Finite Group Databasebeta

← #13138836

Subgroups — #13138836

Order 1728.

406 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 73 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#11721129C211yesmaximal, characteristic
2#11721107C211yesmaximal, characteristic
3#11720549C211yesmaximal, characteristic, complemented
4#63647⋊C3C211yesmaximal, characteristic
5#63645⋊C3C211yesmaximal, characteristic, complemented
6#11720326C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10590689—31nomaximal
9#10590682—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×((C3⋊D3)⋊C3)⋊C4—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18#88450—21no—
19#88448—21no—
20#88448—21no—
21#63900—31no—
22D3×C4⋊Dic3—31no—
23#63871—31no—
24#63856—31no—
25#63647D311yescharacteristic
26#63645D311yescharacteristic, complemented
27C3×(C2×D3)⋊C8—31no—
28C3×#818—31no—
29C3×#816—31no—
30C3×#816—31no—
31C3×#816—61no—
32#63576—31no—
33#63575—31no—
34#63566—31no—
35#63565—31no—
36C22×(C3⋊D3)⋊C3D411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C2×#1084—21no—
40#6229—21no—
41#6229—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C4×(C3⋊D3)⋊C3—21no—
46C4×(C3⋊D3)⋊C3—21no—
47((C3⋊D3)⋊C3)⋊C4—42no—
48C8×H(3,𝔽3)—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50#4884—91no—
51#4811—91no—
52#4722—91no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×C3×C4×D3—31no—
55C2×C3×Dic6—31no—
56C2×C3×Dic6—31no—
57C2×C3×Dic6—61no—
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
65C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
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—
72#3724—61no—
73#3724—61no—
74Dic3⋊Dic3—61no—
75Dic3⋊Dic3—61no—
76#3722—61no—
77(C2×D3)⋊Dic3—61no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×(C3⋊D3)⋊C3—21no—
80C2×(C3⋊D3)⋊C3—21no—
81C2×(C3⋊D3)⋊C3Q1612yes—
82C4×H(3,𝔽3)#4111yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84#1084—41no—
85#1084—41no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3—41no—
89#946—91no—
90C2×C4⋊Dic3—91no—
91#924—91no—
92#897—91no—
93D3×C4⋊C4—91no—
94D3×C4⋊C4—91no—
95C3×#107—91no—
96C3×#103—91no—
97C3×C22⋊C8—91no—
98#835—91no—
99(C2×D3)⋊C8—31no—
100(C2×D3)⋊C8—91no—
101#818—31no—
102#818—91no—
103#818—91no—
104#816—31no—
105#816—31no—
106#816—61no—
107#816—91no—
108#810—91no—
109#807—91no—
110#807—91no—
111C22×C3⋊D3D1211yescharacteristic
112C22×C3×D3—31no—
113C2×C32×C4—21no—
114C2×C32×C4C22×D311yescharacteristic
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3D1211yescharacteristic
119C4×C3⋊D3—21no—
120C4×C3⋊D3—21no—
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×C4×D3—61no—
131C3×C4×D3—61no—
132C3×Dic6—62no—
133C3×Dic6—61no—
134C3×Dic6—61no—
135C3×Dic6—61no—
136C3×Dic6—61no—
137C3×Dic6—121no—
138#624—122no—
139D3×Dic3—122no—
140C32×C8—21no—
141C32×C8—21no—
142C32×C8—41no—
143C3⋊(C3⋊C8)—21no—
144C3×C3⋊C8—61no—
145#484—271noSylow
146C2×H(3,𝔽3)#12311yescharacteristic
147C2×H(3,𝔽3)C2×Q1611yescharacteristic
148C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
149(C3⋊D3)⋊C3—22no—
150C2×C3×Q8—91no—
151C22×C3×C4—91no—
152C22×Dic3—181no—
153C2×C4×D3—31no—
154C2×C4×D3—91no—
155C2×C4×D3—181no—
156C2×C4×D3—181no—
157C2×Dic6—31no—
158C2×Dic6—31no—
159C2×Dic6—61no—
160C2×C3×C8—31no—
161C2×C3×C8—31no—
162C2×C3×C8—61no—
163C2×C3×C8—91no—
164C2×C3×C8D3211yescharacteristic
165C3×C4⋊C4—91no—
166C3×C4⋊C4—91no—
167C3×C4⋊C4—91no—
168#217—181no—
169(C2×D3)⋊C4—181no—
170(C2×D3)⋊C4—181no—
171C4⋊Dic3—31no—
172C4⋊Dic3—31no—
173C4⋊Dic3—31no—
174C4⋊Dic3—31no—
175C4⋊Dic3—61no—
176C4⋊Dic3—91no—
177C4⋊Dic3—91no—
178C4⋊Dic3—181no—
179C4⋊Dic3—181no—
180Dic3⋊C4—181no—
181Dic3⋊C4—181no—
182Dic3⋊C4—181no—
183Dic3⋊C4—181no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—91no—
186C22×C32—21no—
187C22×C32C2×D1211yescharacteristic
188C22×C32D3×D411yescharacteristic
189C2×C3⋊D3—21no—
190C2×C3⋊D3—21no—
191C2×C3⋊D3Dic1212yes—
192C2×C3×D3—32no—
193C2×C3×D3—61no—
194C2×C3×D3—61no—
195C32×C4—21no—
196C32×C4—21no—
197C32×C4#23511yescharacteristic
198C32×C4#23711yescharacteristic
199C32×C4#23711yescharacteristic
200C32×C4#23911yescharacteristic
201C3⋊Dic3—21no—
202C3⋊Dic3—41no—
203C3⋊Dic3—41no—
204C3×Dic3—61no—
205C3×Dic3—122no—
206C3×Dic3—121no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209C3×Dic3—121no—
210C3×Dic3—121no—
211C3×Dic3—121no—
212C3×Dic3—241no—
213C22⋊Q8—271no—
214C2×C4⋊C4—271no—
215#107—91no—
216#107—271no—
217#103—91no—
218#103—271no—
219C22⋊C8—91no—
220H(3,𝔽3)#48411yescharacteristic, complemented, Sylow
221C23×C3—91no—
222C22×D3—31no—
223C22×D3—91no—
224C3×Q8—181no—
225C3×Q8—181no—
226C2×C3×C4—31no—
227C2×C3×C4—31no—
228C2×C3×C4—61no—
229C2×C3×C4—91no—
230C2×C3×C4—181no—
231C2×C3×C4—181no—
232C2×C3×C4—181no—
233C2×C3×C4—181no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4C2×D3211yescharacteristic
237C2×Dic3—31no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—91no—
245C2×Dic3—121no—
246C2×Dic3—121no—
247C2×Dic3—181no—
248C2×Dic3—181no—
249C2×Dic3—362no—
250C4×D3—61no—
251C4×D3—61no—
252C4×D3—181no—
253C4×D3—181no—
254C4×D3—362no—
255C4×D3—362no—
256Dic6—61no—
257Dic6—61no—
258Dic6—61no—
259Dic6—61no—
260Dic6—62no—
261Dic6—121no—
262C3×C8—21no—
263C3×C8—61no—
264C3×C8—61no—
265C3×C8—121no—
266C3×C8—181no—
267C3⋊C8—61no—
268C3⋊C8—181no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32#88611yescharacteristic
273C2×C32#89411yescharacteristic
274C2×C32C2×Dic1211yescharacteristic
275C2×C32#90811yescharacteristic
276C2×C32#91111yescharacteristic
277C2×C32D3×Q1611yescharacteristic
278C3⋊D3—22no—
279C3×D3—62no—
280C2×Q8—91no—
281C22×C4—91no—
282C22×C4—541no—
283C2×C8—91no—
284C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
285C4⋊C4—91no—
286C4⋊C4—91no—
287C4⋊C4—91no—
288C4⋊C4—541no—
289C4⋊C4—541no—
290C22⋊C4—541no—
291C22×C3—31no—
292C22×C3—31no—
293C22×C3—61no—
294C22×C3—92no—
295C22×C3—181no—
296C22×C3—181no—
297C22×C3D3×D1211yescharacteristic
298C2×D3—32no—
299C2×D3—61no—
300C2×D3—61no—
301C2×D3—92no—
302C2×D3—181no—
303C2×D3—181no—
304C3×C4—31no—
305C3×C4—31no—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—61no—
309C3×C4—61no—
310C3×C4—181no—
311C3×C4—361no—
312C3×C4—361no—
313C3×C4—361no—
314C3×C4—361no—
315C3×C4#379611yescharacteristic
316C3×C4#380411yescharacteristic
317Dic3—61no—
318Dic3—121no—
319Dic3—121no—
320Dic3—121no—
321Dic3—121no—
322Dic3—122no—
323Dic3—121no—
324Dic3—121no—
325Dic3—181no—
326Dic3—241no—
327Dic3—361no—
328Dic3—361no—
329C32—21no—
330C32#472211yescharacteristic, complemented
331C32#488411yescharacteristic, complemented
332C23—91no—
333Q8—181no—
334Q8—181no—
335C2×C4—91no—
336C2×C4—181no—
337C2×C4—181no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—181no—
341C2×C4—181no—
342C2×C4—1082no—
343C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
344C8—21no—
345C8—181no—
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—61no—
353C2×C3—61no—
354C2×C3—61no—
355C2×C3—182no—
356C2×C3#6379511yescharacteristic
357C2×C3D3×Dic1211yescharacteristic
358C2×C3#6389011yescharacteristic
359D3—62no—
360D3—182no—
361C22—92no—
362C22—181no—
363C22—181no—
364C22#8865411yescharacteristic
365C4—181no—
366C4—361no—
367C4—361no—
368C4—361no—
369C4—361no—
370C4#8864911yescharacteristic
371C4#8867111yescharacteristic
372C3—31no—
373C3—31no—
374C3—61no—
375C3#1059068411yescharacteristic
376C2—182no—
377C2#1172084911yescharacteristic
378C2#1172085111yescharacteristic
379C2#1172113011yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC22×(C3⋊D3)⋊C3D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#484
LCSG ▷ C2×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. #11720549C2semidirect
5. #63645⋊C3C2semidirect
6. #11720326C2semidirect
7. #11720306C2semidirect
26. #63645D3semidirect
220. H(3,𝔽3)#484semidirect
330. C32#4722semidirect
331. C32#4884semidirect

Maximal subgroups

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

subgroupindex
#117211292
#117211072
#117205492
#63647⋊C32
#63645⋊C32
#117203262
#117203062
#105906893
#105906823