Finite Group Databasebeta

← #13140197

Subgroups — #13140197

Order 1728.

592 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 93 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×#634⋊C3C211yesmaximal, characteristic
2#11721824C211yesmaximal, characteristic
3C2×#88439C212yesmaximal, complemented
4#63657⋊C3C211yesmaximal, characteristic
5#11720309C212yesmaximal, complemented
6#10591190—31nomaximal
7#10591185—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#6229C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3—21no—
11C2×#634⋊C3—21no—
12C2×#634⋊C3—24no—
13C2×#634⋊C3C2212yes—
14#88552—21no—
15C4⋊#1084C2211yescharacteristic
16#88540—21no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18#88439—24no—
19C22×#634D311yescharacteristic
20C22×C3×Dic6—31no—
21C3×#924—31no—
22C3×#924—31no—
23C3×#924—61no—
24C2×#3719—32no—
25C2×#3718—32no—
26#63657D311yescharacteristic
27C3×#830—31no—
28#63567—32no—
29#63565—32no—
30C23×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)D411yescharacteristic
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×#1084—21no—
35C2×#1084—21no—
36#6229—22no—
37C2×(C3⋊Dic3)⋊C3—24no—
38C2×(C3⋊Dic3)⋊C3—22no—
39#634⋊C3—24no—
40#634⋊C3—22no—
41#634⋊C3D414yes—
42(C3⋊(C3⋊C8))⋊C3—22no—
43#5228—91no—
44#5050—91no—
45#4739—91no—
46C22×C32×C4—21no—
47C22×C32×C4C2×D311yescharacteristic
48C22×C32×C4C2×D311yescharacteristic
49C22×C3⋊Dic3—21no—
50C2×#634—21no—
51C2×#634—24no—
52C2×#634C2×D312yes—
53C22×C3×Dic3—61no—
54C2×C3×Dic6—32no—
55C2×C3×Dic6—31no—
56C2×C3×Dic6—31no—
57C2×C3×Dic6—64no—
58C2×C3×Dic6—61no—
59C2×C3×Dic6—61no—
60C2×C3⋊(C3⋊C8)C2×D312yes—
61C3×#217—62no—
62C3×#217—61no—
63C3×#217—61no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—61no—
67C3×Dic3⋊C4—62no—
68C3×Dic3⋊C4—61no—
69C3×Dic3⋊C4—61no—
70C2×C3×C3⋊C8—32no—
71#3719—64no—
72#3718—64no—
73C22×H(3,𝔽3)—21no—
74C22×H(3,𝔽3)—21no—
75C22×H(3,𝔽3)Q1612yescomplemented
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C4×H(3,𝔽3)—21no—
78C4×H(3,𝔽3)C2×D412yes—
79#1084—41no—
80#1084—41no—
81(C3⋊Dic3)⋊C3—24no—
82C22×C3×Q8—91no—
83C22×Dic6—31no—
84C22×Dic6—91no—
85C3×C22⋊Q8—91no—
86C2×#216—92no—
87C2×#216—92no—
88#924—31no—
89#924—31no—
90#924—61no—
91C2×Dic12—92no—
92C3×C22⋊C8—91no—
93#835—92no—
94#830—31no—
95#830—91no—
96#816—92no—
97#810—92no—
98C23×C32—21no—
99C23×C32D1211yescharacteristic
100C23×C32#6711yescharacteristic
101C2×C32×C4—22no—
102C2×C32×C4—22no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4—42no—
106C2×C32×C4D1211yescharacteristic
107C2×C32×C4#6711yescharacteristic
108C2×C32×C4C22×D311yescharacteristic
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—24no—
111C2×C3⋊Dic3—22no—
112#634—24no—
113#634—22no—
114#634#6714yes—
115C2×C3×Dic3—64no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C3×Dic6—34no—
124C3×Dic6—64no—
125C3×Dic6—62no—
126C3×Dic6—62no—
127C3×Dic6—62no—
128C3×Dic6—122no—
129C3⋊(C3⋊C8)—22no—
130C3×C3⋊C8—62no—
131#451—271noSylow
132C2×H(3,𝔽3)—22no—
133C2×H(3,𝔽3)#12011yescharacteristic
134C2×H(3,𝔽3)C2×Q1611yescharacteristic
135C2×H(3,𝔽3)#13711yescharacteristic
136C2×C3×Q8—92no—
137C2×C3×Q8—91no—
138C2×C3×Q8—181no—
139C2×C3×Q8—184no—
140C22×C3×C4—31no—
141C22×C3×C4—31no—
142C22×C3×C4—61no—
143C22×C3×C4—181no—
144C22×C3×C4D3211yescharacteristic
145C22×Dic3—61no—
146C22×Dic3—181no—
147C2×Dic6—32no—
148C2×Dic6—31no—
149C2×Dic6—31no—
150C2×Dic6—61no—
151C2×Dic6—64no—
152C2×Dic6—61no—
153C2×Dic6—92no—
154C2×Dic6—184no—
155C2×Dic6—181no—
156C2×C3×C8—92no—
157C3×C4⋊C4—91no—
158C3×C4⋊C4—181no—
159C3×C22⋊C4—181no—
160#217—61no—
161#217—61no—
162#217—62no—
163#216—184no—
164#216—184no—
165C4⋊Dic3—31no—
166C4⋊Dic3—31no—
167C4⋊Dic3—61no—
168Dic3⋊C4—61no—
169Dic3⋊C4—61no—
170Dic3⋊C4—62no—
171C2×C3⋊C8—32no—
172C2×C3⋊C8—92no—
173Dic12—184no—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32—22no—
179C22×C32—21no—
180C22×C32—41no—
181C22×C32—41no—
182C22×C32Dic1212yescomplemented
183C22×C32#21612yescomplemented
184C22×C32C2×D1211yescharacteristic
185C22×C32C2×#6711yescharacteristic
186C32×C4—21no—
187C32×C4—21no—
188C32×C4—22no—
189C32×C4—41no—
190C32×C4D3×D412yes—
191C32×C4C2×#6712yes—
192C3⋊Dic3—24no—
193C3×Dic3—64no—
194C3×Dic3—122no—
195C3×Dic3—122no—
196C3×Dic3—121no—
197C3×Dic3—121no—
198C3×Dic3—121no—
199C3×Dic3—121no—
200C22×Q8—91no—
201C2×Q16—272no—
202C22⋊Q8—91no—
203#103—272no—
204C22⋊C8—91no—
205H(3,𝔽3)#45111yescharacteristic, complemented, Sylow
206C23×C3—31no—
207C23×C3—31no—
208C23×C3—61no—
209C23×C3#62611yescharacteristic
210C3×Q8—94no—
211C3×Q8—182no—
212C3×Q8—184no—
213C3×Q8—182no—
214C2×C3×C4—22no—
215C2×C3×C4—31no—
216C2×C3×C4—31no—
217C2×C3×C4—31no—
218C2×C3×C4—31no—
219C2×C3×C4—62no—
220C2×C3×C4—62no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—122no—
224C2×C3×C4—182no—
225C2×C3×C4—184no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4#62611yescharacteristic
229C2×C3×C4C2×D3211yescharacteristic
230C2×Dic3—64no—
231C2×Dic3—62no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—182no—
239C2×Dic3—184no—
240Dic6—34no—
241Dic6—64no—
242Dic6—62no—
243Dic6—62no—
244Dic6—62no—
245Dic6—94no—
246Dic6—122no—
247Dic6—182no—
248Dic6—184no—
249C3×C8—182no—
250C3⋊C8—62no—
251C3⋊C8—182no—
252C2×C32—22no—
253C2×C32—22no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—42no—
258C2×C32#88211yescharacteristic
259C2×C32C2×Dic1211yescharacteristic
260C2×C32#90911yescharacteristic
261C2×C32#94211yescharacteristic
262C2×C32C2×#21611yescharacteristic
263C2×C32#95311yescharacteristic
264C2×Q8—92no—
265C2×Q8—91no—
266C2×Q8—184no—
267C2×Q8—181no—
268C22×C4—181no—
269C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
270Q16—544no—
271C2×C8—92no—
272C4⋊C4—91no—
273C4⋊C4—181no—
274C22⋊C4—181no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—31no—
278C22×C3—31no—
279C22×C3—32no—
280C22×C3—32no—
281C22×C3—61no—
282C22×C3—61no—
283C22×C3—61no—
284C22×C3—61no—
285C22×C3—61no—
286C22×C3—62no—
287C22×C3—121no—
288C22×C3—121no—
289C22×C3#371912yes—
290C22×C3C2×#62611yescharacteristic
291C3×C4—21no—
292C3×C4—32no—
293C3×C4—32no—
294C3×C4—61no—
295C3×C4—61no—
296C3×C4—62no—
297C3×C4—121no—
298C3×C4—184no—
299C3×C4—361no—
300C3×C4—361no—
301C3×C4D3×#6712yes—
302Dic3—64no—
303Dic3—121no—
304Dic3—121no—
305Dic3—121no—
306Dic3—121no—
307Dic3—122no—
308Dic3—122no—
309Dic3—184no—
310C32—21no—
311C32#473911yescharacteristic, complemented
312C32#522811yescharacteristic, complemented
313C23#620011yescharacteristic
314Q8—94no—
315Q8—184no—
316Q8—182no—
317Q8—182no—
318C2×C4—22no—
319C2×C4—184no—
320C2×C4—182no—
321C2×C4—181no—
322C2×C4—181no—
323C2×C4#620011yescharacteristic
324C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
325C8—182no—
326C2×C3—22no—
327C2×C3—31no—
328C2×C3—31no—
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—62no—
334C2×C3—62no—
335C2×C3—61no—
336C2×C3—61no—
337C2×C3—61no—
338C2×C3—122no—
339C2×C3#6383311yescharacteristic
340C2×C3C2×#371911yescharacteristic
341C2×C3#6397711yescharacteristic
342C22—21no—
343C22—21no—
344C22#8843912yescomplemented
345C22C2×#620011yescharacteristic
346C4—21no—
347C4—184no—
348C4—361no—
349C4—361no—
350C4#8867612yes—
351C3—31no—
352C3—31no—
353C3—61no—
354C3#1059118611yescharacteristic
355C2—22no—
356C2#1172106911yescharacteristic
357C2C2×#8843911yescharacteristic
358C2#1172132111yescharacteristic
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 radicalC23×H(3,𝔽3)D4
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#451
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
3. C2×#88439C2semidirect
5. #11720309C2semidirect
75. C22×H(3,𝔽3)Q16semidirect
182. C22×C32Dic12semidirect
183. C22×C32#216semidirect
205. H(3,𝔽3)#451semidirect
311. C32#4739semidirect
312. C32#5228semidirect
344. C22#88439semidirect

Maximal subgroups

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

subgroupindex
C22×#634⋊C32
#117218242
C2×#884392
#63657⋊C32
#117203092
#105911903
#105911853