Finite Group Databasebeta

← #13137909

Subgroups — #13137909

Order 1728.

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

normal subgroups only

This group has 393 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#64102⋊C3C211yesmaximal, characteristic
2#11721093C211yesmaximal, characteristic, complemented
3#11720529C211yesmaximal, characteristic, complemented
4#63635⋊C3C211yesmaximal, characteristic, complemented
5#11720318C211yesmaximal, characteristic
6#11720311C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic
8#10590542—31nomaximal
9#10590435—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3—21no—
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15#3752⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88449—21no—
20#88448—21no—
21#64102D311yescharacteristic
22C3×#893—31no—
23#63872—31no—
24#63869—31no—
25#63635D311yescharacteristic, complemented
26C3×#808—31no—
27C3×#807—31no—
28C3×#807—31no—
29C3×#807—61no—
30#63572—31no—
31Dic3⋊(C3⋊C8)—31no—
32#63564—31no—
33#63564—31no—
34#63563—31no—
35#63562—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3D411yescharacteristic
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43C4⋊((C3⋊D3)⋊C3)—21no—
44C4⋊((C3⋊D3)⋊C3)Q812yes—
45C4×(C3⋊D3)⋊C3—42no—
46#6180—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48#5009—91no—
49#4891—91no—
50#4871—91no—
51C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
52C2×C4×C3⋊D3—21no—
53C2×C3×D12—31no—
54C2×C3×C4×D3—61no—
55C32×C4⋊C4—21no—
56C32×C4⋊C4C2×D311yescharacteristic
57C32×C4⋊C4C2×D311yescharacteristic
58#3752—21no—
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60C4×C3⋊Dic3C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
62C3×(C2×D3)⋊C4—61no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—61no—
67C3×C4×Dic3—31no—
68C2×C3×C3⋊C8—31no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—61no—
72#3724—61no—
73Dic3⋊Dic3—61no—
74Dic3⋊Dic3—61no—
75Dic3⋊Dic3—61no—
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C2×(C3⋊D3)⋊C3—22no—
78C2×(C3⋊D3)⋊C3—41no—
79C2×(C3⋊D3)⋊C3—41no—
80C4×H(3,𝔽3)—41no—
81C4×H(3,𝔽3)C2×Q811yescharacteristic
82C4×H(3,𝔽3)#4111yescharacteristic
83#1084—41no—
84(C3⋊Dic3)⋊C3—21no—
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—41no—
87C3×C4×D4—91no—
88#893—31no—
89#893—91no—
90#890—91no—
91#890—91no—
92#870—91no—
93C3×#107—91no—
94C3×#106—91no—
95#832—91no—
96#832—91no—
97#821—91no—
98#821—91no—
99Dic3⋊C8—91no—
100Dic3⋊C8—91no—
101#809—91no—
102#809—91no—
103#808—31no—
104#808—91no—
105#807—31no—
106#807—31no—
107#807—61no—
108C4⋊(C3⋊C8)—91no—
109C22×C3⋊D3—21no—
110C22×C3×D3—61no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—41no—
115C2×C32×C4C22×D311yescharacteristic
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3#6711yescharacteristic
119C2×C3⋊Dic3#6711yescharacteristic
120C4⋊(C3⋊D3)—21no—
121C4⋊(C3⋊D3)Dic612yes—
122C4×C3⋊D3—42no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—121no—
129C3×D12—32no—
130C3×D12—61no—
131C3×C4×D3—122no—
132C3⋊(C3⋊C8)—21no—
133C3×C3⋊C8—62no—
134C3×C3⋊C8—61no—
135C3×C3⋊C8—61no—
136C3×C3⋊C8—61no—
137#478—271noSylow
138C2×H(3,𝔽3)C22⋊Q811yescharacteristic
139C2×H(3,𝔽3)#13511yescharacteristic
140C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
141(C3⋊D3)⋊C3—42no—
142C2×C3×D4—91no—
143C22×C3×C4—181no—
144C2×D12—31no—
145C2×D12—91no—
146C2×C4×D3—61no—
147C2×C4×D3—181no—
148C2×C3×C8—91no—
149C2×C3×C8—91no—
150C3×C4⋊C4—31no—
151C3×C4⋊C4—31no—
152C3×C4⋊C4—61no—
153C3×C4⋊C4—91no—
154C3×C4⋊C4—91no—
155C3×C4⋊C4D3211yescharacteristic
156C3×C22⋊C4—181no—
157C3×C42—91no—
158(C2×D3)⋊C4—61no—
159(C2×D3)⋊C4—181no—
160C4⋊Dic3—31no—
161C4⋊Dic3—31no—
162C4⋊Dic3—31no—
163C4⋊Dic3—61no—
164C4⋊Dic3—91no—
165C4⋊Dic3—181no—
166C4⋊Dic3—181no—
167Dic3⋊C4—181no—
168Dic3⋊C4—181no—
169Dic3⋊C4—181no—
170Dic3⋊C4—181no—
171C4×Dic3—31no—
172C4×Dic3—91no—
173C2×C3⋊C8—31no—
174C2×C3⋊C8—31no—
175C2×C3⋊C8—31no—
176C2×C3⋊C8—61no—
177C2×C3⋊C8—91no—
178C22×C32—21no—
179C22×C32D3×D411yescharacteristic
180C22×C32C2×#6711yescharacteristic
181C2×C3⋊D3—22no—
182C2×C3⋊D3—41no—
183C2×C3⋊D3—41no—
184C2×C3×D3—62no—
185C2×C3×D3—121no—
186C2×C3×D3—121no—
187C32×C4—21no—
188C32×C4—21no—
189C32×C4—41no—
190C32×C4—41no—
191C32×C4—81no—
192C32×C4C2×Dic611yescharacteristic
193C32×C4#23511yescharacteristic
194C32×C4#23511yescharacteristic
195C32×C4D3×Q811yescharacteristic
196C3⋊Dic3—21no—
197C3⋊Dic3—21no—
198C3⋊Dic3—41no—
199C3×Dic3—61no—
200C3×Dic3—61no—
201C3×Dic3—121no—
202C3×Dic3—121no—
203C3×Dic3—121no—
204C3×Dic3—241no—
205#125—271no—
206C4×D4—91no—
207#107—91no—
208#106—91no—
209C4⋊C8—271no—
210#102—271no—
211#102—271no—
212H(3,𝔽3)#47811yescharacteristic, complemented, Sylow
213C23×C3—181no—
214C22×D3—61no—
215C22×D3—181no—
216C3×D4—92no—
217C3×D4—181no—
218C2×C3×C4—21no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—91no—
225C2×C3×C4—91no—
226C2×C3×C4—121no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4—362no—
230C2×C3×C4C2×D3211yescharacteristic
231C2×Dic3—31no—
232C2×Dic3—31no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—91no—
237C2×Dic3—91no—
238C2×Dic3—121no—
239C2×Dic3—181no—
240D12—32no—
241D12—61no—
242D12—92no—
243D12—181no—
244C4×D3—122no—
245C4×D3—362no—
246C3×C8—181no—
247C3×C8—181no—
248C3⋊C8—61no—
249C3⋊C8—61no—
250C3⋊C8—62no—
251C3⋊C8—61no—
252C3⋊C8—181no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32#89611yescharacteristic
257C2×C32#91411yescharacteristic
258C2×C32#91611yescharacteristic
259C2×C32#92411yescharacteristic
260C2×C32#93211yescharacteristic
261C2×C32#95011yescharacteristic
262C3⋊D3—42no—
263C3×D3—122no—
264C2×D4—91no—
265C22×C4—181no—
266C2×C8—91no—
267C2×C8—91no—
268C4⋊C4—91no—
269C4⋊C4—91no—
270C4⋊C4—541no—
271C4⋊C4—541no—
272C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
273C22⋊C4—181no—
274C42—91no—
275C22×C3—31no—
276C22×C3—31no—
277C22×C3—61no—
278C22×C3—182no—
279C22×C3—361no—
280C22×C3—361no—
281C22×C3D3×#6711yescharacteristic
282C2×D3—62no—
283C2×D3—121no—
284C2×D3—121no—
285C2×D3—182no—
286C2×D3—361no—
287C2×D3—361no—
288C3×C4—31no—
289C3×C4—31no—
290C3×C4—31no—
291C3×C4—31no—
292C3×C4—41no—
293C3×C4—61no—
294C3×C4—61no—
295C3×C4—121no—
296C3×C4—121no—
297C3×C4—181no—
298C3×C4—181no—
299C3×C4—241no—
300C3×C4—361no—
301C3×C4—361no—
302C3×C4D3×Dic611yescharacteristic
303C3×C4#379811yescharacteristic
304Dic3—61no—
305Dic3—61no—
306Dic3—121no—
307Dic3—121no—
308Dic3—121no—
309Dic3—181no—
310Dic3—181no—
311Dic3—241no—
312Dic3—361no—
313C32—21no—
314C32#487111yescharacteristic, complemented
315C32#500911yescharacteristic, complemented
316C23—181no—
317D4—92no—
318D4—181no—
319C2×C4—21no—
320C2×C4—91no—
321C2×C4—91no—
322C2×C4—181no—
323C2×C4—181no—
324C2×C4—362no—
325C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
326C8—181no—
327C8—181no—
328C2×C3—31no—
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—31no—
334C2×C3—61no—
335C2×C3—61no—
336C2×C3—61no—
337C2×C3—362no—
338C2×C3#6385111yescharacteristic
339C2×C3#6393411yescharacteristic
340C2×C3#6394711yescharacteristic
341D3—122no—
342D3—362no—
343C22—182no—
344C22—361no—
345C22—361no—
346C22#8867611yescharacteristic
347C4—41no—
348C4—181no—
349C4—181no—
350C4—361no—
351C4—361no—
352C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
353C4#8865011yescharacteristic
354C3—31no—
355C3—31no—
356C3—61no—
357C3#1059054511yescharacteristic
358C2—362no—
359C2#1172110111yescharacteristic
360C2#1172119611yescharacteristic
361C2#1172123911yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#478
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11721093C2semidirect
3. #11720529C2semidirect
4. #63635⋊C3C2semidirect
6. #11720311C2semidirect
25. #63635D3semidirect
212. H(3,𝔽3)#478semidirect
314. C32#4871semidirect
315. C32#5009semidirect

Maximal subgroups

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

subgroupindex
#64102⋊C32
#117210932
#117205292
#63635⋊C32
#117203182
#117203112
#117203082
#105905423
#105904353