Finite Group Databasebeta

← #13144296

Subgroups — #13144296

Order 1728.

935 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 935 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#88552C211yesmaximal, characteristic
3C2×#88552C211yesmaximal, characteristic
4C2×#88540C211yesmaximal, characteristic, complemented
5#11720574C211yesmaximal, characteristic, complemented
6#11720574C211yesmaximal, characteristic, complemented
7#11720574C211yesmaximal, characteristic, complemented
8C3×#4952—34nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C22×#1084C2211yescharacteristic
16C22⋊C4×H(3,𝔽3)—22no—
17#88552—22no—
18#88552—22no—
19#88540—22no—
20C2×C32×C22⋊C4D314yes—
21C2×C3×#217—34no—
22C2×C3×#217—34no—
23C2×C3×Dic3⋊C4—34no—
24C3×C3⋊#95—34no—
25C3×C3⋊#95—34no—
26C3×C3⋊#95—34no—
27C23×H(3,𝔽3)—21no—
28C23×H(3,𝔽3)—21no—
29C23×H(3,𝔽3)—21no—
30C23×H(3,𝔽3)—21no—
31C23×H(3,𝔽3)—21no—
32C23×H(3,𝔽3)—21no—
33C23×H(3,𝔽3)D412yes—
34C23×H(3,𝔽3)C2311yescharacteristic
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×#1084—21no—
42C2×#1084—21no—
43C2×#1084—21no—
44C2×#1084—21no—
45C2×#1084—21no—
46C2×#1084—21no—
47C2×#1084—21no—
48C2×#1084—21no—
49C2×#1084—21no—
50C2×#1084—21no—
51C2×#1084—21no—
52C2×#1084D412yes—
53C3×#397—91no—
54#4952—34no—
55C24×C32C2×D314yes—
56C22×C32×C4C2×D314yes—
57C22×C32×C4C2×D314yes—
58C22×C3×Dic3—34no—
59C22×C3×Dic3—34no—
60C22×C3×Dic3—34no—
61C22×C3×Dic3—34no—
62C32×C22⋊C4—28no—
63C3×#217—68no—
64C3×#217—68no—
65C3×Dic3⋊C4—68no—
66C22×H(3,𝔽3)—22no—
67C22×H(3,𝔽3)—22no—
68C22×H(3,𝔽3)—22no—
69C22×H(3,𝔽3)—41no—
70C22×H(3,𝔽3)—41no—
71C22×H(3,𝔽3)—41no—
72C22×H(3,𝔽3)—41no—
73C22×H(3,𝔽3)C2×D411yescharacteristic
74C22×H(3,𝔽3)C2×D411yescharacteristic
75C22×H(3,𝔽3)#4111yescharacteristic
76C22×H(3,𝔽3)#4111yescharacteristic
77C22×H(3,𝔽3)#4111yescharacteristic
78C22×H(3,𝔽3)#4111yescharacteristic
79C22×H(3,𝔽3)#4111yescharacteristic
80C4×H(3,𝔽3)—41no—
81C4×H(3,𝔽3)—41no—
82#1084—22no—
83#1084—41no—
84#1084—41no—
85#1084—41no—
86C2×C3×C4⋊C4—91no—
87C2×C3×C22⋊C4—34no—
88C2×C3×C22⋊C4—91no—
89C2×C3×C22⋊C4—91no—
90C2×C3×C22⋊C4C3⋊D311yescharacteristic
91C2×#217—34no—
92C2×#217—34no—
93C2×Dic3⋊C4—34no—
94C3×#95—91no—
95C3×#95—91no—
96C3×#95—91no—
97C3⋊#95—34no—
98C3⋊#95—34no—
99C3⋊#95—34no—
100C23×C32—24no—
101C23×C32—24no—
102C23×C32—24no—
103C23×C32—24no—
104C23×C32—24no—
105C23×C32—24no—
106C23×C32#6718yes—
107C23×C32C22×D314yes—
108C2×C32×C4—24no—
109C2×C32×C4—24no—
110C2×C32×C4—24no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C3×Dic3—38no—
115C2×C3×Dic3—64no—
116C2×C3×Dic3—64no—
117C2×C3×Dic3—64no—
118C2×C3×Dic3—64no—
119C2×C3×Dic3—64no—
120C2×C3×Dic3—64no—
121C2×C3×Dic3—64no—
122C2×C3×Dic3—64no—
123C2×C3×Dic3—64no—
124C2×C3×Dic3—64no—
125C2×C3×Dic3—64no—
126#397—91noSylow
127C2×H(3,𝔽3)—42no—
128C2×H(3,𝔽3)#12111yescharacteristic
129C2×H(3,𝔽3)#12311yescharacteristic
130C2×H(3,𝔽3)#12311yescharacteristic
131C2×H(3,𝔽3)#12311yescharacteristic
132C2×H(3,𝔽3)#12411yescharacteristic
133C2×H(3,𝔽3)#12611yescharacteristic
134C2×H(3,𝔽3)#12611yescharacteristic
135C24×C3—34no—
136C24×C3C2×C3⋊D311yescharacteristic
137C22×C3×C4—34no—
138C22×C3×C4—34no—
139C22×C3×C4—91no—
140C22×C3×C4—91no—
141C22×C3×C4—91no—
142C22×C3×C4—91no—
143C22×C3×C4C2×C3⋊D311yescharacteristic
144C22×C3×C4C2×C3⋊D311yescharacteristic
145C22×Dic3—34no—
146C22×Dic3—34no—
147C22×Dic3—34no—
148C22×Dic3—34no—
149C3×C4⋊C4—182no—
150C3×C22⋊C4—22no—
151C3×C22⋊C4—68no—
152C3×C22⋊C4—182no—
153C3×C22⋊C4—182no—
154#217—68no—
155#217—68no—
156Dic3⋊C4—68no—
157C22×C32—28no—
158C22×C32—28no—
159C22×C32—28no—
160C22×C32—44no—
161C22×C32—44no—
162C22×C32—44no—
163C22×C32—44no—
164C22×C32#23514yes—
165C22×C32#23514yes—
166C22×C32D3×D414yes—
167C22×C32#23714yes—
168C22×C32#23714yes—
169C22×C32#23714yes—
170C22×C32C2×#6714yes—
171C32×C4—44no—
172C32×C4—44no—
173C3×Dic3—68no—
174C3×Dic3—124no—
175C3×Dic3—124no—
176C3×Dic3—124no—
177C2×C4⋊C4—91no—
178C2×C22⋊C4—91no—
179C2×C22⋊C4—91no—
180C2×C22⋊C4#27211yescharacteristic
181#95—91no—
182#95—91no—
183#95—91no—
184H(3,𝔽3)#39711yescharacteristic, complemented, Sylow
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—38no—
192C23×C3—34no—
193C23×C3—64no—
194C23×C3—64no—
195C23×C3—64no—
196C23×C3—64no—
197C23×C3—64no—
198C23×C3—64no—
199C23×C3#63812yes—
200C23×C3C22×C3⋊D311yescharacteristic
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—64no—
208C2×C3×C4—64no—
209C2×C3×C4—64no—
210C2×C3×C4—64no—
211C2×C3×C4—64no—
212C2×C3×C4—64no—
213C2×C3×C4—92no—
214C2×C3×C4—181no—
215C2×C3×C4—181no—
216C2×C3×C4—181no—
217C2×C3×C4—181no—
218C2×C3×C4—181no—
219C2×C3×C4—181no—
220C2×C3×C4—181no—
221C2×C3×C4—181no—
222C2×C3×C4—181no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×Dic3—38no—
226C2×Dic3—64no—
227C2×Dic3—64no—
228C2×Dic3—64no—
229C2×Dic3—64no—
230C2×Dic3—64no—
231C2×Dic3—64no—
232C2×Dic3—64no—
233C2×Dic3—64no—
234C2×Dic3—64no—
235C2×Dic3—64no—
236C2×Dic3—64no—
237C2×C32—48no—
238C2×C32#87914yes—
239C2×C32#87914yes—
240C2×C32#88314yes—
241C2×C32#88514yes—
242C2×C32#92914yes—
243C2×C32#93514yes—
244C2×C32#93914yes—
245C24C2×#27211yescharacteristic
246C22×C4—91no—
247C22×C4—91no—
248C22×C4—91no—
249C22×C4—91no—
250C22×C4C2×#27211yescharacteristic
251C22×C4C2×#27211yescharacteristic
252C4⋊C4—182no—
253C22⋊C4—22no—
254C22⋊C4—182no—
255C22⋊C4—182no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—22no—
259C22×C3—34no—
260C22×C3—34no—
261C22×C3—34no—
262C22×C3—34no—
263C22×C3—34no—
264C22×C3—34no—
265C22×C3—34no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—68no—
271C22×C3—68no—
272C22×C3—68no—
273C22×C3—124no—
274C22×C3—124no—
275C22×C3—124no—
276C22×C3—124no—
277C22×C3#382811yescharacteristic
278C22×C3#382811yescharacteristic
279C22×C3D4×C3⋊D311yescharacteristic
280C22×C3#383011yescharacteristic
281C22×C3#383011yescharacteristic
282C22×C3#383011yescharacteristic
283C22×C3C2×#63811yescharacteristic
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—124no—
287C3×C4—124no—
288C3×C4—182no—
289C3×C4—361no—
290C3×C4—361no—
291C3×C4—361no—
292Dic3—68no—
293Dic3—124no—
294Dic3—124no—
295Dic3—124no—
296C32#495214yescomplemented
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23#623512yes—
304C23C22×#27211yescharacteristic
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—92no—
312C2×C4—181no—
313C2×C4—181no—
314C2×C4—181no—
315C2×C4—181no—
316C2×C4—181no—
317C2×C4—181no—
318C2×C4—181no—
319C2×C4—181no—
320C2×C4—181no—
321C2×C4—181no—
322C2×C4—181no—
323C2×C3—34no—
324C2×C3—34no—
325C2×C3—34no—
326C2×C3—34no—
327C2×C3—34no—
328C2×C3—34no—
329C2×C3—34no—
330C2×C3—42no—
331C2×C3—128no—
332C2×C3#6408811yescharacteristic
333C2×C3#6408811yescharacteristic
334C2×C3#6409211yescharacteristic
335C2×C3#6409411yescharacteristic
336C2×C3#6413811yescharacteristic
337C2×C3#6414411yescharacteristic
338C2×C3#6414811yescharacteristic
339C22—22no—
340C22—22no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22#8874011yescharacteristic
347C22#8874011yescharacteristic
348C22D4×#27211yescharacteristic
349C22#8874411yescharacteristic
350C22#8874411yescharacteristic
351C22#8874411yescharacteristic
352C22C2×#623511yescharacteristic
353C4—41no—
354C4—41no—
355C4—182no—
356C4—361no—
357C4—361no—
358C4—361no—
359C3—34no—
360C3#1059265111yescharacteristic
361C2—42no—
362C2#1172173511yescharacteristic
363C2#1172173511yescharacteristic
364C2#1172173911yescharacteristic
365C2#1172174111yescharacteristic
366C2#1172182911yescharacteristic
367C2#1172184511yescharacteristic
368C2#1172184911yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, cosocleC23×H(3,𝔽3)C23
FittingC2×C22⋊C4×H(3,𝔽3)C2
prime radicalC24×H(3,𝔽3)C22
hypercenterC2×C3×C22⋊C4C3⋊D3
hypocenterH(3,𝔽3)#397
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×#88540C2semidirect
5. #11720574C2semidirect
6. #11720574C2semidirect
7. #11720574C2semidirect
184. H(3,𝔽3)#397semidirect
296. C32#4952semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×H(3,𝔽3)2
C2×#885522
C2×#885402
#117205742
C3×#49523