Finite Group Databasebeta

← #13137991

Subgroups — #13137991

Order 1728.

514 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 81 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#11721761C211yesmaximal, characteristic
2#11721084C211yesmaximal, characteristic, complemented
3C2×#88437C211yesmaximal, characteristic
4#63637⋊C3C211yesmaximal, characteristic, complemented
5#63634⋊C3C211yesmaximal, characteristic, complemented
6#11720324C211yesmaximal, characteristic
7#11720303C211yesmaximal, characteristic, complemented
8#10590590—31nomaximal
9#10590478—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×#272—21no—
12C2×#634⋊C3C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#88542—21no—
15C4×#1084C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88448—21no—
20#88447—21no—
21#88437C414yescomplemented
22C3×#893—31no—
23C3×#893—31no—
24C3×#893—61no—
25#63844—31no—
26Dic3×Dic6—31no—
27C2×#3716—31no—
28C2×#3715—31no—
29#63637D311yescharacteristic, complemented
30#63634D311yescharacteristic, complemented
31C3×#810—31no—
32C3×#807—31no—
33#63563—31no—
34#63562—31no—
35#63554—31no—
36#63554—31no—
37C22×#272—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084D411yescharacteristic
41C2×#1084D411yescharacteristic
42C4⋊#272—21no—
43C4⋊#272C2×C412yes—
44C4×#272—42no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47#634⋊C3—21no—
48#634⋊C3C2×C412yescomplemented
49(C3⋊(C3⋊C8))⋊C3—21no—
50(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
51#5018—91no—
52#4889—91no—
53#4817—91no—
54C2×#634C2×D311yescharacteristic
55C2×C3×D12—31no—
56C2×C3×D12—31no—
57C2×C3×D12—61no—
58C2×C3×C4×D3—61no—
59C2×C3×C4×D3—61no—
60C2×C3×C4×D3—62no—
61C2×C3×Dic6—31no—
62C32×C4⋊C4—21no—
63C32×C4⋊C4C2×D311yescharacteristic
64C32×C4⋊C4C2×D311yescharacteristic
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×(C2×D3)⋊C4—61no—
69C3×(C2×D3)⋊C4—61no—
70C3×(C2×D3)⋊C4—62no—
71C3×C4⋊Dic3—31no—
72C3×C4×Dic3—31no—
73C3×C4×Dic3—31no—
74C3×C4×Dic3—61no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77#3724—61no—
78Dic3⋊Dic3—61no—
79Dic32—61no—
80Dic32—61no—
81#3716—34no—
82#3715—34no—
83C22×H(3,𝔽3)C2×D411yescharacteristic
84C2×#272—22no—
85C2×#272—41no—
86C2×#272—41no—
87C4×H(3,𝔽3)—41no—
88C4×H(3,𝔽3)C22×C411yescharacteristic
89C4×H(3,𝔽3)#4111yescharacteristic
90#1084—21no—
91#1084—21no—
92(C3⋊Dic3)⋊C3—22no—
93(C3⋊Dic3)⋊C3—41no—
94C3×C4×D4—91no—
95Q8×Dic3—91no—
96C2×#215—91no—
97C2×#214—91no—
98C2×#204—91no—
99#893—31no—
100#893—31no—
101#893—61no—
102#887—91no—
103C4×Dic6—91no—
104C3×#107—91no—
105C3×#103—91no—
106#832—91no—
107#821—91no—
108#815—91no—
109#815—91no—
110#810—31no—
111#810—91no—
112#809—91no—
113#807—31no—
114#807—91no—
115#803—91no—
116C22×C3×D3—62no—
117C22×C3×D3—61no—
118C22×C3×D3—61no—
119C2×C32×C4—22no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—21no—
127#634—21no—
128#634C4×D312yescomplemented
129C2×C3×Dic3—31no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—31no—
132C2×C3×Dic3—31no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C3×D12—32no—
138C3×D12—32no—
139C3×D12—62no—
140C3×D12—62no—
141C3×D12—61no—
142C3×D12—61no—
143C3×C4×D3—124no—
144C3×C4×D3—122no—
145C3×C4×D3—122no—
146C3×Dic6—32no—
147C3×Dic6—61no—
148C3⋊(C3⋊C8)—21no—
149C3⋊(C3⋊C8)C4×D312yescomplemented
150C3×C3⋊C8—32no—
151C3×C3⋊C8—61no—
152#440—271noSylow
153C2×H(3,𝔽3)C4×D411yescharacteristic
154C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
155C2×H(3,𝔽3)#13711yescharacteristic
156#272—42no—
157C2×C3×Q8—91no—
158C2×C3×D4—91no—
159C22×C3×C4—181no—
160C2×D12—31no—
161C2×D12—31no—
162C2×D12—61no—
163C2×C4×D3—61no—
164C2×C4×D3—61no—
165C2×C4×D3—62no—
166C2×Dic6—31no—
167C2×Dic6—91no—
168C2×C3×C8—91no—
169C2×C3×C8—91no—
170C3×C4⋊C4—31no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—61no—
173C3×C4⋊C4—91no—
174C3×C4⋊C4D3211yescharacteristic
175C3×C22⋊C4—181no—
176C3×C42—91no—
177#215—94no—
178#214—94no—
179(C2×D3)⋊C4—61no—
180(C2×D3)⋊C4—61no—
181(C2×D3)⋊C4—62no—
182C4⋊Dic3—31no—
183C4⋊Dic3—91no—
184C4⋊Dic3—181no—
185Dic3⋊C4—181no—
186Dic3⋊C4—181no—
187C4×Dic3—31no—
188C4×Dic3—31no—
189C4×Dic3—61no—
190C4×Dic3—181no—
191C4×Dic3—181no—
192C4×Dic3—181no—
193C2×C3⋊C8—31no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—91no—
196C2×C3⋊C8—91no—
197#204—94no—
198C22×C32—21no—
199C22×C32D3×D411yescharacteristic
200C22×C32D3×D411yescharacteristic
201C2×C3×D3—64no—
202C2×C3×D3—62no—
203C2×C3×D3—62no—
204C2×C3×D3—122no—
205C2×C3×D3—122no—
206C2×C3×D3—121no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—42no—
213C32×C4—41no—
214C32×C4—41no—
215C32×C4C2×C4×D311yescharacteristic
216C32×C4C2×C4×D311yescharacteristic
217C32×C4#23711yescharacteristic
218C32×C4#23911yescharacteristic
219C3⋊Dic3—22no—
220C3⋊Dic3—41no—
221C3×Dic3—62no—
222C3×Dic3—62no—
223C3×Dic3—61no—
224C3×Dic3—61no—
225C3×Dic3—61no—
226C3×Dic3—61no—
227C3×Dic3—121no—
228C3×Dic3—121no—
229C2×SD16—271no—
230C4×Q8—271no—
231C4×D4—91no—
232#107—91no—
233#103—91no—
234#102—271no—
235#97—271no—
236H(3,𝔽3)#44011yescharacteristic, complemented, Sylow
237C23×C3—181no—
238C22×D3—61no—
239C22×D3—61no—
240C22×D3—62no—
241C3×Q8—92no—
242C3×Q8—181no—
243C3×D4—92no—
244C3×D4—181no—
245C2×C3×C4—21no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—62no—
251C2×C3×C4—61no—
252C2×C3×C4—91no—
253C2×C3×C4—91no—
254C2×C3×C4—181no—
255C2×C3×C4—181no—
256C2×C3×C4—362no—
257C2×C3×C4C2×D3211yescharacteristic
258C2×Dic3—31no—
259C2×Dic3—31no—
260C2×Dic3—31no—
261C2×Dic3—31no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—181no—
267C2×Dic3—181no—
268D12—32no—
269D12—32no—
270D12—62no—
271D12—61no—
272D12—61no—
273D12—62no—
274C4×D3—122no—
275C4×D3—122no—
276C4×D3—124no—
277Dic6—32no—
278Dic6—61no—
279Dic6—92no—
280Dic6—181no—
281C3×C8—92no—
282C3×C8—181no—
283C3⋊C8—32no—
284C3⋊C8—61no—
285C3⋊C8—92no—
286C3⋊C8—181no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—21no—
290C2×C32#67⋊C411yescharacteristic
291C2×C32#89311yescharacteristic
292C2×C32#91111yescharacteristic
293C2×C32#91411yescharacteristic
294C2×C32#91811yescharacteristic
295C2×C32#91811yescharacteristic
296C3×D3—124no—
297C3×D3—122no—
298C3×D3—122no—
299C2×Q8—91no—
300C2×D4—91no—
301C22×C4—181no—
302SD16—274no—
303C2×C8—91no—
304C2×C8—91no—
305C4⋊C4—91no—
306C4⋊C4—541no—
307C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
308C22⋊C4—181no—
309C42—91no—
310C42—541no—
311C22×C3—31no—
312C22×C3—31no—
313C22×C3—61no—
314C22×C3—182no—
315C22×C3—361no—
316C22×C3—361no—
317C22×C3#381111yescharacteristic
318C2×D3—62no—
319C2×D3—62no—
320C2×D3—64no—
321C2×D3—121no—
322C2×D3—121no—
323C2×D3—121no—
324C2×D3—121no—
325C2×D3—122no—
326C2×D3—122no—
327C3×C4—31no—
328C3×C4—31no—
329C3×C4—31no—
330C3×C4—31no—
331C3×C4—41no—
332C3×C4—61no—
333C3×C4—61no—
334C3×C4—121no—
335C3×C4—121no—
336C3×C4—122no—
337C3×C4—182no—
338C3×C4—181no—
339C3×C4—181no—
340C3×C4—361no—
341C3×C4#379611yescharacteristic
342C3×C4C4×D3211yescharacteristic
343Dic3—61no—
344Dic3—61no—
345Dic3—61no—
346Dic3—61no—
347Dic3—62no—
348Dic3—62no—
349Dic3—121no—
350Dic3—121no—
351Dic3—182no—
352Dic3—361no—
353C32—21no—
354C32#481711yescharacteristic, complemented
355C32#488911yescharacteristic, complemented
356C23—181no—
357Q8—92no—
358Q8—181no—
359D4—92no—
360D4—181no—
361C2×C4—21no—
362C2×C4—91no—
363C2×C4—91no—
364C2×C4—181no—
365C2×C4—181no—
366C2×C4—362no—
367C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
368C8—92no—
369C8—181no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—61no—
377C2×C3—61no—
378C2×C3—61no—
379C2×C3—362no—
380C2×C3#6388111yescharacteristic
381C2×C3#6393711yescharacteristic
382C2×C3#6394511yescharacteristic
383D3—122no—
384D3—122no—
385D3—124no—
386C22—182no—
387C22—361no—
388C22—361no—
389C22#8867711yescharacteristic
390C4—41no—
391C4—181no—
392C4—182no—
393C4—181no—
394C4—361no—
395C4#8864911yescharacteristic
396C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
397C3—31no—
398C3—31no—
399C3—61no—
400C3#1059048411yescharacteristic
401C2—362no—
402C2#1172112411yescharacteristic
403C2#1172119311yescharacteristic
404C2#1172122711yescharacteristic
subgroupG/H
centerC22#88677
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×C4⋊#272C22
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#440
LCSG ▷ 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
2. #11721084C2semidirect
4. #63637⋊C3C2semidirect
5. #63634⋊C3C2semidirect
7. #11720303C2semidirect
21. #88437C4semidirect
29. #63637D3semidirect
30. #63634D3semidirect
48. #634⋊C3C2×C4semidirect
50. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
128. #634C4×D3semidirect
149. C3⋊(C3⋊C8)C4×D3semidirect
236. H(3,𝔽3)#440semidirect
354. C32#4817semidirect
355. C32#4889semidirect

Maximal subgroups

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

subgroupindex
#117217612
#117210842
C2×#884372
#63637⋊C32
#63634⋊C32
#117203242
#117203032
#105905903
#105904783