Finite Group Databasebeta

← #13138911

Subgroups — #13138911

Order 1728.

386 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 79 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#11721097C211yesmaximal, characteristic
2#11720553C211yesmaximal, characteristic
3#63641⋊C3C211yesmaximal, characteristic, complemented
4C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
5#11720323C211yesmaximal, characteristic
6#11720312C211yesmaximal, characteristic, complemented
7#11720311C211yesmaximal, characteristic, complemented
8#10590698—31nomaximal
9#10590696—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C2×#6180C2211yescharacteristic
13C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17#88451C411yescharacteristic
18#88451C411yescharacteristic
19#88449C411yescharacteristic
20#88449C411yescharacteristic
21#63855—31no—
22#63855—31no—
23#63641D311yescharacteristic, complemented
24C8×C3⋊Dic3D311yescharacteristic, complemented
25C3×(C2×D3)⋊C8—31no—
26C3×(C2×D3)⋊C8—31no—
27C3×(C2×D3)⋊C8—61no—
28C3×Dic3⋊C8—31no—
29C3×C8×Dic3—31no—
30#63572—31no—
31Dic3⋊(C3⋊C8)—31no—
32(C2×D3)⋊(C3⋊C8)—31no—
33(C2×D3)⋊(C3⋊C8)—31no—
34#63555—31no—
35#63554—31no—
36C22×#272C2×C411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084C2×C411yescharacteristic
39C4×#272—21no—
40C4×#272—21no—
41C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
42C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
43C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
45C8×H(3,𝔽3)—21no—
46#6180—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49#4840—91no—
50#4720—91no—
51#4690—91no—
52C2×C3×C4×D3—31no—
53C2×C3×C4×D3—31no—
54C2×C3×C4×D3—61no—
55C2×C32×C8—21no—
56C2×C32×C8C2×D311yescharacteristic
57C2×C32×C8C2×D311yescharacteristic
58C4×C3⋊Dic3C2×D311yescharacteristic
59C4×C3⋊Dic3C2×D311yescharacteristic
60C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
62C3×C4×Dic3—31no—
63C3×C4×Dic3—31no—
64C2×C3×C3⋊C8—31no—
65C2×C3×C3⋊C8—31no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—61no—
69Dic3⋊Dic3—31no—
70Dic3⋊Dic3—31no—
71Dic3⋊Dic3—31no—
72Dic3⋊Dic3—31no—
73(C2×D3)⋊Dic3—31no—
74(C2×D3)⋊Dic3—31no—
75(C2×D3)⋊Dic3—31no—
76(C2×D3)⋊Dic3—31no—
77C22×H(3,𝔽3)C22×C411yescharacteristic
78C2×#272—21no—
79C2×#272—21no—
80C2×#272—21no—
81C4×H(3,𝔽3)#4111yescharacteristic
82C4×H(3,𝔽3)#4111yescharacteristic
83#1084—21no—
84(C3⋊Dic3)⋊C3—21no—
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—21no—
88#926—91no—
89#872—91no—
90#872—91no—
91C3×C4⋊C8—91no—
92C3×C22⋊C8—91no—
93C3×C4×C8—91no—
94#830—91no—
95(C2×D3)⋊C8—31no—
96(C2×D3)⋊C8—31no—
97(C2×D3)⋊C8—61no—
98(C2×D3)⋊C8—91no—
99(C2×D3)⋊C8—91no—
100#815—91no—
101#815—91no—
102Dic3⋊C8—31no—
103Dic3⋊C8—91no—
104Dic3⋊C8—91no—
105Dic3⋊C8—91no—
106C8×Dic3—31no—
107C8×Dic3—91no—
108C4⋊(C3⋊C8)—91no—
109#803—91no—
110C22×C3×D3—31no—
111C22×C3×D3—31no—
112C22×C3×D3—61no—
113C2×C32×C4—21no—
114C2×C32×C4C22×D311yescharacteristic
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3C4×D311yescharacteristic
117C2×C3⋊Dic3C4×D311yescharacteristic
118C2×C3⋊Dic3C4×D311yescharacteristic
119C2×C3⋊Dic3C4×D311yescharacteristic
120C2×C3×Dic3—31no—
121C2×C3×Dic3—31no—
122C2×C3×Dic3—31no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—61no—
127C3×C4×D3—61no—
128C3×C4×D3—61no—
129C3×C4×D3—61no—
130C3×C4×D3—61no—
131C3×C4×D3—121no—
132C3×C4×D3—121no—
133C32×C8—21no—
134C32×C8—21no—
135C32×C8—41no—
136C3⋊(C3⋊C8)—21no—
137C3⋊(C3⋊C8)—21no—
138C3×C3⋊C8—61no—
139C3×C3⋊C8—61no—
140C3×C3⋊C8—61no—
141C3×C3⋊C8—61no—
142C3×C3⋊C8—121no—
143#433—271noSylow
144C2×H(3,𝔽3)#11711yescharacteristic
145C2×H(3,𝔽3)#13111yescharacteristic
146C2×H(3,𝔽3)#13111yescharacteristic
147#272—41no—
148C22×C3×C4—91no—
149C2×C4×D3—31no—
150C2×C4×D3—31no—
151C2×C4×D3—61no—
152C2×C3×C8—31no—
153C2×C3×C8—31no—
154C2×C3×C8—61no—
155C2×C3×C8—91no—
156C2×C3×C8—91no—
157C2×C3×C8—91no—
158C2×C3×C8D3211yescharacteristic
159C3×C42—91no—
160C3×C42—91no—
161#217—91no—
162#217—91no—
163(C2×D3)⋊C4—91no—
164(C2×D3)⋊C4—91no—
165(C2×D3)⋊C4—91no—
166(C2×D3)⋊C4—91no—
167C4⋊Dic3—91no—
168C4⋊Dic3—91no—
169Dic3⋊C4—91no—
170Dic3⋊C4—91no—
171Dic3⋊C4—91no—
172Dic3⋊C4—91no—
173C4×Dic3—31no—
174C4×Dic3—31no—
175C4×Dic3—91no—
176C4×Dic3—91no—
177C2×C3⋊C8—31no—
178C2×C3⋊C8—31no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—61no—
182C2×C3⋊C8—91no—
183C2×C3⋊C8—91no—
184C22×C32—21no—
185C22×C32C2×C4×D311yescharacteristic
186C22×C32C2×C4×D311yescharacteristic
187C2×C3×D3—62no—
188C2×C3×D3—61no—
189C2×C3×D3—61no—
190C2×C3×D3—61no—
191C2×C3×D3—61no—
192C2×C3×D3—61no—
193C2×C3×D3—61no—
194C2×C3×D3—121no—
195C2×C3×D3—121no—
196C32×C4—21no—
197C32×C4—21no—
198C32×C4#23511yescharacteristic
199C32×C4#23511yescharacteristic
200C32×C4#23711yescharacteristic
201C32×C4#23911yescharacteristic
202C3⋊Dic3—21no—
203C3⋊Dic3—21no—
204C3⋊Dic3—21no—
205C3⋊Dic3—21no—
206C3×Dic3—61no—
207C3×Dic3—61no—
208C3×Dic3—61no—
209C3×Dic3—61no—
210C3×Dic3—61no—
211C3×Dic3—61no—
212C3×Dic3—121no—
213#117—271no—
214C4⋊C8—91no—
215C4⋊C8—271no—
216C22⋊C8—91no—
217C22⋊C8—271no—
218#97—271no—
219C4×C8—91no—
220H(3,𝔽3)#43311yescharacteristic, complemented, Sylow
221C23×C3—91no—
222C22×D3—31no—
223C22×D3—31no—
224C22×D3—61no—
225C2×C3×C4—31no—
226C2×C3×C4—31no—
227C2×C3×C4—61no—
228C2×C3×C4—91no—
229C2×C3×C4—91no—
230C2×C3×C4—91no—
231C2×C3×C4—91no—
232C2×C3×C4—91no—
233C2×C3×C4—181no—
234C2×C3×C4—181no—
235C2×C3×C4C2×D3211yescharacteristic
236C2×Dic3—31no—
237C2×Dic3—31no—
238C2×Dic3—31no—
239C2×Dic3—31no—
240C2×Dic3—31no—
241C2×Dic3—31no—
242C2×Dic3—61no—
243C2×Dic3—91no—
244C2×Dic3—91no—
245C2×Dic3—91no—
246C2×Dic3—91no—
247C4×D3—61no—
248C4×D3—61no—
249C4×D3—61no—
250C4×D3—61no—
251C4×D3—121no—
252C4×D3—121no—
253C3×C8—21no—
254C3×C8—61no—
255C3×C8—61no—
256C3×C8—121no—
257C3×C8—181no—
258C3×C8—181no—
259C3×C8—181no—
260C3⋊C8—61no—
261C3⋊C8—61no—
262C3⋊C8—61no—
263C3⋊C8—61no—
264C3⋊C8—121no—
265C3⋊C8—181no—
266C3⋊C8—181no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32#87211yescharacteristic
271C2×C32#89211yescharacteristic
272C2×C32#90111yescharacteristic
273C2×C32#90111yescharacteristic
274C2×C32#90111yescharacteristic
275C2×C32#90711yescharacteristic
276C3×D3—122no—
277C3×D3—121no—
278C3×D3—121no—
279C22×C4—91no—
280C2×C8—91no—
281C2×C8—91no—
282C2×C8—91no—
283C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
284C4⋊C4—271no—
285C4⋊C4—271no—
286C22⋊C4—271no—
287C22⋊C4—271no—
288C42—91no—
289C42—91no—
290C22×C3—31no—
291C22×C3—31no—
292C22×C3—61no—
293C22×C3—181no—
294C22×C3—181no—
295C22×C3—181no—
296C22×C3C4×D3211yescharacteristic
297C2×D3—61no—
298C2×D3—61no—
299C2×D3—61no—
300C2×D3—61no—
301C2×D3—61no—
302C2×D3—61no—
303C2×D3—62no—
304C2×D3—121no—
305C2×D3—121no—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—61no—
311C3×C4—61no—
312C3×C4—181no—
313C3×C4—181no—
314C3×C4—181no—
315C3×C4—181no—
316C3×C4—181no—
317C3×C4#379911yescharacteristic
318C3×C4#380411yescharacteristic
319Dic3—61no—
320Dic3—61no—
321Dic3—61no—
322Dic3—61no—
323Dic3—61no—
324Dic3—61no—
325Dic3—121no—
326Dic3—181no—
327Dic3—181no—
328Dic3—181no—
329Dic3—181no—
330C32—21no—
331C32#469011yescharacteristic, complemented
332C32#484011yescharacteristic, complemented
333C23—91no—
334C2×C4—91no—
335C2×C4—91no—
336C2×C4—91no—
337C2×C4—91no—
338C2×C4—91no—
339C2×C4—181no—
340C2×C4—181no—
341C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
342C8—21no—
343C8—181no—
344C8—181no—
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—361no—
356C2×C3#6380311yescharacteristic
357C2×C3#6380411yescharacteristic
358C2×C3#6383611yescharacteristic
359D3—121no—
360D3—121no—
361D3—122no—
362C22—181no—
363C22—181no—
364C22—181no—
365C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
366C4—181no—
367C4—181no—
368C4—181no—
369C4—181no—
370C4—181no—
371C4#8865211yescharacteristic
372C4#8867111yescharacteristic
373C3—31no—
374C3—31no—
375C3—61no—
376C3#1059064811yescharacteristic
377C2—361no—
378C2#1172084011yescharacteristic
379C2#1172084111yescharacteristic
380C2#1172107611yescharacteristic
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×#272C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#433
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
3. #63641⋊C3C2semidirect
4. C8×(C3⋊Dic3)⋊C3C2semidirect
6. #11720312C2semidirect
7. #11720311C2semidirect
23. #63641D3semidirect
24. C8×C3⋊Dic3D3semidirect
220. H(3,𝔽3)#433semidirect
331. C32#4690semidirect
332. C32#4840semidirect

Maximal subgroups

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

subgroupindex
#117210972
#117205532
#63641⋊C32
C8×(C3⋊Dic3)⋊C32
#117203232
#117203122
#117203112
#105906983
#105906963