Finite Group Databasebeta

← #13140189

Subgroups — #13140189

Order 1728.

894 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 894 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C22×C4⋊#272C211yesmaximal, characteristic
2#64133⋊C3C211yesmaximal, characteristic, complemented
3C2×#88437C212yesmaximal, complemented
4#63657⋊C3C211yesmaximal, characteristic, complemented
5#11720324C212yesmaximal, complemented
6#10591172—31nomaximal
7#10591169—31nomaximal
8C23×#272—21no—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×C4⋊#272—21no—
11C2×C4⋊#272—24no—
12C2×C4⋊#272C2212yes—
13C2×#634⋊C3C2211yescharacteristic, complemented
14#3757⋊C3—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
16#3750⋊C3—21no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
18#88437—24no—
19C22×C3×D12—31no—
20C22×C3×D12—31no—
21C22×C3×D12—61no—
22#64133D311yescharacteristic, complemented
23C3×#924—31no—
24C2×#3716—32no—
25C2×#3715—32no—
26#63657D311yescharacteristic, complemented
27C3×#830—31no—
28#63563—32no—
29#63562—32no—
30C23×H(3,𝔽3)D411yescharacteristic
31C22×#272—24no—
32C22×#272—22no—
33C22×#272—42no—
34C22×#272—42no—
35C2×C4×H(3,𝔽3)—22no—
36C2×C4×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C4⋊#272—24no—
39C4⋊#272—22no—
40C4⋊#272D414yes—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43#634⋊C3—22no—
44(C3⋊(C3⋊C8))⋊C3—22no—
45#5218—91no—
46#5046—91no—
47#4731—91no—
48C23×C3×D3—62no—
49C23×C3×D3—61no—
50C23×C3×D3—61no—
51C22×C32×C4—21no—
52C22×C32×C4C2×D311yescharacteristic
53C22×C32×C4C2×D311yescharacteristic
54C2×#634C2×D311yescharacteristic, complemented
55C2×C3×D12—32no—
56C2×C3×D12—32no—
57C2×C3×D12—64no—
58C2×C3×D12—64no—
59C2×C3×D12—62no—
60C2×C3×D12—68no—
61C2×C3×D12—62no—
62C2×C3×D12—61no—
63C2×C3×D12—61no—
64C2×C3×Dic6—31no—
65#3757—21no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
67#3750—21no—
68C2×C3⋊(C3⋊C8)C2×D312yescomplemented
69C3×#217—61no—
70C3×C4⋊Dic3—31no—
71C3×Dic3⋊C4—61no—
72C2×C3×C3⋊C8—32no—
73#3716—64no—
74#3715—64no—
75C22×H(3,𝔽3)—21no—
76C22×H(3,𝔽3)—21no—
77C22×H(3,𝔽3)SD1612yescomplemented
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×#272—24no—
80C2×#272—48no—
81C2×#272—42no—
82C2×#272—42no—
83C4×H(3,𝔽3)—21no—
84C4×H(3,𝔽3)C2×D412yes—
85(C3⋊Dic3)⋊C3—41no—
86(C3⋊Dic3)⋊C3—41no—
87C22×C3×D4—91no—
88C22×D12—31no—
89C22×D12—31no—
90C22×D12—61no—
91C3×C22⋊Q8—91no—
92C2×#215—92no—
93C2×#214—92no—
94#924—31no—
95#924—91no—
96C2×#204—92no—
97C3×C22⋊C8—91no—
98#832—92no—
99#830—31no—
100#830—91no—
101#821—92no—
102#809—92no—
103C23×C32—21no—
104C23×C32D1211yescharacteristic
105C23×C32#6711yescharacteristic
106C22×C3×D3—68no—
107C22×C3×D3—64no—
108C22×C3×D3—64no—
109C22×C3×D3—64no—
110C22×C3×D3—62no—
111C22×C3×D3—62no—
112C22×C3×D3—124no—
113C22×C3×D3—122no—
114C22×C3×D3—122no—
115C22×C3×D3—124no—
116C22×C3×D3—122no—
117C22×C3×D3—122no—
118C2×C32×C4—24no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4D1211yescharacteristic
124C2×C32×C4#6711yescharacteristic
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—21no—
129#634—22no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C3×D12—34no—
133C3×D12—34no—
134C3×D12—64no—
135C3×D12—68no—
136C3×D12—64no—
137C3×D12—64no—
138C3×D12—64no—
139C3×D12—62no—
140C3×D12—62no—
141C3×Dic6—62no—
142C3⋊(C3⋊C8)—22no—
143C3×C3⋊C8—62no—
144#450—271noSylow
145C2×H(3,𝔽3)—22no—
146C2×H(3,𝔽3)#12011yescharacteristic
147C2×H(3,𝔽3)C2×SD1611yescharacteristic
148C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
149#272—44no—
150C24×C3—181no—
151C23×D3—61no—
152C23×D3—61no—
153C23×D3—62no—
154C2×C3×Q8—91no—
155C2×C3×D4—92no—
156C2×C3×D4—184no—
157C2×C3×D4—181no—
158C22×C3×C4—31no—
159C22×C3×C4—31no—
160C22×C3×C4—61no—
161C22×C3×C4D3211yescharacteristic
162C2×D12—32no—
163C2×D12—32no—
164C2×D12—62no—
165C2×D12—64no—
166C2×D12—64no—
167C2×D12—61no—
168C2×D12—61no—
169C2×D12—68no—
170C2×D12—62no—
171C2×Dic6—31no—
172C2×Dic6—91no—
173C2×C3×C8—92no—
174C3×C4⋊C4—91no—
175C3×C4⋊C4—181no—
176C3×C22⋊C4—181no—
177#217—61no—
178#217—181no—
179#215—184no—
180#214—184no—
181C4⋊Dic3—31no—
182C4⋊Dic3—91no—
183Dic3⋊C4—61no—
184Dic3⋊C4—181no—
185C2×C3⋊C8—32no—
186C2×C3⋊C8—92no—
187#204—184no—
188C22×C32—22no—
189C22×C32—22no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—22no—
195C22×C32—21no—
196C22×C32#20412yescomplemented
197C22×C32#21512yescomplemented
198C22×C32C2×D1211yescharacteristic
199C22×C32C2×#6711yescharacteristic
200C2×C3×D3—68no—
201C2×C3×D3—64no—
202C2×C3×D3—64no—
203C2×C3×D3—1216no—
204C2×C3×D3—128no—
205C2×C3×D3—128no—
206C2×C3×D3—124no—
207C2×C3×D3—124no—
208C2×C3×D3—122no—
209C2×C3×D3—122no—
210C2×C3×D3—122no—
211C2×C3×D3—122no—
212C32×C4—22no—
213C32×C4—21no—
214C32×C4—21no—
215C32×C4—22no—
216C32×C4D3×D412yes—
217C32×C4D3×D412yes—
218C3⋊Dic3—41no—
219C3⋊Dic3—41no—
220C3×Dic3—121no—
221C3×Dic3—121no—
222C22×D4—91no—
223C2×SD16—272no—
224C22⋊Q8—91no—
225#102—272no—
226C22⋊C8—91no—
227H(3,𝔽3)#45011yescharacteristic, complemented, Sylow
228C23×C3—31no—
229C23×C3—31no—
230C23×C3—61no—
231C23×C3—184no—
232C23×C3—182no—
233C23×C3—362no—
234C23×C3—362no—
235C23×C3#62611yescharacteristic
236C22×D3—62no—
237C22×D3—62no—
238C22×D3—64no—
239C22×D3—64no—
240C22×D3—68no—
241C22×D3—64no—
242C22×D3—122no—
243C22×D3—122no—
244C22×D3—122no—
245C22×D3—122no—
246C22×D3—124no—
247C22×D3—124no—
248C3×Q8—182no—
249C3×D4—94no—
250C3×D4—184no—
251C3×D4—182no—
252C2×C3×C4—22no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—31no—
257C2×C3×C4—62no—
258C2×C3×C4—62no—
259C2×C3×C4—64no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—181no—
263C2×C3×C4—181no—
264C2×C3×C4#62611yescharacteristic
265C2×C3×C4C2×D3211yescharacteristic
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—181no—
269C2×Dic3—181no—
270D12—34no—
271D12—34no—
272D12—64no—
273D12—62no—
274D12—62no—
275D12—64no—
276D12—64no—
277D12—68no—
278D12—64no—
279Dic6—62no—
280Dic6—182no—
281C3×C8—182no—
282C3⋊C8—62no—
283C3⋊C8—182no—
284C2×C32—24no—
285C2×C32—22no—
286C2×C32—22no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—21no—
290C2×C32#88211yescharacteristic
291C2×C32C2×#20411yescharacteristic
292C2×C32#90811yescharacteristic
293C2×C32#93711yescharacteristic
294C2×C32C2×#21511yescharacteristic
295C2×C32#94911yescharacteristic
296C3×D3—128no—
297C3×D3—124no—
298C3×D3—124no—
299C24—181no—
300C2×Q8—91no—
301C2×D4—92no—
302C2×D4—184no—
303C2×D4—181no—
304C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
305SD16—544no—
306C2×C8—92no—
307C4⋊C4—91no—
308C4⋊C4—181no—
309C22⋊C4—181no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—31no—
313C22×C3—31no—
314C22×C3—32no—
315C22×C3—32no—
316C22×C3—61no—
317C22×C3—61no—
318C22×C3—61no—
319C22×C3—61no—
320C22×C3—62no—
321C22×C3—62no—
322C22×C3—61no—
323C22×C3—62no—
324C22×C3—184no—
325C22×C3—368no—
326C22×C3—362no—
327C22×C3—362no—
328C22×C3#371712yes—
329C22×C3C2×#62611yescharacteristic
330C2×D3—64no—
331C2×D3—64no—
332C2×D3—68no—
333C2×D3—122no—
334C2×D3—122no—
335C2×D3—122no—
336C2×D3—122no—
337C2×D3—128no—
338C2×D3—128no—
339C2×D3—1216no—
340C2×D3—124no—
341C2×D3—124no—
342C3×C4—21no—
343C3×C4—32no—
344C3×C4—32no—
345C3×C4—61no—
346C3×C4—61no—
347C3×C4—62no—
348C3×C4—62no—
349C3×C4—361no—
350C3×C4—361no—
351C3×C4#381112yes—
352Dic3—121no—
353Dic3—121no—
354Dic3—361no—
355Dic3—361no—
356C32—21no—
357C32#473111yescharacteristic, complemented
358C32#504611yescharacteristic, complemented
359C23—184no—
360C23—182no—
361C23—362no—
362C23—362no—
363C23#620011yescharacteristic
364Q8—182no—
365D4—94no—
366D4—184no—
367D4—182no—
368C2×C4—22no—
369C2×C4—181no—
370C2×C4—181no—
371C2×C4#620011yescharacteristic
372C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
373C8—182no—
374C2×C3—22no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—31no—
381C2×C3—62no—
382C2×C3—62no—
383C2×C3—64no—
384C2×C3—61no—
385C2×C3—61no—
386C2×C3—61no—
387C2×C3—364no—
388C2×C3#6382511yescharacteristic
389C2×C3C2×#371711yescharacteristic
390C2×C3#6398111yescharacteristic
391D3—124no—
392D3—124no—
393D3—128no—
394C22—21no—
395C22—21no—
396C22—184no—
397C22—368no—
398C22—362no—
399C22—362no—
400C22#8843712yescomplemented
401C22C2×#620011yescharacteristic
402C4—21no—
403C4—361no—
404C4—361no—
405C4#8867712yes—
406C3—31no—
407C3—31no—
408C3—61no—
409C3#1059118311yescharacteristic
410C2—22no—
411C2—364no—
412C2#1172106811yescharacteristic
413C2C2×#8843711yescharacteristic
414C2#1172132311yescharacteristic
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 radicalC22×C4⋊#272C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#450
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
2. #64133⋊C3C2semidirect
3. C2×#88437C2semidirect
4. #63657⋊C3C2semidirect
5. #11720324C2semidirect
13. C2×#634⋊C3C22semidirect
15. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
17. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
22. #64133D3semidirect
26. #63657D3semidirect
54. C2×#634C2×D3semidirect
66. C4⋊(C3⋊Dic3)C2×D3semidirect
68. C2×C3⋊(C3⋊C8)C2×D3semidirect
77. C22×H(3,𝔽3)SD16semidirect
196. C22×C32#204semidirect
197. C22×C32#215semidirect
227. H(3,𝔽3)#450semidirect
357. C32#4731semidirect
358. C32#5046semidirect
400. C22#88437semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#2722
#64133⋊C32
C2×#884372
#63657⋊C32
#117203242
#105911723
#105911693