Finite Group Databasebeta

← #13138994

Subgroups — #13138994

Order 1728.

427 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 87 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#11721083C211yesmaximal, characteristic
2#11721082C211yesmaximal, characteristic
3C2×#88439C211yesmaximal, characteristic
4#11720549C211yesmaximal, characteristic
5#63643⋊C3C211yesmaximal, characteristic
6C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
7#11720306C211yesmaximal, characteristic
8#10590793—31nomaximal
9#10590790—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17#88449—21no—
18#88448—21no—
19#88447—21no—
20#88447—21no—
21#88439C414yescomplemented
22#63844—31no—
23#63843—31no—
24#63843—31no—
25Dic3×Dic6—31no—
26C2×#3719—31no—
27C2×#3718—31no—
28#63643D311yescharacteristic
29C8×C3⋊Dic3D311yescharacteristic
30C3×#816—31no—
31C3×#816—31no—
32C3×#816—31no—
33C3×#816—61no—
34C3×C8×Dic3—31no—
35#63576—31no—
36#63575—31no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C2×#1084—21no—
40#6229—21no—
41#6229C2×C412yes—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46#634⋊C3—21no—
47#634⋊C3C2×C412yes—
48C8×H(3,𝔽3)—21no—
49(C3⋊(C3⋊C8))⋊C3C2×C412yes—
50#4790—91no—
51#4790—91no—
52C4×Dic12—91no—
53C2×#634C2×D311yescharacteristic
54C2×C3×Dic6—31no—
55C2×C3×Dic6—31no—
56C2×C3×Dic6—31no—
57C2×C3×Dic6—61no—
58C2×C32×C8—21no—
59C2×C32×C8C2×D311yescharacteristic
60C2×C32×C8C2×D311yescharacteristic
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62C4×C3⋊Dic3C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—61no—
68C3×C4×Dic3—31no—
69C2×C3×C3⋊C8—31no—
70#3724—61no—
71Dic3⋊Dic3—61no—
72Dic3⋊Dic3—61no—
73Dic3⋊Dic3—61no—
74Dic32—61no—
75Dic32—61no—
76Dic32—61no—
77Dic32—61no—
78#3719—34no—
79#3718—34no—
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C4×H(3,𝔽3)C22×C411yescharacteristic
82C4×H(3,𝔽3)#4111yescharacteristic
83#1084—22no—
84#1084—41no—
85(C3⋊Dic3)⋊C3—22no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3Q1612yescomplemented
89Q8×Dic3—91no—
90Q8×Dic3—91no—
91C2×#216—91no—
92C2×#216—91no—
93C2×Dic12—91no—
94#887—91no—
95#887—91no—
96C4×Dic6—91no—
97C4×Dic6—91no—
98C3×#103—91no—
99C3×#103—91no—
100C3×C4×C8—91no—
101#818—91no—
102#816—31no—
103#816—31no—
104#816—31no—
105#816—61no—
106#816—91no—
107C8×Dic3—31no—
108C8×Dic3—91no—
109#807—91no—
110#807—91no—
111C2×C32×C4—21no—
112C2×C32×C4C22×D311yescharacteristic
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3D1211yescharacteristic
117C2×C3⋊Dic3D1211yescharacteristic
118#634—21no—
119#634C4×D312yes—
120C2×C3×Dic3—31no—
121C2×C3×Dic3—31no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—121no—
129C2×C3×Dic3—121no—
130C3×Dic6—32no—
131C3×Dic6—32no—
132C3×Dic6—32no—
133C3×Dic6—62no—
134C3×Dic6—61no—
135C3×Dic6—61no—
136C3×Dic6—61no—
137C3×Dic6—121no—
138C32×C8—21no—
139C32×C8—21no—
140C32×C8—41no—
141C3⋊(C3⋊C8)C4×D312yes—
142C3×C3⋊C8—32no—
143C4×Q16—271noSylow
144C2×H(3,𝔽3)C4×D411yescharacteristic
145C2×H(3,𝔽3)C2×Q1611yescharacteristic
146C2×H(3,𝔽3)#13511yescharacteristic
147C2×C3×Q8—91no—
148C2×C3×Q8—91no—
149C2×Dic6—31no—
150C2×Dic6—31no—
151C2×Dic6—31no—
152C2×Dic6—61no—
153C2×Dic6—91no—
154C2×C3×C8—31no—
155C2×C3×C8—31no—
156C2×C3×C8—61no—
157C2×C3×C8—91no—
158C2×C3×C8D3211yescharacteristic
159C3×C4⋊C4—91no—
160C3×C4⋊C4—91no—
161C3×C42—91no—
162#216—94no—
163#216—94no—
164C4⋊Dic3—31no—
165C4⋊Dic3—31no—
166C4⋊Dic3—31no—
167C4⋊Dic3—61no—
168C4⋊Dic3—91no—
169C4⋊Dic3—181no—
170C4⋊Dic3—181no—
171Dic3⋊C4—181no—
172Dic3⋊C4—181no—
173Dic3⋊C4—181no—
174Dic3⋊C4—181no—
175C4×Dic3—31no—
176C4×Dic3—91no—
177C4×Dic3—181no—
178C4×Dic3—181no—
179C4×Dic3—181no—
180C4×Dic3—181no—
181C4×Dic3—181no—
182C4×Dic3—181no—
183C2×C3⋊C8—31no—
184C2×C3⋊C8—91no—
185Dic12—94no—
186C22×C32—21no—
187C22×C32C2×D1211yescharacteristic
188C22×C32D3×D411yescharacteristic
189C32×C4—21no—
190C32×C4—21no—
191C32×C4C2×C4×D311yescharacteristic
192C32×C4C2×C4×D311yescharacteristic
193C32×C4#23511yescharacteristic
194C32×C4#23711yescharacteristic
195C3⋊Dic3—22no—
196C3⋊Dic3—21no—
197C3⋊Dic3—41no—
198C3⋊Dic3Dic1212yescomplemented
199C3×Dic3—32no—
200C3×Dic3—62no—
201C3×Dic3—62no—
202C3×Dic3—62no—
203C3×Dic3—61no—
204C3×Dic3—122no—
205C3×Dic3—121no—
206C3×Dic3—121no—
207C3×Dic3—121no—
208C3×Dic3—241no—
209C2×Q16—271no—
210C4×Q8—271no—
211C4×Q8—271no—
212#107—271no—
213#103—91no—
214#103—91no—
215C4×C8—91no—
216H(3,𝔽3)C4×Q1611yescharacteristic, complemented, Sylow
217C3×Q8—92no—
218C3×Q8—92no—
219C3×Q8—181no—
220C3×Q8—181no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—61no—
224C2×C3×C4—91no—
225C2×C3×C4—91no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4C2×D3211yescharacteristic
231C2×Dic3—31no—
232C2×Dic3—31no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—91no—
240C2×Dic3—91no—
241C2×Dic3—121no—
242C2×Dic3—121no—
243C2×Dic3—181no—
244C2×Dic3—181no—
245Dic6—32no—
246Dic6—32no—
247Dic6—32no—
248Dic6—62no—
249Dic6—61no—
250Dic6—61no—
251Dic6—61no—
252Dic6—92no—
253Dic6—121no—
254Dic6—181no—
255C3×C8—21no—
256C3×C8—61no—
257C3×C8—61no—
258C3×C8—92no—
259C3×C8—121no—
260C3⋊C8—32no—
261C3⋊C8—92no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32C4×D1211yescharacteristic
266C2×C32#67⋊C411yescharacteristic
267C2×C32#90411yescharacteristic
268C2×C32C2×Dic1211yescharacteristic
269C2×C32#91611yescharacteristic
270C2×C32D3×Q1611yescharacteristic
271C2×Q8—91no—
272C2×Q8—91no—
273Q16—274no—
274C2×C8—91no—
275C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
276C4⋊C4—91no—
277C4⋊C4—91no—
278C4⋊C4—541no—
279C4⋊C4—541no—
280C42—91no—
281C42—541no—
282C42—541no—
283C22×C3—31no—
284C22×C3—31no—
285C22×C3—61no—
286C22×C3D3×D1211yescharacteristic
287C3×C4—31no—
288C3×C4—31no—
289C3×C4—31no—
290C3×C4—31no—
291C3×C4—61no—
292C3×C4—61no—
293C3×C4—92no—
294C3×C4—181no—
295C3×C4—182no—
296C3×C4—182no—
297C3×C4—361no—
298C3×C4—361no—
299C3×C4C4×D3211yescharacteristic
300C3×C4#380411yescharacteristic
301Dic3—32no—
302Dic3—62no—
303Dic3—62no—
304Dic3—61no—
305Dic3—62no—
306Dic3—92no—
307Dic3—122no—
308Dic3—121no—
309Dic3—121no—
310Dic3—121no—
311Dic3—182no—
312Dic3—181no—
313Dic3—241no—
314Dic3—361no—
315C32—21no—
316C32C4×Dic1211yescharacteristic, complemented
317C32#479011yescharacteristic, complemented
318Q8—92no—
319Q8—92no—
320Q8—181no—
321Q8—181no—
322C2×C4—91no—
323C2×C4—91no—
324C2×C4—181no—
325C2×C4—181no—
326C2×C4—181no—
327C2×C4—181no—
328C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
329C8—21no—
330C8—92no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—31no—
334C2×C3—31no—
335C2×C3—31no—
336C2×C3—31no—
337C2×C3—61no—
338C2×C3—61no—
339C2×C3—61no—
340C2×C3D3×Dic1211yescharacteristic
341C2×C3#6380611yescharacteristic
342C2×C3#6388011yescharacteristic
343C22#8865411yescharacteristic
344C4—92no—
345C4—181no—
346C4—182no—
347C4—182no—
348C4—361no—
349C4—361no—
350C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
351C4#8867111yescharacteristic
352C3—31no—
353C3—31no—
354C3—61no—
355C3#1059079311yescharacteristic
356C2#1172085111yescharacteristic
357C2#1172085411yescharacteristic
358C2#1172112311yescharacteristic
subgroupG/H
centerC22#88654
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC22×H(3,𝔽3)C2×D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×Q16
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
21. #88439C4semidirect
88. (C3⋊Dic3)⋊C3Q16semidirect
198. C3⋊Dic3Dic12semidirect
216. H(3,𝔽3)C4×Q16semidirect
316. C32C4×Dic12semidirect
317. C32#4790semidirect

Maximal subgroups

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

subgroupindex
#117210832
#117210822
C2×#884392
#117205492
#63643⋊C32
C8×(C3⋊Dic3)⋊C32
#117203062
#105907933
#105907903