Finite Group Databasebeta

← #13140550

Subgroups — #13140550

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#64147⋊C3C211yesmaximal, characteristic
2#11721081C211yesmaximal, characteristic, complemented
3#11720578C211yesmaximal, characteristic, complemented
4#63659⋊C3C211yesmaximal, characteristic, complemented
5#11720317C211yesmaximal, characteristic, complemented
6#11720315C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10591425—31nomaximal
9#10591423—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15#3757⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88450—21no—
20#88447—21no—
21#64147D311yescharacteristic
22C3×#938—31no—
23#63849—31no—
24#63841—31no—
25#63659D311yescharacteristic, complemented
26C3×#832—31no—
27C3×#832—31no—
28C3×#832—31no—
29C3×#832—61no—
30#63575—31no—
31#63575—31no—
32#63574—31no—
33#63573—31no—
34#63557—31no—
35(C2×D3)⋊(C3⋊C8)—31no—
36C23×H(3,𝔽3)—21no—
37C22×(C3⋊D3)⋊C3D411yescharacteristic
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42#638⋊C3—42no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C4×(C3⋊D3)⋊C3—21no—
47C4×(C3⋊D3)⋊C3—21no—
48#6180—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50#5035—91no—
51#4784—91no—
52#4779—91no—
53C2×C32×D4—21no—
54C2×C32×D4C2×D311yescharacteristic
55C2×C32×D4C2×D311yescharacteristic
56C2×#638—21no—
57C2×C4×C3⋊D3C2×D311yescharacteristic
58C2×C3×#67—61no—
59C2×C3×C4×D3—31no—
60#3757—21no—
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
64C3×#217—61no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—61no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—61no—
74#3722—61no—
75(C2×D3)⋊Dic3—61no—
76Dic32—61no—
77Dic32—61no—
78C22×H(3,𝔽3)—41no—
79C22×H(3,𝔽3)—41no—
80C22×H(3,𝔽3)—41no—
81C22×H(3,𝔽3)C2×D411yescharacteristic
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—
91C3×#121—91no—
92#938—31no—
93#938—91no—
94#926—91no—
95#892—91no—
96#892—91no—
97C3×#102—91no—
98C3×#102—91no—
99#832—31no—
100#832—31no—
101#832—31no—
102#832—61no—
103#832—91no—
104#830—91no—
105(C2×D3)⋊C8—91no—
106(C2×D3)⋊C8—91no—
107#818—91no—
108#817—91no—
109#808—91no—
110#808—91no—
111#807—91no—
112#807—91no—
113C23×C32—21no—
114C23×C32—21no—
115C23×C32—41no—
116C22×C3⋊D3#6711yescharacteristic
117C22×C3×D3—31no—
118C32×D4—22no—
119C32×D4—21no—
120C32×D4—21no—
121C32×D4—21no—
122C32×D4—21no—
123C32×D4—41no—
124C2×C32×C4—21no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127#638—42no—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3#6711yescharacteristic
131C4×C3⋊D3—21no—
132C4×C3⋊D3—21no—
133C3×#67—122no—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—121no—
140C3×C4×D3—61no—
141C3×C4×D3—61no—
142C3⋊(C3⋊C8)—21no—
143C3×C3⋊C8—61no—
144C3×C3⋊C8—61no—
145C3×C3⋊C8—61no—
146C3×C3⋊C8—121no—
147#482—271noSylow
148C2×H(3,𝔽3)—81no—
149C2×H(3,𝔽3)#12311yescharacteristic
150C2×H(3,𝔽3)#13511yescharacteristic
151C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
152(C3⋊D3)⋊C3—41no—
153C2×C3×D4—31no—
154C2×C3×D4—31no—
155C2×C3×D4—61no—
156C2×C3×D4—181no—
157C2×C3×D4D3211yescharacteristic
158C22×C3×C4—91no—
159C2×#67—61no—
160C2×#67—181no—
161C2×C4×D3—31no—
162C2×C4×D3—91no—
163C2×C3×C8—91no—
164C2×C3×C8—91no—
165C3×C4⋊C4—91no—
166C3×C4⋊C4—91no—
167C3×C4⋊C4—91no—
168C3×C22⋊C4—181no—
169#217—61no—
170#217—181no—
171#217—181no—
172(C2×D3)⋊C4—181no—
173(C2×D3)⋊C4—181no—
174C4⋊Dic3—31no—
175C4⋊Dic3—31no—
176C4⋊Dic3—31no—
177C4⋊Dic3—31no—
178C4⋊Dic3—61no—
179C4⋊Dic3—91no—
180C4⋊Dic3—91no—
181C4×Dic3—181no—
182C4×Dic3—181no—
183C4×Dic3—181no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—61no—
188C2×C3⋊C8—91no—
189C22×C32—21no—
190C22×C32—42no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32—81no—
198C22×C32—81no—
199C22×C32D3×D411yescharacteristic
200C22×C32C2×#6711yescharacteristic
201C2×C3⋊D3—21no—
202C2×C3⋊D3—21no—
203C2×C3⋊D3—21no—
204C2×C3×D3—61no—
205C2×C3×D3—61no—
206C2×C3×D3—61no—
207C32×C4—21no—
208C32×C4—21no—
209C32×C4#23511yescharacteristic
210C32×C4#23711yescharacteristic
211C32×C4#23711yescharacteristic
212C32×C4#23711yescharacteristic
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—91no—
223#117—271no—
224#107—271no—
225#106—271no—
226#102—91no—
227#102—91no—
228C22⋊C8—271no—
229H(3,𝔽3)#48211yescharacteristic, complemented, Sylow
230C23×C3—21no—
231C23×C3—61no—
232C23×C3—61no—
233C23×C3—91no—
234C23×C3—121no—
235C22×D3—31no—
236C22×D3—91no—
237C3×D4—21no—
238C3×D4—21no—
239C3×D4—61no—
240C3×D4—61no—
241C3×D4—61no—
242C3×D4—61no—
243C3×D4—62no—
244C3×D4—121no—
245C3×D4—362no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—61no—
249C2×C3×C4—91no—
250C2×C3×C4—181no—
251C2×C3×C4—181no—
252C2×C3×C4—181no—
253C2×C3×C4—181no—
254C2×C3×C4—181no—
255C2×C3×C4C2×D3211yescharacteristic
256#67—122no—
257#67—362no—
258C2×Dic3—31no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—91no—
264C2×Dic3—121no—
265C2×Dic3—181no—
266C2×Dic3—181no—
267C4×D3—61no—
268C4×D3—61no—
269C4×D3—181no—
270C4×D3—181no—
271C3×C8—181no—
272C3×C8—181no—
273C3⋊C8—61no—
274C3⋊C8—61no—
275C3⋊C8—61no—
276C3⋊C8—121no—
277C3⋊C8—181no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—82no—
282C2×C32—81no—
283C2×C32—81no—
284C2×C32#88311yescharacteristic
285C2×C32#91111yescharacteristic
286C2×C32#91611yescharacteristic
287C2×C32#93211yescharacteristic
288C2×C32#93511yescharacteristic
289C2×C32#95011yescharacteristic
290C3⋊D3—41no—
291C3×D3—121no—
292C2×D4—181no—
293C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
294C22×C4—91no—
295C2×C8—91no—
296C2×C8—91no—
297C4⋊C4—91no—
298C4⋊C4—91no—
299C4⋊C4—91no—
300C22⋊C4—181no—
301C22⋊C4—541no—
302C42—541no—
303C22×C3—31no—
304C22×C3—31no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—61no—
309C22×C3—121no—
310C22×C3—121no—
311C22×C3—121no—
312C22×C3—121no—
313C22×C3—121no—
314C22×C3—121no—
315C22×C3—122no—
316C22×C3—181no—
317C22×C3—181no—
318C22×C3—181no—
319C22×C3—241no—
320C22×C3—241no—
321C22×C3D3×#6711yescharacteristic
322C2×D3—61no—
323C2×D3—61no—
324C2×D3—61no—
325C2×D3—181no—
326C2×D3—181no—
327C2×D3—181no—
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#380411yescharacteristic
339C3×C4#380511yescharacteristic
340Dic3—61no—
341Dic3—121no—
342Dic3—121no—
343Dic3—121no—
344Dic3—121no—
345Dic3—181no—
346Dic3—241no—
347Dic3—361no—
348Dic3—361no—
349C32—21no—
350C32#477911yescharacteristic, complemented
351C32#503511yescharacteristic, complemented
352C23—21no—
353C23—91no—
354D4—21no—
355D4—21no—
356D4—362no—
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—181no—
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—81no—
376C2×C3—241no—
377C2×C3—241no—
378C2×C3—242no—
379C2×C3—361no—
380C2×C3#6393211yescharacteristic
381C2×C3#6393511yescharacteristic
382C2×C3#6396911yescharacteristic
383D3—121no—
384D3—361no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—181no—
389C22—181no—
390C22—181no—
391C22#8867611yescharacteristic
392C4—181no—
393C4—361no—
394C4—361no—
395C4—361no—
396C4#8867111yescharacteristic
397C4#8867211yescharacteristic
398C3—31no—
399C3—31no—
400C3—61no—
401C3#1059142211yescharacteristic
402C2—81no—
403C2—361no—
404C2#1172119711yescharacteristic
405C2#1172120011yescharacteristic
406C2#1172131111yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radical#64147⋊C3C2
hypercenterC2×D4((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×D4

Complements

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

NH ≅ G/Ntype
2. #11721081C2semidirect
3. #11720578C2semidirect
4. #63659⋊C3C2semidirect
5. #11720317C2semidirect
6. #11720315C2semidirect
7. #11720307C2semidirect
25. #63659D3semidirect
229. H(3,𝔽3)#482semidirect
350. C32#4779semidirect
351. C32#5035semidirect

Maximal subgroups

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

subgroupindex
#64147⋊C32
#117210812
#117205782
#63659⋊C32
#117203172
#117203152
#117203072
#105914253
#105914233