Finite Group Databasebeta

← #13140200

Subgroups — #13140200

Order 1728.

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

normal subgroups only

This group has 674 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×#6229C211yesmaximal, characteristic
2#64133⋊C3C211yesmaximal, characteristic
3C2×#88439C212yesmaximal, complemented
4#63657⋊C3C211yesmaximal, characteristic
5#11720326C212yesmaximal, complemented
6#10591186—31nomaximal
7#10591185—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084—21no—
10C2×#6229—21no—
11C2×#6229—24no—
12C2×#6229C2212yes—
13C2×#634⋊C3C2211yescharacteristic
14#3757⋊C3—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16#3750⋊C3—21no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18#88439—24no—
19C22×C3×Dic6—31no—
20C22×C3×Dic6—31no—
21C22×C3×Dic6—61no—
22#64133D311yescharacteristic
23C3×#924—31no—
24C2×#3719—32no—
25C2×#3718—32no—
26#63657D311yescharacteristic
27C3×#830—31no—
28#63566—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—24no—
35C2×#1084—22no—
36#6229—24no—
37#6229—22no—
38#6229D414yes—
39C2×(C3⋊Dic3)⋊C3—21no—
40C2×(C3⋊Dic3)⋊C3—21no—
41#634⋊C3—22no—
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
49C2×#634C2×D311yescharacteristic
50C22×C3×Dic3—62no—
51C22×C3×Dic3—61no—
52C22×C3×Dic3—61no—
53C2×C3×Dic6—32no—
54C2×C3×Dic6—32no—
55C2×C3×Dic6—31no—
56C2×C3×Dic6—64no—
57C2×C3×Dic6—64no—
58C2×C3×Dic6—62no—
59C2×C3×Dic6—68no—
60C2×C3×Dic6—62no—
61C2×C3×Dic6—61no—
62C2×C3×Dic6—61no—
63#3757—21no—
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65#3750—21no—
66C2×C3⋊(C3⋊C8)C2×D312yes—
67C3×#217—61no—
68C3×C4⋊Dic3—31no—
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—24no—
80(C3⋊Dic3)⋊C3—41no—
81(C3⋊Dic3)⋊C3—41no—
82C22×C3×Q8—91no—
83C22×Dic6—31no—
84C22×Dic6—31no—
85C22×Dic6—61no—
86C3×C22⋊Q8—91no—
87C2×#216—92no—
88C2×#216—92no—
89#924—31no—
90#924—91no—
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—24no—
102C2×C32×C4—22no—
103C2×C32×C4—22no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4D1211yescharacteristic
107C2×C32×C4#6711yescharacteristic
108C2×C32×C4C22×D311yescharacteristic
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—21no—
112#634—22no—
113C2×C3×Dic3—68no—
114C2×C3×Dic3—64no—
115C2×C3×Dic3—64no—
116C2×C3×Dic3—64no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C3×Dic6—34no—
122C3×Dic6—34no—
123C3×Dic6—64no—
124C3×Dic6—68no—
125C3×Dic6—64no—
126C3×Dic6—64no—
127C3×Dic6—64no—
128C3×Dic6—62no—
129C3×Dic6—62no—
130C3×Dic6—62no—
131C3⋊(C3⋊C8)—22no—
132C3×C3⋊C8—62no—
133#451—271noSylow
134C2×H(3,𝔽3)—22no—
135C2×H(3,𝔽3)#12011yescharacteristic
136C2×H(3,𝔽3)C2×Q1611yescharacteristic
137C2×H(3,𝔽3)#13711yescharacteristic
138C2×C3×Q8—91no—
139C2×C3×Q8—92no—
140C2×C3×Q8—184no—
141C2×C3×Q8—181no—
142C22×C3×C4—31no—
143C22×C3×C4—31no—
144C22×C3×C4—61no—
145C22×C3×C4—181no—
146C22×C3×C4D3211yescharacteristic
147C22×Dic3—61no—
148C22×Dic3—61no—
149C22×Dic3—62no—
150C2×Dic6—32no—
151C2×Dic6—32no—
152C2×Dic6—31no—
153C2×Dic6—62no—
154C2×Dic6—64no—
155C2×Dic6—64no—
156C2×Dic6—61no—
157C2×Dic6—61no—
158C2×Dic6—68no—
159C2×Dic6—62no—
160C2×Dic6—91no—
161C2×C3×C8—92no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4—181no—
164C3×C22⋊C4—181no—
165#217—61no—
166#217—181no—
167#216—184no—
168#216—184no—
169C4⋊Dic3—31no—
170C4⋊Dic3—91no—
171Dic3⋊C4—61no—
172Dic3⋊C4—181no—
173C2×C3⋊C8—32no—
174C2×C3⋊C8—92no—
175Dic12—184no—
176C22×C32—22no—
177C22×C32—22no—
178C22×C32—21no—
179C22×C32—21no—
180C22×C32—21no—
181C22×C32—21no—
182C22×C32—22no—
183C22×C32—21no—
184C22×C32Dic1212yescomplemented
185C22×C32#21612yescomplemented
186C22×C32C2×D1211yescharacteristic
187C22×C32C2×#6711yescharacteristic
188C32×C4—22no—
189C32×C4—21no—
190C32×C4—21no—
191C32×C4—22no—
192C32×C4D3×D412yes—
193C32×C4D3×D412yes—
194C3⋊Dic3—41no—
195C3⋊Dic3—41no—
196C3×Dic3—68no—
197C3×Dic3—64no—
198C3×Dic3—64no—
199C3×Dic3—121no—
200C3×Dic3—121no—
201C22×Q8—91no—
202C2×Q16—272no—
203C22⋊Q8—91no—
204#103—272no—
205C22⋊C8—91no—
206H(3,𝔽3)#45111yescharacteristic, complemented, Sylow
207C23×C3—31no—
208C23×C3—31no—
209C23×C3—61no—
210C23×C3#62611yescharacteristic
211C3×Q8—94no—
212C3×Q8—184no—
213C3×Q8—182no—
214C3×Q8—182no—
215C2×C3×C4—22no—
216C2×C3×C4—31no—
217C2×C3×C4—31no—
218C2×C3×C4—31no—
219C2×C3×C4—31no—
220C2×C3×C4—62no—
221C2×C3×C4—62no—
222C2×C3×C4—64no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—184no—
226C2×C3×C4—182no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4#62611yescharacteristic
230C2×C3×C4C2×D3211yescharacteristic
231C2×Dic3—62no—
232C2×Dic3—62no—
233C2×Dic3—64no—
234C2×Dic3—64no—
235C2×Dic3—68no—
236C2×Dic3—64no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—181no—
240C2×Dic3—181no—
241Dic6—34no—
242Dic6—34no—
243Dic6—64no—
244Dic6—62no—
245Dic6—62no—
246Dic6—64no—
247Dic6—64no—
248Dic6—68no—
249Dic6—64no—
250Dic6—62no—
251Dic6—182no—
252C3×C8—182no—
253C3⋊C8—62no—
254C3⋊C8—182no—
255C2×C32—24no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32#88211yescharacteristic
262C2×C32C2×Dic1211yescharacteristic
263C2×C32#90911yescharacteristic
264C2×C32#93711yescharacteristic
265C2×C32C2×#21611yescharacteristic
266C2×C32#95111yescharacteristic
267C2×Q8—92no—
268C2×Q8—91no—
269C2×Q8—184no—
270C2×Q8—181no—
271C22×C4—181no—
272C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
273Q16—544no—
274C2×C8—92no—
275C4⋊C4—91no—
276C4⋊C4—181no—
277C22⋊C4—181no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—31no—
281C22×C3—31no—
282C22×C3—32no—
283C22×C3—32no—
284C22×C3—61no—
285C22×C3—61no—
286C22×C3—61no—
287C22×C3—61no—
288C22×C3—62no—
289C22×C3—62no—
290C22×C3—61no—
291C22×C3—62no—
292C22×C3#371912yes—
293C22×C3C2×#62611yescharacteristic
294C3×C4—21no—
295C3×C4—32no—
296C3×C4—32no—
297C3×C4—61no—
298C3×C4—61no—
299C3×C4—62no—
300C3×C4—62no—
301C3×C4—184no—
302C3×C4—361no—
303C3×C4—361no—
304C3×C4#381112yes—
305Dic3—64no—
306Dic3—64no—
307Dic3—68no—
308Dic3—121no—
309Dic3—121no—
310Dic3—361no—
311Dic3—361no—
312C32—21no—
313C32#473911yescharacteristic, complemented
314C32#505011yescharacteristic, complemented
315C23#620011yescharacteristic
316Q8—94no—
317Q8—184no—
318Q8—182no—
319Q8—182no—
320C2×C4—22no—
321C2×C4—184no—
322C2×C4—182no—
323C2×C4—181no—
324C2×C4—181no—
325C2×C4#620011yescharacteristic
326C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
327C8—182no—
328C2×C3—22no—
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—31no—
334C2×C3—31no—
335C2×C3—62no—
336C2×C3—62no—
337C2×C3—64no—
338C2×C3—61no—
339C2×C3—61no—
340C2×C3—61no—
341C2×C3#6383111yescharacteristic
342C2×C3C2×#371911yescharacteristic
343C2×C3#6398111yescharacteristic
344C22—21no—
345C22—21no—
346C22#8843912yescomplemented
347C22C2×#620011yescharacteristic
348C4—21no—
349C4—184no—
350C4—361no—
351C4—361no—
352C4#8867712yes—
353C3—31no—
354C3—31no—
355C3—61no—
356C3#1059119011yescharacteristic
357C2—22no—
358C2#1172107411yescharacteristic
359C2C2×#8843911yescharacteristic
360C2#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 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. #11720326C2semidirect
75. C22×H(3,𝔽3)Q16semidirect
184. C22×C32Dic12semidirect
185. C22×C32#216semidirect
206. H(3,𝔽3)#451semidirect
313. C32#4739semidirect
314. C32#5050semidirect
346. C22#88439semidirect

Maximal subgroups

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

subgroupindex
C22×#62292
#64133⋊C32
C2×#884392
#63657⋊C32
#117203262
#105911863
#105911853