Finite Group Databasebeta

← #13139947

Subgroups — #13139947

Order 1728.

447 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 75 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#11721826C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic
3#63657⋊C3C211yesmaximal, characteristic, complemented
4#11720308C211yesmaximal, characteristic, complemented
5#11720308C211yesmaximal, characteristic, complemented
6#11720307C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10591136—31nomaximal
9#10591090—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13#88552—21no—
14C4⋊#1084C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16C4×#1084—21no—
17#3752⋊C3—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64139D311yescharacteristic
22C3×#930—31no—
23C3×#926—31no—
24C3×#926—31no—
25C3×#926—61no—
26#63657D311yescharacteristic, complemented
27C3×#830—31no—
28#63576—31no—
29#63575—31no—
30#63574—31no—
31#63573—31no—
32#63564—31no—
33#63564—31no—
34#63562—31no—
35#63562—31no—
36C23×H(3,𝔽3)D411yescharacteristic
37C22×(C3⋊D3)⋊C3—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)D411yescharacteristic
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—21no—
43C2×#1084—21no—
44#638⋊C3—42no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50#5035—91no—
51#5003—91no—
52#4737—91no—
53C22×C32×C4—21no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C2×D311yescharacteristic
56C2×#638—21no—
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×C3×#67—61no—
59C2×C3×D12—31no—
60#3752—21no—
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
64C3×#217—62no—
65C3×#217—61no—
66C3×#217—61no—
67C3×(C2×D3)⋊C4—61no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—61no—
74C3×C4⋊Dic3—61no—
75C3×C4×Dic3—62no—
76C3×C4×Dic3—61no—
77C3×C4×Dic3—61no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80C22×H(3,𝔽3)—21no—
81C22×H(3,𝔽3)—21no—
82C22×H(3,𝔽3)—21no—
83C22×H(3,𝔽3)C2×D411yescharacteristic
84C2×(C3⋊D3)⋊C3—41no—
85C2×(C3⋊D3)⋊C3—41no—
86C2×(C3⋊D3)⋊C3—41no—
87C4×H(3,𝔽3)—21no—
88C4×H(3,𝔽3)#4111yescharacteristic
89C4×H(3,𝔽3)#4111yescharacteristic
90#1084—41no—
91#1084—41no—
92(C3⋊Dic3)⋊C3—41no—
93C3×#121—91no—
94C3×#117—91no—
95#930—31no—
96#930—91no—
97#926—31no—
98#926—31no—
99#926—61no—
100C3×C22⋊C8—91no—
101#832—91no—
102#832—91no—
103#830—31no—
104#830—91no—
105#821—91no—
106#821—91no—
107#818—91no—
108#817—91no—
109#809—91no—
110#809—91no—
111#808—91no—
112#808—91no—
113#807—91no—
114#807—91no—
115C23×C32—21no—
116C23×C32D1211yescharacteristic
117C23×C32#6711yescharacteristic
118C22×C3⋊D3—21no—
119C22×C3×D3—61no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4D1211yescharacteristic
129C2×C32×C4#6711yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132#638—42no—
133C2×C3⋊Dic3—21no—
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)—21no—
136C3×#67—122no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C3×D12—61no—
145C3×D12—61no—
146C3⋊(C3⋊C8)—21no—
147C3⋊(C3⋊C8)—21no—
148C3×C3⋊C8—61no—
149C3×C3⋊C8—61no—
150#482—271noSylow
151C2×H(3,𝔽3)—41no—
152C2×H(3,𝔽3)#12311yescharacteristic
153C2×H(3,𝔽3)#13511yescharacteristic
154C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
155(C3⋊D3)⋊C3—81no—
156C2×C3×D4—91no—
157C2×C3×D4—181no—
158C22×C3×C4—31no—
159C22×C3×C4—31no—
160C22×C3×C4—61no—
161C22×C3×C4D3211yescharacteristic
162C2×#67—61no—
163C2×#67—181no—
164C2×D12—31no—
165C2×D12—91no—
166C2×C3×C8—91no—
167C2×C3×C8—91no—
168C3×C4⋊C4—91no—
169C3×C4⋊C4—91no—
170C3×C4⋊C4—91no—
171C3×C22⋊C4—181no—
172C3×C22⋊C4—181no—
173C3×C42—181no—
174#217—61no—
175#217—61no—
176#217—62no—
177(C2×D3)⋊C4—61no—
178(C2×D3)⋊C4—181no—
179C4⋊Dic3—31no—
180C4⋊Dic3—31no—
181C4⋊Dic3—31no—
182C4⋊Dic3—31no—
183C4⋊Dic3—31no—
184C4⋊Dic3—61no—
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—22no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—21no—
201C22×C32—21no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32C2×D1211yescharacteristic
205C22×C32C2×#6711yescharacteristic
206C2×C3⋊D3—41no—
207C2×C3⋊D3—41no—
208C2×C3⋊D3—41no—
209C2×C3×D3—121no—
210C2×C3×D3—121no—
211C2×C3×D3—121no—
212C32×C4—22no—
213C32×C4—21no—
214C32×C4—21no—
215C32×C4—21no—
216C32×C4—21no—
217C32×C4#23711yescharacteristic
218C32×C4#23711yescharacteristic
219C32×C4#23711yescharacteristic
220C32×C4#23711yescharacteristic
221C3⋊Dic3—41no—
222C3×Dic3—122no—
223C3×Dic3—122no—
224C3×Dic3—121no—
225C3×Dic3—121no—
226C3×Dic3—121no—
227C3×Dic3—121no—
228C3×Dic3—121no—
229#121—91no—
230#117—91no—
231#107—271no—
232#106—271no—
233#102—271no—
234#102—271no—
235C22⋊C8—91no—
236H(3,𝔽3)#48211yescharacteristic, complemented, Sylow
237C23×C3—31no—
238C23×C3—31no—
239C23×C3—61no—
240C23×C3—181no—
241C23×C3#62611yescharacteristic
242C22×D3—61no—
243C22×D3—181no—
244C3×D4—181no—
245C3×D4—181no—
246C3×D4—362no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—62no—
258C2×C3×C4—62no—
259C2×C3×C4—61no—
260C2×C3×C4—61no—
261C2×C3×C4—181no—
262C2×C3×C4—181no—
263C2×C3×C4—181no—
264C2×C3×C4#62611yescharacteristic
265C2×C3×C4C2×D3211yescharacteristic
266#67—122no—
267#67—362no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—62no—
273C2×Dic3—62no—
274C2×Dic3—61no—
275C2×Dic3—181no—
276D12—61no—
277D12—61no—
278D12—181no—
279D12—181no—
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—42no—
290C2×C32—41no—
291C2×C32—41no—
292C2×C32#88611yescharacteristic
293C2×C32#90411yescharacteristic
294C2×C32#90811yescharacteristic
295C2×C32#93211yescharacteristic
296C2×C32#93511yescharacteristic
297C2×C32#95011yescharacteristic
298C3⋊D3—81no—
299C3×D3—241no—
300C2×D4—91no—
301C2×D4—181no—
302C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
303C2×C8—91no—
304C2×C8—91no—
305C4⋊C4—91no—
306C4⋊C4—91no—
307C4⋊C4—91no—
308C22⋊C4—181no—
309C22⋊C4—181no—
310C42—181no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—31no—
315C22×C3—31no—
316C22×C3—61no—
317C22×C3—61no—
318C22×C3—61no—
319C22×C3—61no—
320C22×C3—61no—
321C22×C3—61no—
322C22×C3—62no—
323C22×C3—61no—
324C22×C3—121no—
325C22×C3—121no—
326C22×C3—361no—
327C22×C3—361no—
328C22×C3—361no—
329C22×C3C2×#62611yescharacteristic
330C2×D3—121no—
331C2×D3—121no—
332C2×D3—121no—
333C2×D3—361no—
334C2×D3—361no—
335C2×D3—361no—
336C3×C4—21no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—31no—
340C3×C4—31no—
341C3×C4—61no—
342C3×C4—61no—
343C3×C4—62no—
344C3×C4—61no—
345C3×C4—61no—
346C3×C4—361no—
347C3×C4—361no—
348C3×C4—361no—
349C3×C4#380511yescharacteristic
350C3×C4#380511yescharacteristic
351Dic3—121no—
352Dic3—121no—
353Dic3—121no—
354Dic3—121no—
355Dic3—122no—
356Dic3—122no—
357Dic3—121no—
358Dic3—361no—
359C32—21no—
360C32#473711yescharacteristic, complemented
361C32#503511yescharacteristic, complemented
362C23—181no—
363C23#620011yescharacteristic
364D4—181no—
365D4—181no—
366D4—362no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—181no—
370C2×C4—181no—
371C2×C4—181no—
372C2×C4#620011yescharacteristic
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—41no—
383C2×C3—61no—
384C2×C3—61no—
385C2×C3—61no—
386C2×C3—121no—
387C2×C3—121no—
388C2×C3—122no—
389C2×C3—721no—
390C2×C3#6382911yescharacteristic
391C2×C3#6383011yescharacteristic
392C2×C3#6396211yescharacteristic
393D3—241no—
394D3—721no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—361no—
399C22—361no—
400C22—361no—
401C22C2×#620011yescharacteristic
402C4—21no—
403C4—361no—
404C4—361no—
405C4—361no—
406C4#8867211yescharacteristic
407C4#8867211yescharacteristic
408C3—31no—
409C3—31no—
410C3—61no—
411C3#1059109311yescharacteristic
412C2—41no—
413C2—721no—
414C2#1172107111yescharacteristic
415C2#1172107311yescharacteristic
416C2#1172130711yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radical#64139⋊C3C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#482
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11721826C2semidirect
3. #63657⋊C3C2semidirect
4. #11720308C2semidirect
5. #11720308C2semidirect
6. #11720307C2semidirect
7. #11720306C2semidirect
26. #63657D3semidirect
236. H(3,𝔽3)#482semidirect
360. C32#4737semidirect
361. C32#5035semidirect

Maximal subgroups

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

subgroupindex
#117218262
#64139⋊C32
#63657⋊C32
#117203082
#117203072
#117203062
#105911363
#105910903