C2semidirect
7. #11720313C2semidirect
18. #3744⋊C3C4semidirect
28. #63644D3semidirect
47. #634⋊C3C2×C4semidirect
49. C8×H(3,𝔽3)C2×C4semidirect
65. #3744Dic3semidirect
88. #1084SD16semidirect
125. #634C2×Dic3semidirect
142. C32×C8C4×D3semidirect
143. C32×C8C2×Dic3semidirect
225. H(3,𝔽3)C4×SD16semidirect
335. C32#4852semidirect
336. C32Dic3×SD16semidirect
350. C8((C3⋊D3)⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3744⋊C32
#117211362
#117210842
C8×#10842
#63644⋊C32
#117203182
#117203132
#105907583
Dic3×#2043