Finite Group Databasebeta

← #13137983

Subgroups — #13137983

Order 1728.

447 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 67 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#11721760C211yesmaximal, characteristic, complemented
2#11721159C211yesmaximal, characteristic
3#63636⋊C3C211yesmaximal, characteristic, complemented
4#63634⋊C3C211yesmaximal, characteristic, complemented
5#11720323C211yesmaximal, characteristic, complemented
6#11720308C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10590502—31nomaximal
9#10590424—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#6200—21no—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#88542—21no—
15C4⋊#1084C2211yescharacteristic
16C4×#1084—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88450—21no—
21C3×#892—31no—
22C3×#892—31no—
23C3×#892—61no—
24#63909—31no—
25#63908—31no—
26#63636D311yescharacteristic, complemented
27#63634D311yescharacteristic, complemented
28C3×#809—31no—
29C3×#807—31no—
30#63574—31no—
31#63573—31no—
32#63564—31no—
33#63562—31no—
34(C2×D3)⋊(C3⋊C8)—31no—
35(C2×D3)⋊(C3⋊C8)—31no—
36C22×#272D411yescharacteristic
37C22×(C3⋊D3)⋊C3—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—21no—
41C2×#1084D411yescharacteristic
42C4×#272—21no—
43C4×#272—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47#6200—42no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50#5003—91no—
51#4885—91no—
52#4779—91no—
53C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
54C2×C3×D12—31no—
55C2×C3×C4×D3—31no—
56C2×C3×C4×D3—31no—
57C2×C3×C4×D3—61no—
58C2×#626—61no—
59C2×#625—61no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C4⋊C4C2×D311yescharacteristic
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
65C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
66C3×(C2×D3)⋊C4—61no—
67C3×(C2×D3)⋊C4—61no—
68C3×(C2×D3)⋊C4—62no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—61no—
73C3×C4×Dic3—61no—
74C3×C4×Dic3—61no—
75C3×C4×Dic3—62no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78#3722—61no—
79(C2×D3)⋊Dic3—61no—
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C2×#272—21no—
82C2×#272—21no—
83C2×#272—21no—
84C2×(C3⋊D3)⋊C3—41no—
85C2×(C3⋊D3)⋊C3—41no—
86C2×(C3⋊D3)⋊C3—41no—
87C4×H(3,𝔽3)—41no—
88C4×H(3,𝔽3)#4111yescharacteristic
89C4×H(3,𝔽3)#4111yescharacteristic
90#1084—21no—
91#1084—41no—
92(C3⋊Dic3)⋊C3—41no—
93C3×#117—91no—
94#938—91no—
95#930—91no—
96#895—91no—
97#892—31no—
98#892—31no—
99#892—61no—
100C3×#107—91no—
101C3×#102—91no—
102#832—91no—
103#830—91no—
104#821—91no—
105(C2×D3)⋊C8—91no—
106(C2×D3)⋊C8—91no—
107#817—91no—
108#809—31no—
109#809—91no—
110#809—91no—
111#808—91no—
112#808—91no—
113#807—31no—
114#807—91no—
115C22×C3⋊D3—21no—
116C22×C3×D3—31no—
117C22×C3×D3—31no—
118C22×C3×D3—61no—
119C22×C3×D3—61no—
120C2×C32×C4—22no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4C22×D311yescharacteristic
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3—21no—
127C4⋊(C3⋊D3)—21no—
128C4⋊(C3⋊D3)—21no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C3×D12—61no—
137C3×D12—61no—
138C3×C4×D3—62no—
139C3×C4×D3—62no—
140C3×C4×D3—61no—
141C3×C4×D3—61no—
142C3×C4×D3—61no—
143C3×C4×D3—61no—
144#626—122no—
145#625—122no—
146C3⋊(C3⋊C8)—21no—
147C3⋊(C3⋊C8)—21no—
148C3×C3⋊C8—61no—
149C3×C3⋊C8—61no—
150#482—271noSylow
151C2×H(3,𝔽3)#12311yescharacteristic
152C2×H(3,𝔽3)#13511yescharacteristic
153C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
154#272—41no—
155(C3⋊D3)⋊C3—81no—
156C2×C3×D4—91no—
157C22×C3×C4—91no—
158C2×#67—181no—
159C2×#67—181no—
160C2×D12—31no—
161C2×D12—91no—
162C2×D12—181no—
163C2×C4×D3—31no—
164C2×C4×D3—31no—
165C2×C4×D3—61no—
166C2×C3×C8—91no—
167C2×C3×C8—91no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4—31no—
170C3×C4⋊C4—61no—
171C3×C4⋊C4—91no—
172C3×C4⋊C4—91no—
173C3×C4⋊C4D3211yescharacteristic
174C3×C22⋊C4—181no—
175C3×C42—181no—
176#217—181no—
177(C2×D3)⋊C4—61no—
178(C2×D3)⋊C4—61no—
179(C2×D3)⋊C4—62no—
180(C2×D3)⋊C4—181no—
181(C2×D3)⋊C4—181no—
182C4⋊Dic3—31no—
183C4⋊Dic3—31no—
184C4⋊Dic3—31no—
185C4⋊Dic3—61no—
186C4⋊Dic3—91no—
187C4×Dic3—61no—
188C4×Dic3—61no—
189C4×Dic3—62no—
190C2×C3⋊C8—31no—
191C2×C3⋊C8—31no—
192C2×C3⋊C8—91no—
193C2×C3⋊C8—91no—
194C22×C32—21no—
195C22×C32D3×D411yescharacteristic
196C22×C32D3×D411yescharacteristic
197C2×C3⋊D3—41no—
198C2×C3⋊D3—41no—
199C2×C3⋊D3—41no—
200C2×C3×D3—62no—
201C2×C3×D3—61no—
202C2×C3×D3—61no—
203C2×C3×D3—61no—
204C2×C3×D3—61no—
205C2×C3×D3—61no—
206C2×C3×D3—61no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C2×C3×D3—121no—
211C2×C3×D3—121no—
212C32×C4—21no—
213C32×C4—21no—
214C32×C4—42no—
215C32×C4—41no—
216C32×C4—41no—
217C32×C4#23511yescharacteristic
218C32×C4#23511yescharacteristic
219C32×C4#23711yescharacteristic
220C32×C4#23911yescharacteristic
221C3⋊Dic3—41no—
222C3×Dic3—62no—
223C3×Dic3—61no—
224C3×Dic3—61no—
225C3×Dic3—122no—
226C3×Dic3—121no—
227C3×Dic3—121no—
228C3×Dic3—121no—
229#121—271no—
230#117—91no—
231#107—91no—
232#106—271no—
233#102—91no—
234#102—271no—
235C22⋊C8—271no—
236H(3,𝔽3)#48211yescharacteristic, complemented, Sylow
237C23×C3—91no—
238C23×C3—181no—
239C22×D3—31no—
240C22×D3—31no—
241C22×D3—61no—
242C22×D3—61no—
243C22×D3—181no—
244C3×D4—181no—
245C3×D4—181no—
246C2×C3×C4—21no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—62no—
252C2×C3×C4—61no—
253C2×C3×C4—91no—
254C2×C3×C4—181no—
255C2×C3×C4—181no—
256C2×C3×C4—181no—
257C2×C3×C4—181no—
258C2×C3×C4C2×D3211yescharacteristic
259#67—362no—
260#67—362no—
261C2×Dic3—31no—
262C2×Dic3—31no—
263C2×Dic3—61no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—62no—
267C2×Dic3—61no—
268C2×Dic3—181no—
269D12—61no—
270D12—61no—
271D12—181no—
272D12—181no—
273D12—362no—
274C4×D3—61no—
275C4×D3—61no—
276C4×D3—61no—
277C4×D3—61no—
278C4×D3—62no—
279C4×D3—62no—
280C3×C8—181no—
281C3×C8—181no—
282C3⋊C8—61no—
283C3⋊C8—61no—
284C3⋊C8—181no—
285C3⋊C8—181no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32#88311yescharacteristic
290C2×C32#89411yescharacteristic
291C2×C32#91111yescharacteristic
292C2×C32#91111yescharacteristic
293C2×C32#91611yescharacteristic
294C2×C32#91911yescharacteristic
295C3⋊D3—81no—
296C3×D3—122no—
297C3×D3—121no—
298C3×D3—121no—
299C3×D3—241no—
300C2×D4—91no—
301C2×D4—541no—
302C22×C4—91no—
303C2×C8—91no—
304C2×C8—91no—
305C4⋊C4—91no—
306C4⋊C4—91no—
307C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
308C22⋊C4—181no—
309C22⋊C4—541no—
310C42—181no—
311C22×C3—31no—
312C22×C3—31no—
313C22×C3—61no—
314C22×C3—181no—
315C22×C3—181no—
316C22×C3—181no—
317C22×C3—361no—
318C22×C3—361no—
319C22×C3—361no—
320C22×C3#381111yescharacteristic
321C2×D3—61no—
322C2×D3—61no—
323C2×D3—61no—
324C2×D3—61no—
325C2×D3—61no—
326C2×D3—61no—
327C2×D3—62no—
328C2×D3—121no—
329C2×D3—121no—
330C2×D3—121no—
331C2×D3—121no—
332C2×D3—121no—
333C2×D3—361no—
334C2×D3—361no—
335C2×D3—361no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—31no—
340C3×C4—41no—
341C3×C4—61no—
342C3×C4—61no—
343C3×C4—121no—
344C3×C4—121no—
345C3×C4—122no—
346C3×C4—181no—
347C3×C4—361no—
348C3×C4—361no—
349C3×C4#379611yescharacteristic
350C3×C4#379811yescharacteristic
351Dic3—61no—
352Dic3—61no—
353Dic3—62no—
354Dic3—121no—
355Dic3—121no—
356Dic3—122no—
357Dic3—121no—
358Dic3—361no—
359C32—21no—
360C32#477911yescharacteristic, complemented
361C32#488511yescharacteristic, complemented
362C23—91no—
363C23—181no—
364D4—181no—
365D4—181no—
366D4—1082no—
367C2×C4—21no—
368C2×C4—91no—
369C2×C4—181no—
370C2×C4—181no—
371C2×C4—181no—
372C2×C4—181no—
373C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
374C8—181no—
375C8—181no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—61no—
383C2×C3—61no—
384C2×C3—61no—
385C2×C3—361no—
386C2×C3—721no—
387C2×C3#6385311yescharacteristic
388C2×C3#6392711yescharacteristic
389C2×C3#6395011yescharacteristic
390D3—121no—
391D3—121no—
392D3—122no—
393D3—241no—
394D3—721no—
395C22—181no—
396C22—181no—
397C22—181no—
398C22—361no—
399C22—361no—
400C22—361no—
401C22#8867711yescharacteristic
402C4—41no—
403C4—181no—
404C4—361no—
405C4—361no—
406C4#8864911yescharacteristic
407C4#8865011yescharacteristic
408C3—31no—
409C3—31no—
410C3—61no—
411C3#1059051211yescharacteristic
412C2—361no—
413C2—721no—
414C2#1172110411yescharacteristic
415C2#1172119111yescharacteristic
416C2#1172123011yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radical#11721159C2
hypercenterC4⋊C4((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 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721760C2semidirect
3. #63636⋊C3C2semidirect
4. #63634⋊C3C2semidirect
5. #11720323C2semidirect
6. #11720308C2semidirect
7. #11720307C2semidirect
26. #63636D3semidirect
27. #63634D3semidirect
236. H(3,𝔽3)#482semidirect
360. C32#4779semidirect
361. C32#4885semidirect

Maximal subgroups

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

subgroupindex
#117217602
#117211592
#63636⋊C32
#63634⋊C32
#117203232
#117203082
#117203072
#105905023
#105904243