Finite Group Databasebeta

← #13138978

Subgroups — #13138978

Order 1728.

421 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 69 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#11721156C211yesmaximal, characteristic
2#11721081C211yesmaximal, characteristic, complemented
3#11720554C211yesmaximal, characteristic, complemented
4#63647⋊C3C211yesmaximal, characteristic, complemented
5#63645⋊C3C211yesmaximal, characteristic, complemented
6#11720324C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10590834—31nomaximal
9#10590651—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×#6200—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#88451—21no—
19#88450—21no—
20#88447—21no—
21#63916—31no—
22#63908—31no—
23#63849—31no—
24#63841—31no—
25#63647D311yescharacteristic, complemented
26#63645D311yescharacteristic, complemented
27C3×#821—31no—
28C3×#821—31no—
29C3×#821—61no—
30C3×(C2×D3)⋊C8—31no—
31C3×#818—31no—
32#63574—31no—
33#63573—31no—
34#63563—31no—
35#63562—31no—
36C22×#272—21no—
37C22×(C3⋊D3)⋊C3D411yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C4⋊#272—21no—
41C4⋊#272—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#6200—42no—
48C8×H(3,𝔽3)—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50#4885—91no—
51#4784—91no—
52#4737—91no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×C3×D12—31no—
55C2×C3×D12—31no—
56C2×C3×D12—61no—
57C2×C3×C4×D3—31no—
58C2×#626—61no—
59C2×#625—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#3722—61no—
73(C2×D3)⋊Dic3—61no—
74(C2×D3)⋊Dic3—61no—
75(C2×D3)⋊Dic3—61no—
76Dic32—61no—
77Dic32—61no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×#272—41no—
80C2×#272—41no—
81C2×#272—41no—
82C2×(C3⋊D3)⋊C3—21no—
83C2×(C3⋊D3)⋊C3—21no—
84C2×(C3⋊D3)⋊C3—21no—
85C4×H(3,𝔽3)#4111yescharacteristic
86C4×H(3,𝔽3)#4111yescharacteristic
87#1084—41no—
88(C3⋊Dic3)⋊C3—21no—
89(C3⋊Dic3)⋊C3—41no—
90(C3⋊Dic3)⋊C3—41no—
91#938—91no—
92#930—91no—
93#926—91no—
94#895—91no—
95#892—91no—
96#892—91no—
97C3×#107—91no—
98C3×#102—91no—
99C3×C22⋊C8—91no—
100#832—91no—
101#821—31no—
102#821—31no—
103#821—61no—
104#821—91no—
105(C2×D3)⋊C8—31no—
106(C2×D3)⋊C8—91no—
107#818—31no—
108#818—91no—
109#817—91no—
110#809—91no—
111#808—91no—
112#808—91no—
113C22×C3⋊D3D1211yescharacteristic
114C22×C3×D3—31no—
115C22×C3×D3—61no—
116C22×C3×D3—61no—
117C22×C3×D3—121no—
118C2×C32×C4—21no—
119C2×C32×C4C22×D311yescharacteristic
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3D1211yescharacteristic
124C4×C3⋊D3—21no—
125C4×C3⋊D3—21no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—121no—
132C3×D12—62no—
133C3×D12—61no—
134C3×D12—61no—
135C3×D12—61no—
136C3×D12—61no—
137C3×D12—121no—
138C3×C4×D3—61no—
139C3×C4×D3—61no—
140#626—122no—
141#625—122no—
142C32×C8—21no—
143C32×C8—21no—
144C32×C8—41no—
145C3⋊(C3⋊C8)—21no—
146C3×C3⋊C8—61no—
147#482—271noSylow
148C2×H(3,𝔽3)#12311yescharacteristic
149C2×H(3,𝔽3)#13511yescharacteristic
150C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
151#272—81no—
152(C3⋊D3)⋊C3—41no—
153C2×C3×D4—91no—
154C22×C3×C4—91no—
155C2×#67—181no—
156C2×#67—181no—
157C2×D12—31no—
158C2×D12—31no—
159C2×D12—61no—
160C2×D12—181no—
161C2×C4×D3—31no—
162C2×C4×D3—91no—
163C2×C3×C8—31no—
164C2×C3×C8—31no—
165C2×C3×C8—61no—
166C2×C3×C8—91no—
167C2×C3×C8D3211yescharacteristic
168C3×C4⋊C4—91no—
169C3×C4⋊C4—91no—
170C3×C4⋊C4—91no—
171#217—181no—
172#217—181no—
173(C2×D3)⋊C4—181no—
174(C2×D3)⋊C4—181no—
175(C2×D3)⋊C4—181no—
176(C2×D3)⋊C4—181no—
177C4⋊Dic3—31no—
178C4⋊Dic3—31no—
179C4⋊Dic3—31no—
180C4⋊Dic3—31no—
181C4⋊Dic3—61no—
182C4⋊Dic3—91no—
183C4⋊Dic3—91no—
184C4×Dic3—181no—
185C4×Dic3—181no—
186C4×Dic3—181no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—91no—
189C22×C32—21no—
190C22×C32C2×D1211yescharacteristic
191C22×C32D3×D411yescharacteristic
192C2×C3⋊D3—21no—
193C2×C3⋊D3—21no—
194C2×C3⋊D3—21no—
195C2×C3×D3—61no—
196C2×C3×D3—61no—
197C2×C3×D3—61no—
198C2×C3×D3—122no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201C2×C3×D3—121no—
202C2×C3×D3—121no—
203C2×C3×D3—121no—
204C2×C3×D3—121no—
205C2×C3×D3—241no—
206C2×C3×D3—241no—
207C32×C4—21no—
208C32×C4—21no—
209C32×C4#23511yescharacteristic
210C32×C4#23711yescharacteristic
211C32×C4#23711yescharacteristic
212C32×C4#23911yescharacteristic
213C3⋊Dic3—21no—
214C3⋊Dic3—41no—
215C3⋊Dic3—41no—
216C3×Dic3—61no—
217C3×Dic3—121no—
218C3×Dic3—121no—
219C3×Dic3—121no—
220C3×Dic3—121no—
221C3×Dic3—241no—
222#121—271no—
223#117—271no—
224#107—91no—
225#106—271no—
226#102—91no—
227#102—271no—
228C22⋊C8—91no—
229H(3,𝔽3)#48211yescharacteristic, complemented, Sylow
230C23×C3—91no—
231C23×C3—181no—
232C22×D3—31no—
233C22×D3—61no—
234C22×D3—61no—
235C22×D3—91no—
236C22×D3—121no—
237C3×D4—181no—
238C3×D4—181no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—61no—
242C2×C3×C4—91no—
243C2×C3×C4—181no—
244C2×C3×C4—181no—
245C2×C3×C4—181no—
246C2×C3×C4—181no—
247C2×C3×C4—181no—
248C2×C3×C4C2×D3211yescharacteristic
249#67—362no—
250#67—362no—
251C2×Dic3—31no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—61no—
255C2×Dic3—61no—
256C2×Dic3—91no—
257C2×Dic3—121no—
258C2×Dic3—181no—
259C2×Dic3—181no—
260D12—61no—
261D12—61no—
262D12—61no—
263D12—61no—
264D12—62no—
265D12—121no—
266D12—362no—
267C4×D3—61no—
268C4×D3—61no—
269C4×D3—181no—
270C4×D3—181no—
271C3×C8—21no—
272C3×C8—61no—
273C3×C8—61no—
274C3×C8—121no—
275C3×C8—181no—
276C3⋊C8—61no—
277C3⋊C8—181no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32#88611yescharacteristic
282C2×C32#89411yescharacteristic
283C2×C32#90411yescharacteristic
284C2×C32#90811yescharacteristic
285C2×C32#91111yescharacteristic
286C2×C32#91911yescharacteristic
287C3⋊D3—41no—
288C3×D3—121no—
289C3×D3—242no—
290C3×D3—241no—
291C3×D3—241no—
292C2×D4—91no—
293C2×D4—541no—
294C22×C4—91no—
295C2×C8—91no—
296C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
297C4⋊C4—91no—
298C4⋊C4—91no—
299C4⋊C4—91no—
300C22⋊C4—541no—
301C22⋊C4—541no—
302C42—541no—
303C22×C3—31no—
304C22×C3—31no—
305C22×C3—61no—
306C22×C3—181no—
307C22×C3—181no—
308C22×C3—181no—
309C22×C3—361no—
310C22×C3—361no—
311C22×C3—361no—
312C22×C3D3×D1211yescharacteristic
313C2×D3—61no—
314C2×D3—61no—
315C2×D3—61no—
316C2×D3—121no—
317C2×D3—121no—
318C2×D3—121no—
319C2×D3—121no—
320C2×D3—121no—
321C2×D3—121no—
322C2×D3—122no—
323C2×D3—181no—
324C2×D3—181no—
325C2×D3—181no—
326C2×D3—241no—
327C2×D3—241no—
328C3×C4—31no—
329C3×C4—31no—
330C3×C4—31no—
331C3×C4—31no—
332C3×C4—61no—
333C3×C4—61no—
334C3×C4—181no—
335C3×C4—361no—
336C3×C4—361no—
337C3×C4—361no—
338C3×C4#379611yescharacteristic
339C3×C4#380411yescharacteristic
340Dic3—61no—
341Dic3—121no—
342Dic3—121no—
343Dic3—121no—
344Dic3—121no—
345Dic3—181no—
346Dic3—241no—
347Dic3—361no—
348Dic3—361no—
349C32—21no—
350C32#473711yescharacteristic, complemented
351C32#488511yescharacteristic, complemented
352C23—91no—
353C23—181no—
354D4—181no—
355D4—181no—
356D4—1082no—
357C2×C4—91no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4—181no—
361C2×C4—181no—
362C2×C4—181no—
363C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
364C8—21no—
365C8—181no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—61no—
373C2×C3—61no—
374C2×C3—61no—
375C2×C3—361no—
376C2×C3—721no—
377C2×C3#6379511yescharacteristic
378C2×C3#6380811yescharacteristic
379C2×C3#6389011yescharacteristic
380D3—121no—
381D3—241no—
382D3—241no—
383D3—242no—
384D3—361no—
385C22—181no—
386C22—181no—
387C22—181no—
388C22—361no—
389C22—361no—
390C22—361no—
391C22#8865411yescharacteristic
392C4—181no—
393C4—361no—
394C4—361no—
395C4—361no—
396C4#8864911yescharacteristic
397C4#8867111yescharacteristic
398C3—31no—
399C3—31no—
400C3—61no—
401C3#1059070511yescharacteristic
402C2—361no—
403C2—721no—
404C2#1172084911yescharacteristic
405C2#1172085011yescharacteristic
406C2#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 radical#11721156C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#482
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
2. #11721081C2semidirect
3. #11720554C2semidirect
4. #63647⋊C3C2semidirect
5. #63645⋊C3C2semidirect
6. #11720324C2semidirect
7. #11720307C2semidirect
25. #63647D3semidirect
26. #63645D3semidirect
229. H(3,𝔽3)#482semidirect
350. C32#4737semidirect
351. C32#4885semidirect

Maximal subgroups

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

subgroupindex
#117211562
#117210812
#117205542
#63647⋊C32
#63645⋊C32
#117203242
#117203072
#105908343
#105906513