Finite Group Databasebeta

← #13138848

Subgroups — #13138848

Order 1728.

463 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 99 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#11721132C211yesmaximal, characteristic
3#11721082C211yesmaximal, characteristic, complemented
4#11720550C211yesmaximal, characteristic, complemented
5C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
6#11720309C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic
8#10590775—31nomaximal
9#10590758—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×((C3⋊D3)⋊C3)⋊C4—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18#3744⋊C3C414yescomplemented
19#88450—21no—
20#88448—21no—
21#88447—21no—
22C2×#3744D311yescharacteristic
23C2×C3×#204—31no—
24#63898—31no—
25#63894—31no—
26#63844—31no—
27#63843—31no—
28C8×C3⋊Dic3D311yescharacteristic, complemented
29C3×#817—31no—
30C3×#817—31no—
31C3×#817—61no—
32C3×C8×Dic3—31no—
33#63567—31no—
34#63565—31no—
35#63564—31no—
36#63562—31no—
37C22×(C3⋊D3)⋊C3—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C2×#1084—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
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(C3⋊(C3⋊C8))⋊C3—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⋊Dic3C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
65#3744Dic314yescomplemented
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—61no—
71C3×C4⋊Dic3—61no—
72C3×C4×Dic3—31no—
73C2×C3×C3⋊C8—31no—
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—41no—
88#1084—41no—
89(C3⋊Dic3)⋊C3—22no—
90(C3⋊Dic3)⋊C3—21no—
91(C3⋊Dic3)⋊C3SD1612yescomplemented
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—31no—
108#817—61no—
109#816—91no—
110C8×Dic3—31no—
111C8×Dic3—91no—
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⋊Dic3D1211yescharacteristic
121C2×C3⋊Dic3D1211yescharacteristic
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—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—121no—
134C2×C3×Dic3—121no—
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)—21no—
145C3×C3⋊C8—61no—
146C4×SD16—271noSylow
147C2×H(3,𝔽3)C4×D411yescharacteristic
148C2×H(3,𝔽3)C2×SD1611yescharacteristic
149C2×H(3,𝔽3)#13511yescharacteristic
150(C3⋊D3)⋊C3—42no—
151C2×C3×Q8—91no—
152C2×C3×D4—91no—
153C22×Dic3—181no—
154C2×D12—31no—
155C2×D12—91no—
156C2×C4×D3—181no—
157C2×C4×D3—181no—
158C2×Dic6—31no—
159C2×Dic6—91no—
160C3×SD16—94no—
161C2×C3×C8—31no—
162C2×C3×C8—31no—
163C2×C3×C8—61no—
164C2×C3×C8—91no—
165C2×C3×C8D3211yescharacteristic
166C3×C4⋊C4—91no—
167C3×C4⋊C4—91no—
168C3×C42—91no—
169#217—181no—
170(C2×D3)⋊C4—181no—
171(C2×D3)⋊C4—181no—
172C4⋊Dic3—31no—
173C4⋊Dic3—31no—
174C4⋊Dic3—31no—
175C4⋊Dic3—31no—
176C4⋊Dic3—61no—
177C4⋊Dic3—61no—
178C4⋊Dic3—181no—
179Dic3⋊C4—181no—
180Dic3⋊C4—181no—
181C4×Dic3—31no—
182C4×Dic3—91no—
183C4×Dic3—181no—
184C4×Dic3—181no—
185C4×Dic3—181no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—91no—
188#204—34no—
189#204—94no—
190C22×C32—21no—
191C22×C32C2×D1211yescharacteristic
192C22×C32D3×D411yescharacteristic
193C2×C3⋊D3—22no—
194C2×C3⋊D3—41no—
195C2×C3⋊D3—41no—
196C2×C3×D3—62no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C32×C4—21no—
200C32×C4—21no—
201C32×C4C2×C4×D311yescharacteristic
202C32×C4#23511yescharacteristic
203C32×C4#23711yescharacteristic
204C32×C4C22×Dic311yescharacteristic
205C3⋊Dic3—21no—
206C3⋊Dic3—22no—
207C3⋊Dic3#20412yescomplemented
208C3×Dic3—32no—
209C3×Dic3—62no—
210C3×Dic3—61no—
211C3×Dic3—121no—
212C3×Dic3—121no—
213C3×Dic3—121no—
214C3×Dic3—121no—
215C3×Dic3—241no—
216C3×Dic3—241no—
217C2×SD16—91no—
218C4×Q8—271no—
219C4×D4—271no—
220#106—91no—
221#103—271no—
222#102—271no—
223C4×C8—91no—
224H(3,𝔽3)C4×SD1611yescharacteristic, complemented, Sylow
225C23×C3—181no—
226C22×D3—61no—
227C22×D3—181no—
228C3×Q8—92no—
229C3×Q8—181no—
230C3×D4—92no—
231C3×D4—181no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—61no—
235C2×C3×C4—91no—
236C2×C3×C4—91no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4C2×D3211yescharacteristic
241C2×Dic3—31no—
242C2×Dic3—31no—
243C2×Dic3—61no—
244C2×Dic3—61no—
245C2×Dic3—61no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—91no—
249C2×Dic3—91no—
250C2×Dic3—121no—
251C2×Dic3—121no—
252C2×Dic3—181no—
253C2×Dic3—362no—
254D12—32no—
255D12—61no—
256D12—92no—
257D12—181no—
258C4×D3—362no—
259C4×D3—362no—
260Dic6—32no—
261Dic6—61no—
262Dic6—92no—
263Dic6—181no—
264C3×C8—32no—
265C3×C8—32no—
266C3×C8—62no—
267C3×C8—181no—
268C3×C8D3×Dic312yes—
269C3⋊C8—61no—
270C3⋊C8—181no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32C4×D1211yescharacteristic
275C2×C32C2×#20411yescharacteristic
276C2×C32#90411yescharacteristic
277C2×C32D3×SD1611yescharacteristic
278C2×C32#91611yescharacteristic
279C2×C32D4×Dic311yescharacteristic
280C3⋊D3—42no—
281C3×D3—122no—
282C2×Q8—91no—
283C2×D4—91no—
284C22×C4—541no—
285SD16—94no—
286C2×C8—91no—
287C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
288C4⋊C4—91no—
289C4⋊C4—91no—
290C4⋊C4—541no—
291C22⋊C4—541no—
292C42—91no—
293C42—541no—
294C22×C3—31no—
295C22×C3—31no—
296C22×C3—61no—
297C22×C3—182no—
298C22×C3—361no—
299C22×C3—361no—
300C22×C3D3×D1211yescharacteristic
301C2×D3—62no—
302C2×D3—121no—
303C2×D3—121no—
304C2×D3—182no—
305C2×D3—361no—
306C2×D3—361no—
307C3×C4—31no—
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—31no—
311C3×C4—61no—
312C3×C4—61no—
313C3×C4—92no—
314C3×C4—181no—
315C3×C4—182no—
316C3×C4—361no—
317C3×C4—361no—
318C3×C4#379511yescharacteristic
319C3×C4C2×D3×Dic311yescharacteristic
320Dic3—32no—
321Dic3—61no—
322Dic3—62no—
323Dic3—92no—
324Dic3—121no—
325Dic3—121no—
326Dic3—121no—
327Dic3—121no—
328Dic3—182no—
329Dic3—181no—
330Dic3—241no—
331Dic3—241no—
332C32—21no—
333C32C4×#20411yescharacteristic, complemented
334C32Dic3×SD1611yescharacteristic, complemented
335C23—181no—
336Q8—92no—
337Q8—181no—
338D4—92no—
339D4—181no—
340C2×C4—91no—
341C2×C4—91no—
342C2×C4—181no—
343C2×C4—181no—
344C2×C4—181no—
345C2×C4—1082no—
346C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
347C8—181no—
348C8((C3⋊D3)⋊C3)⋊C412yescomplemented
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—61no—
356C2×C3—61no—
357C2×C3—61no—
358C2×C3—362no—
359C2×C3D3×#20411yescharacteristic
360C2×C3#6380611yescharacteristic
361C2×C3Dic3×D1211yescharacteristic
362D3—122no—
363D3—362no—
364C22—182no—
365C22—361no—
366C22—361no—
367C22#8865411yescharacteristic
368C4—92no—
369C4—181no—
370C4—182no—
371C4—361no—
372C4—361no—
373C4#8864811yescharacteristic
374C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
375C3—31no—
376C3—31no—
377C3—61no—
378C3Dic3×#20411yescharacteristic
379C2—362no—
380C2#1172084611yescharacteristic
381C2#1172085411yescharacteristic
382C2#1172113211yescharacteristic
subgroupG/H
centerC22#88654
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
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. #11721082C2semidirect
4. #11720550C2semidirect
5. C8×(C3⋊Dic3)⋊C3C2semidirect
6. #11720309C2semidirect
18. #3744⋊C3C4semidirect
28. C8×C3⋊Dic3D3semidirect
47. #634⋊C3C2×C4semidirect
49. C8×H(3,𝔽3)C2×C4semidirect
65. #3744Dic3semidirect
91. (C3⋊Dic3)⋊C3SD16semidirect
125. #634C2×Dic3semidirect
142. C32×C8C4×D3semidirect
143. C32×C8C2×Dic3semidirect
207. C3⋊Dic3#204semidirect
224. H(3,𝔽3)C4×SD16semidirect
333. C32C4×#204semidirect
334. C32Dic3×SD16semidirect
348. 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
#117211322
#117210822
#117205502
C8×(C3⋊Dic3)⋊C32
#117203092
#117203082
#105907753
#105907583