Finite Group Databasebeta

← #13161086

Subgroups — #13161086

Order 1728.

1048 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 158 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#11723977C211yesmaximal, characteristic
2C2×C3×#3754C211yesmaximal, characteristic
3C2×C32×#214C211yesmaximal, characteristic
4#11723451C211yesmaximal, characteristic, complemented
5C2×#88800C211yesmaximal, characteristic, complemented
6C2×#88799C211yesmaximal, characteristic, complemented
7#11722021C211yesmaximal, characteristic, complemented
8#10592866—31nomaximal
9#10591493—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14#88864—22no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3754—22no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#214—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—22no—
21#88800—22no—
22#88799—22no—
23C2×C32×SD16—31no—
24#64153—31no—
25C2×#3756—31no—
26#64145—31no—
27#64145—34no—
28#64145—64no—
29C2×#3754D311yescharacteristic
30C2×#3746—31no—
31C2×C3×#214—34no—
32C2×C3×#214D314yes—
33#63873—34no—
34C2×#3719—34no—
35C2×#3719—34no—
36#63642—31no—
37#63555—34no—
38C23×C33—21no—
39C33×D4—21no—
40C33×D4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)D412yes—
44#6308—21no—
45#6308—21no—
46C2×C3×C3⋊Dic3—21no—
47C3×#634—21no—
48C3×#634—21no—
49C2×C32×Dic3—21no—
50C32×Dic6—21no—
51C32×Dic6—21no—
52C3×C3⋊(C3⋊C8)D412yes—
53C32×C3⋊C8D412yes—
54#5167—94no—
55#5167—91no—
56C2×C32×Q8—31no—
57C2×C32×D4—24no—
58C2×C32×D4C2×D314yes—
59C2×C32×D4C2×D311yescharacteristic
60C2×#634—31no—
61C2×#634—34no—
62C2×#634—64no—
63C2×#634C2×D311yescharacteristic
64C2×C3×Dic6—34no—
65C2×C3×Dic6C2×D314yes—
66C32×SD16—62no—
67C2×C32×C8—31no—
68#3757—62no—
69#3757—68no—
70#3757—616no—
71#3756—62no—
72#3754—22no—
73#3750—62no—
74C4×C3⋊Dic3—31no—
75C4×C3⋊Dic3—34no—
76C4×C3⋊Dic3—64no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78#3746—62no—
79C3×#214—28no—
80C3×#214—68no—
81C2×C3×C3⋊C8—34no—
82C2×C3×C3⋊C8C2×D314yes—
83#3724—68no—
84#3719—68no—
85#3719—68no—
86C22×C33—41no—
87C22×C33—42no—
88C22×C33C2×D411yescharacteristic
89C33×C4C2×D411yescharacteristic
90C33×C4C2×D411yescharacteristic
91C3⋊(C3⋊Dic3)—22no—
92C3⋊(C3⋊Dic3)—41no—
93C3×C3⋊Dic3—41no—
94C32×Dic3—41no—
95C2×C3×SD16—34no—
96C2×C3×SD16—91no—
97#944—94no—
98#944—91no—
99C2×#216—94no—
100C2×#216—91no—
101#936—94no—
102#936—91no—
103#936—184no—
104C2×#214—34no—
105C2×#214C3⋊D311yescharacteristic
106C2×Dic12—94no—
107C2×Dic12—91no—
108#815—94no—
109#815—91no—
110C23×C32—28no—
111C23×C32—24no—
112C23×C32—21no—
113C32×Q8—61no—
114C32×Q8—61no—
115C32×D4—21no—
116C32×D4—21no—
117C32×D4—24no—
118C32×D4—24no—
119C32×D4—44no—
120C32×D4—44no—
121C2×C32×C4—24no—
122C2×C32×C4—61no—
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—32no—
127C2×C3⋊Dic3—38no—
128C2×C3⋊Dic3—68no—
129C2×C3⋊Dic3—68no—
130C2×C3⋊Dic3—64no—
131C2×C3⋊Dic3—61no—
132#634—21no—
133#634—21no—
134#634—61no—
135#634—61no—
136#634—64no—
137#634—64no—
138#634—124no—
139#634—124no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3—64no—
142C3×Dic6—24no—
143C3×Dic6—24no—
144C3×Dic6—64no—
145C3×Dic6—64no—
146C32×C8—32no—
147C3⋊(C3⋊C8)#6712yes—
148C3×C3⋊C8—38no—
149C3×C3⋊C8#6718yes—
150#497—271noSylow
151C2×C33—81no—
152C2×C33#12711yescharacteristic
153C2×C33#13712yes—
154C2×C3×Q8—34no—
155C2×C3×Q8—91no—
156C2×C3×D4—24no—
157C2×C3×D4D3214yes—
158C2×C3×D4C2×C3⋊D311yescharacteristic
159C2×Dic6—34no—
160C2×Dic6—94no—
161C2×Dic6—91no—
162C2×Dic6—184no—
163C2×Dic6C2×C3⋊D311yescharacteristic
164C3×SD16—68no—
165C3×SD16—182no—
166C2×C3×C8—34no—
167C2×C3×C8—91no—
168#217—188no—
169#217—182no—
170#217—1816no—
171#216—188no—
172#216—182no—
173#214—22no—
174#214—68no—
175Dic3⋊C4—182no—
176Dic3⋊C4—188no—
177C4×Dic3—94no—
178C4×Dic3—91no—
179C4×Dic3—184no—
180C2×C3⋊C8—34no—
181C2×C3⋊C8C2×C3⋊D311yescharacteristic
182Dic12—188no—
183Dic12—182no—
184C22×C32—24no—
185C22×C32—48no—
186C22×C32—416no—
187C22×C32—48no—
188C22×C32—42no—
189C22×C32—44no—
190C22×C32—41no—
191C22×C32D3×D414yes—
192C22×C32D3×D411yescharacteristic
193C32×C4—24no—
194C32×C4—24no—
195C32×C4—121no—
196C32×C4D3×D414yes—
197C32×C4D3×D411yescharacteristic
198C32×C4C2×#6714yes—
199C32×C4C2×#6711yescharacteristic
200C3⋊Dic3—41no—
201C3⋊Dic3—62no—
202C3⋊Dic3—68no—
203C3⋊Dic3—128no—
204C3⋊Dic3—128no—
205C3⋊Dic3—124no—
206C3⋊Dic3—121no—
207C3×Dic3—44no—
208C3×Dic3—124no—
209C2×Q16—271no—
210C2×Q16—271no—
211C2×SD16—31no—
212C2×SD16—91no—
213C4⋊Q8—271no—
214#124—271no—
215#97—271no—
216C33#49711yescharacteristic, complemented, Sylow
217C23×C3—28no—
218C23×C3—24no—
219C23×C3—21no—
220C3×Q8—64no—
221C3×Q8—64no—
222C3×Q8—181no—
223C3×Q8—181no—
224C3×D4—24no—
225C3×D4—24no—
226C3×D4—21no—
227C3×D4—21no—
228C3×D4—44no—
229C3×D4—44no—
230C2×C3×C4—24no—
231C2×C3×C4—64no—
232C2×C3×C4—181no—
233C2×C3×C4C2×D3214yes—
234C2×C3×C4C22×C3⋊D311yescharacteristic
235C2×Dic3—21no—
236C2×Dic3—64no—
237C2×Dic3—98no—
238C2×Dic3—92no—
239C2×Dic3—188no—
240C2×Dic3—188no—
241C2×Dic3—184no—
242C2×Dic3—181no—
243Dic6—21no—
244Dic6—21no—
245Dic6—64no—
246Dic6—64no—
247Dic6—184no—
248Dic6—184no—
249Dic6—181no—
250Dic6—181no—
251Dic6—364no—
252Dic6—364no—
253C3×C8—38no—
254C3×C8—92no—
255C3⋊C8—38no—
256C3⋊C8#63812yes—
257C2×C32—28no—
258C2×C32—24no—
259C2×C32—88no—
260C2×C32—84no—
261C2×C32—81no—
262C2×C32#91518yes—
263C2×C32#91512yes—
264C2×C32#94014yes—
265C2×C32#94011yescharacteristic
266C2×Q8—31no—
267C2×Q8—91no—
268C2×Q8—271no—
269C2×D4D3×C3⋊D311yescharacteristic
270Q16—542no—
271Q16—542no—
272SD16—62no—
273SD16—182no—
274C2×C8—31no—
275C2×C8—91no—
276C4⋊C4—542no—
277C22⋊C4—542no—
278C42—271no—
279C22×C3—24no—
280C22×C3—48no—
281C22×C3—416no—
282C22×C3—42no—
283C22×C3—41no—
284C22×C3—48no—
285C22×C3—44no—
286C22×C3#381114yes—
287C22×C3D4×C3⋊D311yescharacteristic
288C3×C4—24no—
289C3×C4—24no—
290C3×C4—124no—
291C3×C4—361no—
292C3×C4D3×#6714yes—
293C3×C4D3×#6714yes—
294C3×C4D4×C3⋊D311yescharacteristic
295C3×C4C2×#63811yescharacteristic
296Dic3—41no—
297Dic3—124no—
298Dic3—182no—
299Dic3—188no—
300Dic3—368no—
301Dic3—368no—
302Dic3—361no—
303Dic3—364no—
304C32—24no—
305C32#516714yescomplemented
306C32#516711yescharacteristic, complemented
307C23—21no—
308Q8—61no—
309Q8—61no—
310Q8—181no—
311Q8—181no—
312Q8—541no—
313Q8—541no—
314D4—21no—
315D4—21no—
316C2×C4—61no—
317C2×C4—181no—
318C2×C4—272no—
319C2×C4—541no—
320C2×C4C2×D3×C3⋊D311yescharacteristic
321C8—32no—
322C8—92no—
323C2×C3—28no—
324C2×C3—24no—
325C2×C3—88no—
326C2×C3—84no—
327C2×C3—81no—
328C2×C3#6393118yes—
329C2×C3#6397914yes—
330C2×C3#6412412yes—
331C2×C3#6414911yescharacteristic
332C22—42no—
333C22—41no—
334C22#8903911yescharacteristic
335C4—121no—
336C4—361no—
337C4—542no—
338C4—1081no—
339C4D3×#63811yescharacteristic
340C4C3⋊D3×#6711yescharacteristic
341C3—24no—
342C3#1059149314yescomplemented
343C3#1059286611yescharacteristic, complemented
344C2—81no—
345C2#1172355412yes—
346C2#1172363311yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#497
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
4. #11723451C2semidirect
5. C2×#88800C2semidirect
6. C2×#88799C2semidirect
7. #11722021C2semidirect
216. C33#497semidirect
305. C32#5167semidirect
306. C32#5167semidirect
342. C3#10591493semidirect
343. C3#10592866semidirect

Maximal subgroups

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

subgroupindex
#117239772
C2×C3×#37542
C2×C32×#2142
#117234512
C2×#888002
C2×#887992
#117220212
#105928663
#105914933