Finite Group Databasebeta

← #13138941

Subgroups — #13138941

Order 1728.

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

normal subgroups only

This group has 421 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#11721159C211yesmaximal, characteristic
2#11721089C211yesmaximal, characteristic, complemented
3#11720553C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic, complemented
5#63644⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720315C211yesmaximal, characteristic, complemented
8#10590837—31nomaximal
9#10590704—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#6200—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#88451—21no—
19#88450—21no—
20#88447—21no—
21#63909—31no—
22#63908—31no—
23#63841—31no—
24#63841—31no—
25#63648D311yescharacteristic, complemented
26#63644D311yescharacteristic, complemented
27C3×#821—31no—
28C3×(C2×D3)⋊C8—31no—
29C3×(C2×D3)⋊C8—31no—
30C3×(C2×D3)⋊C8—61no—
31C3×#817—31no—
32#63575—31no—
33#63575—31no—
34#63564—31no—
35#63563—31no—
36C22×#272D411yescharacteristic
37C22×(C3⋊D3)⋊C3—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084D411yescharacteristic
40C4×#272—21no—
41C4×#272—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47#6200—42no—
48C8×H(3,𝔽3)—21no—
49#6180—21no—
50#4866—91no—
51#4779—91no—
52#4737—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—
60C2×C32×C8—21no—
61C2×C32×C8C2×D311yescharacteristic
62C2×C32×C8C2×D311yescharacteristic
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—61no—
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—21no—
80C2×#272—21no—
81C2×#272—21no—
82C2×(C3⋊D3)⋊C3—41no—
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C4×H(3,𝔽3)#4111yescharacteristic
86C4×H(3,𝔽3)#4111yescharacteristic
87#1084—21no—
88(C3⋊Dic3)⋊C3—41no—
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×#106—91no—
98C3×#102—91no—
99C3×C22⋊C8—91no—
100#832—91no—
101#821—31no—
102#821—91no—
103#821—91no—
104(C2×D3)⋊C8—31no—
105(C2×D3)⋊C8—31no—
106(C2×D3)⋊C8—61no—
107#818—91no—
108#817—31no—
109#817—91no—
110#809—91no—
111#807—91no—
112#807—91no—
113C22×C3⋊D3—21no—
114C22×C3×D3—31no—
115C22×C3×D3—31no—
116C22×C3×D3—61no—
117C22×C3×D3—61no—
118C2×C32×C4—21no—
119C2×C32×C4C22×D311yescharacteristic
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—21no—
124C4⋊(C3⋊D3)—21no—
125C4⋊(C3⋊D3)—21no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C3×D12—61no—
133C3×D12—61no—
134C3×C4×D3—61no—
135C3×C4×D3—61no—
136C3×C4×D3—61no—
137C3×C4×D3—61no—
138C3×C4×D3—121no—
139C3×C4×D3—121no—
140#626—122no—
141#625—122no—
142C32×C8—21no—
143C32×C8—21no—
144C32×C8—41no—
145C3×C3⋊C8—61no—
146C3×C3⋊C8—61no—
147C3×C3⋊C8—121no—
148#482—271noSylow
149C2×H(3,𝔽3)#12311yescharacteristic
150C2×H(3,𝔽3)#13511yescharacteristic
151C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
152#272—41no—
153(C3⋊D3)⋊C3—81no—
154C2×C3×D4—91no—
155C22×C3×C4—91no—
156C2×#67—181no—
157C2×#67—181no—
158C2×D12—31no—
159C2×D12—91no—
160C2×D12—181no—
161C2×C4×D3—31no—
162C2×C4×D3—31no—
163C2×C4×D3—61no—
164C2×C3×C8—31no—
165C2×C3×C8—31no—
166C2×C3×C8—61no—
167C2×C3×C8—91no—
168C2×C3×C8D3211yescharacteristic
169C3×C4⋊C4—91no—
170C3×C4⋊C4—91no—
171C3×C4⋊C4—91no—
172#217—181no—
173#217—181no—
174(C2×D3)⋊C4—181no—
175(C2×D3)⋊C4—181no—
176(C2×D3)⋊C4—181no—
177(C2×D3)⋊C4—181no—
178C4⋊Dic3—31no—
179C4⋊Dic3—31no—
180C4⋊Dic3—31no—
181C4⋊Dic3—91no—
182C4⋊Dic3—91no—
183C4⋊Dic3—91no—
184C4×Dic3—181no—
185C4×Dic3—181no—
186C4×Dic3—181no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—31no—
189C2×C3⋊C8—61no—
190C22×C32—21no—
191C22×C32D3×D411yescharacteristic
192C22×C32D3×D411yescharacteristic
193C2×C3⋊D3—41no—
194C2×C3⋊D3—41no—
195C2×C3⋊D3—41no—
196C2×C3×D3—62no—
197C2×C3×D3—61no—
198C2×C3×D3—61no—
199C2×C3×D3—61no—
200C2×C3×D3—61no—
201C2×C3×D3—61no—
202C2×C3×D3—61no—
203C2×C3×D3—121no—
204C2×C3×D3—121no—
205C2×C3×D3—121no—
206C2×C3×D3—121no—
207C2×C3×D3—121no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4#23511yescharacteristic
211C32×C4#23511yescharacteristic
212C32×C4#23711yescharacteristic
213C32×C4#23911yescharacteristic
214C3⋊Dic3—41no—
215C3⋊Dic3—41no—
216C3⋊Dic3—41no—
217C3×Dic3—61no—
218C3×Dic3—61no—
219C3×Dic3—121no—
220C3×Dic3—121no—
221C3×Dic3—121no—
222C3×Dic3—121no—
223#121—271no—
224#117—271no—
225#107—271no—
226#106—91no—
227#102—91no—
228#102—271no—
229C22⋊C8—91no—
230H(3,𝔽3)#48211yescharacteristic, complemented, Sylow
231C23×C3—91no—
232C23×C3—181no—
233C22×D3—31no—
234C22×D3—31no—
235C22×D3—61no—
236C22×D3—61no—
237C22×D3—181no—
238C3×D4—181no—
239C3×D4—181no—
240C2×C3×C4—31no—
241C2×C3×C4—31no—
242C2×C3×C4—61no—
243C2×C3×C4—91no—
244C2×C3×C4—181no—
245C2×C3×C4—181no—
246C2×C3×C4—181no—
247C2×C3×C4—181no—
248C2×C3×C4—181no—
249C2×C3×C4C2×D3211yescharacteristic
250#67—362no—
251#67—362no—
252C2×Dic3—31no—
253C2×Dic3—31no—
254C2×Dic3—61no—
255C2×Dic3—61no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—181no—
259C2×Dic3—181no—
260C2×Dic3—181no—
261D12—61no—
262D12—61no—
263D12—181no—
264D12—181no—
265D12—362no—
266C4×D3—61no—
267C4×D3—61no—
268C4×D3—61no—
269C4×D3—61no—
270C4×D3—121no—
271C4×D3—121no—
272C3×C8—21no—
273C3×C8—61no—
274C3×C8—61no—
275C3×C8—121no—
276C3×C8—181no—
277C3⋊C8—61no—
278C3⋊C8—61no—
279C3⋊C8—121no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32#88311yescharacteristic
284C2×C32#89411yescharacteristic
285C2×C32#91111yescharacteristic
286C2×C32#91411yescharacteristic
287C2×C32#91611yescharacteristic
288C2×C32#91611yescharacteristic
289C3⋊D3—81no—
290C3×D3—122no—
291C3×D3—121no—
292C3×D3—121no—
293C3×D3—241no—
294C2×D4—91no—
295C2×D4—541no—
296C22×C4—91no—
297C2×C8—91no—
298C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
299C4⋊C4—91no—
300C4⋊C4—91no—
301C4⋊C4—91no—
302C22⋊C4—541no—
303C22⋊C4—541no—
304C42—541no—
305C22×C3—31no—
306C22×C3—31no—
307C22×C3—61no—
308C22×C3—181no—
309C22×C3—181no—
310C22×C3—181no—
311C22×C3—361no—
312C22×C3—361no—
313C22×C3—361no—
314C22×C3#380211yescharacteristic
315C2×D3—61no—
316C2×D3—61no—
317C2×D3—61no—
318C2×D3—61no—
319C2×D3—61no—
320C2×D3—61no—
321C2×D3—62no—
322C2×D3—121no—
323C2×D3—121no—
324C2×D3—121no—
325C2×D3—121no—
326C2×D3—121no—
327C2×D3—361no—
328C2×D3—361no—
329C2×D3—361no—
330C3×C4—31no—
331C3×C4—31no—
332C3×C4—31no—
333C3×C4—31no—
334C3×C4—61no—
335C3×C4—61no—
336C3×C4—181no—
337C3×C4—361no—
338C3×C4—361no—
339C3×C4—361no—
340C3×C4#379911yescharacteristic
341C3×C4#380411yescharacteristic
342Dic3—61no—
343Dic3—61no—
344Dic3—121no—
345Dic3—121no—
346Dic3—121no—
347Dic3—121no—
348Dic3—361no—
349Dic3—361no—
350Dic3—361no—
351C32—21no—
352C32#477911yescharacteristic, complemented
353C32#486611yescharacteristic, complemented
354C23—91no—
355C23—181no—
356D4—181no—
357D4—181no—
358D4—1082no—
359C2×C4—91no—
360C2×C4—181no—
361C2×C4—181no—
362C2×C4—181no—
363C2×C4—181no—
364C2×C4—181no—
365C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
366C8—21no—
367C8—181no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—31no—
374C2×C3—61no—
375C2×C3—61no—
376C2×C3—61no—
377C2×C3—361no—
378C2×C3—721no—
379C2×C3#6379811yescharacteristic
380C2×C3#6380911yescharacteristic
381C2×C3#6385411yescharacteristic
382D3—121no—
383D3—121no—
384D3—122no—
385D3—241no—
386D3—721no—
387C22—181no—
388C22—181no—
389C22—181no—
390C22—361no—
391C22—361no—
392C22—361no—
393C22#8865511yescharacteristic
394C4—181no—
395C4—361no—
396C4—361no—
397C4—361no—
398C4#8865211yescharacteristic
399C4#8867111yescharacteristic
400C3—31no—
401C3—31no—
402C3—61no—
403C3#1059065211yescharacteristic
404C2—361no—
405C2—721no—
406C2#1172083911yescharacteristic
407C2#1172084311yescharacteristic
408C2#1172110211yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radical#11721159C2
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. #11721089C2semidirect
3. #11720553C2semidirect
4. #63648⋊C3C2semidirect
5. #63644⋊C3C2semidirect
6. #11720318C2semidirect
7. #11720315C2semidirect
25. #63648D3semidirect
26. #63644D3semidirect
230. H(3,𝔽3)#482semidirect
352. C32#4779semidirect
353. C32#4866semidirect

Maximal subgroups

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

subgroupindex
#117211592
#117210892
#117205532
#63648⋊C32
#63644⋊C32
#117203182
#117203152
#105908373
#105907043