Finite Group Databasebeta

← #13138005

Subgroups — #13138005

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#64101⋊C3C211yesmaximal, characteristic, complemented
2#11721156C211yesmaximal, characteristic
3#11720531C211yesmaximal, characteristic, complemented
4#63634⋊C3C211yesmaximal, characteristic, complemented
5#11720324C211yesmaximal, characteristic, complemented
6#11720317C211yesmaximal, characteristic, complemented
7#11720314C211yesmaximal, characteristic, complemented
8#10590512—31nomaximal
9#10590502—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×#6200—21no—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15#3752⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88451—21no—
21#64101D311yescharacteristic, complemented
22C3×#892—31no—
23#63916—31no—
24#63908—31no—
25#63634D311yescharacteristic, complemented
26C3×#809—31no—
27C3×#809—31no—
28C3×#809—61no—
29C3×#807—31no—
30#63573—31no—
31#63573—31no—
32#63563—31no—
33#63562—31no—
34#63557—31no—
35(C2×D3)⋊(C3⋊C8)—31no—
36C22×#272—21no—
37C22×(C3⋊D3)⋊C3D411yescharacteristic
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
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—
48#6180—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50#5003—91no—
51#4885—91no—
52#4779—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—
60C32×C4⋊C4—21no—
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C4⋊C4C2×D311yescharacteristic
63#3752—21no—
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66C4×C3⋊Dic3—21no—
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×(C2×D3)⋊C4—61no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4×Dic3—61no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—61no—
76(C2×D3)⋊Dic3—61no—
77(C2×D3)⋊Dic3—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)—41no—
86C4×H(3,𝔽3)#4111yescharacteristic
87C4×H(3,𝔽3)#4111yescharacteristic
88(C3⋊Dic3)⋊C3—21no—
89(C3⋊Dic3)⋊C3—41no—
90(C3⋊Dic3)⋊C3—41no—
91C3×#117—91no—
92#938—91no—
93#930—91no—
94#895—91no—
95#892—31no—
96#892—91no—
97C3×#107—91no—
98C3×#102—91no—
99#832—91no—
100#830—91no—
101#821—91no—
102(C2×D3)⋊C8—91no—
103(C2×D3)⋊C8—91no—
104#817—91no—
105#809—31no—
106#809—31no—
107#809—61no—
108#809—91no—
109#808—91no—
110#808—91no—
111#807—31no—
112#807—91no—
113C22×C3⋊D3#6711yescharacteristic
114C22×C3×D3—31no—
115C22×C3×D3—61no—
116C22×C3×D3—61no—
117C22×C3×D3—121no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—41no—
122C2×C32×C4C22×D311yescharacteristic
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3#6711yescharacteristic
127C4×C3⋊D3—21no—
128C4×C3⋊D3—21no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
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—
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)#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—91no—
164C2×C3×C8—91no—
165C3×C4⋊C4—31no—
166C3×C4⋊C4—31no—
167C3×C4⋊C4—61no—
168C3×C4⋊C4—91no—
169C3×C4⋊C4—91no—
170C3×C4⋊C4D3211yescharacteristic
171C3×C22⋊C4—181no—
172C3×C42—181no—
173#217—181no—
174(C2×D3)⋊C4—61no—
175(C2×D3)⋊C4—181no—
176(C2×D3)⋊C4—181no—
177(C2×D3)⋊C4—181no—
178C4⋊Dic3—31no—
179C4⋊Dic3—31no—
180C4⋊Dic3—91no—
181C4⋊Dic3—91no—
182C4×Dic3—61no—
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×C32D3×D411yescharacteristic
191C22×C32C2×#6711yescharacteristic
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—41no—
210C32×C4—41no—
211C32×C4—81no—
212C32×C4#23511yescharacteristic
213C32×C4#23511yescharacteristic
214C32×C4#23511yescharacteristic
215C32×C4#23911yescharacteristic
216C3⋊Dic3—21no—
217C3⋊Dic3—41no—
218C3⋊Dic3—41no—
219C3×Dic3—61no—
220C3×Dic3—121no—
221C3×Dic3—121no—
222#121—271no—
223#117—91no—
224#107—91no—
225#106—271no—
226#102—91no—
227#102—271no—
228C22⋊C8—271no—
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—21no—
240C2×C3×C4—31no—
241C2×C3×C4—31no—
242C2×C3×C4—61no—
243C2×C3×C4—61no—
244C2×C3×C4—61no—
245C2×C3×C4—91no—
246C2×C3×C4—121no—
247C2×C3×C4—181no—
248C2×C3×C4—181no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4C2×D3211yescharacteristic
252#67—362no—
253#67—362no—
254C2×Dic3—31no—
255C2×Dic3—61no—
256C2×Dic3—61no—
257C2×Dic3—91no—
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—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#89411yescharacteristic
282C2×C32#91111yescharacteristic
283C2×C32#91911yescharacteristic
284C2×C32#92911yescharacteristic
285C2×C32#94911yescharacteristic
286C2×C32#95011yescharacteristic
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—91no—
297C4⋊C4—91no—
298C4⋊C4—91no—
299C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
300C22⋊C4—181no—
301C22⋊C4—541no—
302C42—181no—
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×#6711yescharacteristic
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—41no—
333C3×C4—61no—
334C3×C4—61no—
335C3×C4—121no—
336C3×C4—121no—
337C3×C4—181no—
338C3×C4—241no—
339C3×C4—361no—
340C3×C4—361no—
341C3×C4#379811yescharacteristic
342C3×C4#379911yescharacteristic
343Dic3—61no—
344Dic3—121no—
345Dic3—121no—
346Dic3—181no—
347Dic3—361no—
348Dic3—361no—
349C32—21no—
350C32#488511yescharacteristic, complemented
351C32#500311yescharacteristic, complemented
352C23—91no—
353C23—181no—
354D4—181no—
355D4—181no—
356D4—1082no—
357C2×C4—21no—
358C2×C4—91no—
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—361no—
376C2×C3—721no—
377C2×C3#6385011yescharacteristic
378C2×C3#6392511yescharacteristic
379C2×C3#6394911yescharacteristic
380D3—121no—
381D3—241no—
382D3—241no—
383D3—242no—
384D3—361no—
385C22—181no—
386C22—181no—
387C22—181no—
388C22—361no—
389C22—361no—
390C22—361no—
391C22#8867611yescharacteristic
392C4—41no—
393C4—181no—
394C4—361no—
395C4—361no—
396C4#8865011yescharacteristic
397C4#8865211yescharacteristic
398C3—31no—
399C3—31no—
400C3—61no—
401C3#1059042411yescharacteristic
402C2—361no—
403C2—721no—
404C2#1172110011yescharacteristic
405C2#1172119511yescharacteristic
406C2#1172123511yescharacteristic
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 radical#11721156C2
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. #64101⋊C3C2semidirect
3. #11720531C2semidirect
4. #63634⋊C3C2semidirect
5. #11720324C2semidirect
6. #11720317C2semidirect
7. #11720314C2semidirect
21. #64101D3semidirect
25. #63634D3semidirect
229. H(3,𝔽3)#482semidirect
350. C32#4885semidirect
351. C32#5003semidirect

Maximal subgroups

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

subgroupindex
#64101⋊C32
#117211562
#117205312
#63634⋊C32
#117203242
#117203172
#117203142
#105905123
#105905023