Finite Group Databasebeta

← #13161087

Subgroups — #13161087

Order 1728.

976 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 154 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×C32×#214C211yesmaximal, characteristic
3#11723397C211yesmaximal, characteristic, complemented
4C2×#88799C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic
6#11722050C211yesmaximal, characteristic, complemented
7#11722042C211yesmaximal, characteristic, complemented
8#10592865—31nomaximal
9#10591443—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13#88864—21no—
14#88864—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#214—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21Dic3×C3⋊Dic3—21no—
22#88799—22no—
23C2×C32×SD16—31no—
24Q8×C3⋊Dic3—31no—
25C2×#3756—31no—
26#64145—31no—
27#64145—34no—
28#64145—64no—
29C2×C3×#214D314yes—
30#63843—34no—
31C2×#3719—34no—
32#63659D311yescharacteristic
33#63643—31no—
34#63641—31no—
35C3×#832—34no—
36#63572—34no—
37#63567—34no—
38C23×C33—21no—
39C33×D4—21no—
40C33×D4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45#6308—21no—
46#6308—21no—
47C2×C3×C3⋊Dic3—21no—
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6D412yes—
51C3×C3⋊(C3⋊C8)—21no—
52C32×C3⋊C8—21no—
53#5166—94no—
54#4767—91no—
55C2×C32×Q8—31no—
56C2×C32×D4—24no—
57C2×C32×D4C2×D314yes—
58C2×C32×D4C2×D311yescharacteristic
59C2×#634—31no—
60C2×#634—34no—
61C2×#634—64no—
62C2×C3×Dic6C2×D314yes—
63C32×SD16—62no—
64C2×C32×C8—31no—
65#3757—61no—
66#3757—61no—
67#3757—64no—
68#3757—64no—
69#3757—68no—
70#3757—68no—
71#3756—62no—
72C4⋊(C3⋊Dic3)—61no—
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74C4×C3⋊Dic3—31no—
75C4×C3⋊Dic3—34no—
76C4×C3⋊Dic3—61no—
77C4×C3⋊Dic3—64no—
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C3×#214—28no—
80C3×C4⋊Dic3—34no—
81C2×C3×C3⋊C8—34no—
82C2×C3×C3⋊C8C2×D314yes—
83Dic3⋊Dic3—64no—
84Dic32—64no—
85#3719—68no—
86C22×C33—41no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33C2×D411yescharacteristic
90C33×C4C2×D411yescharacteristic
91C33×C4#4111yescharacteristic
92C3⋊(C3⋊Dic3)—21no—
93C3⋊(C3⋊Dic3)—21no—
94C3⋊(C3⋊Dic3)—41no—
95C3×C3⋊Dic3—41no—
96C32×Dic3—22no—
97C2×C3×SD16—34no—
98Q8×Dic3—94no—
99C2×#216—94no—
100#936—94no—
101#936—91no—
102#936—184no—
103C2×#214C3⋊D311yescharacteristic
104C2×Dic12—91no—
105#887—91no—
106C3×#102—91no—
107#832—34no—
108#816—94no—
109Dic3⋊C8—94no—
110Dic3⋊C8—91no—
111#810—91no—
112C23×C32—28no—
113C23×C32—24no—
114C23×C32—21no—
115C32×Q8—32no—
116C32×Q8—61no—
117C32×D4—21no—
118C32×D4—21no—
119C32×D4—24no—
120C32×D4—24no—
121C32×D4—44no—
122C32×D4—44no—
123C2×C32×C4—24no—
124C2×C32×C4—61no—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—31no—
129C2×C3⋊Dic3—31no—
130C2×C3⋊Dic3—34no—
131C2×C3⋊Dic3—34no—
132C2×C3⋊Dic3—64no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—68no—
135C2×C3⋊Dic3—64no—
136C2×C3⋊Dic3—61no—
137#634—61no—
138#634—61no—
139#634—64no—
140#634—64no—
141#634—124no—
142#634—124no—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—64no—
145C3×Dic6—24no—
146C3×Dic6#6718yes—
147C32×C8—61no—
148C3⋊(C3⋊C8)—21no—
149C3×C3⋊C8—24no—
150C3×C3⋊C8—64no—
151#464—271noSylow
152C2×C33—81no—
153C2×C33#12111yescharacteristic
154C2×C33#13511yescharacteristic
155C2×C33#13711yescharacteristic
156C2×C3×Q8—34no—
157C2×C3×D4—24no—
158C2×C3×D4D3214yes—
159C2×C3×D4C2×C3⋊D311yescharacteristic
160C2×Dic6—94no—
161C2×Dic6—91no—
162C2×Dic6—184no—
163C2×Dic6C2×C3⋊D311yescharacteristic
164C3×SD16—68no—
165C2×C3×C8—34no—
166C2×C3×C8—91no—
167C3×C4⋊C4—91no—
168#217—184no—
169#217—184no—
170#217—181no—
171#217—181no—
172#217—188no—
173#217—188no—
174#216—188no—
175#214—22no—
176C4⋊Dic3—34no—
177C4⋊Dic3—184no—
178Dic3⋊C4—181no—
179C4×Dic3—94no—
180C4×Dic3—91no—
181C4×Dic3—184no—
182C4×Dic3—181no—
183C4×Dic3—184no—
184C2×C3⋊C8—34no—
185C2×C3⋊C8C2×C3⋊D311yescharacteristic
186Dic12—182no—
187C22×C32—24no—
188C22×C32—48no—
189C22×C32—48no—
190C22×C32—48no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32—44no—
194C22×C32—41no—
195C22×C32—44no—
196C22×C32—44no—
197C22×C32D3×D414yes—
198C22×C32D3×D411yescharacteristic
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—62no—
202C32×C4#23511yescharacteristic
203C32×C4D3×D411yescharacteristic
204C32×C4#23714yes—
205C32×C4C2×#6714yes—
206C3⋊Dic3—41no—
207C3⋊Dic3—61no—
208C3⋊Dic3—61no—
209C3⋊Dic3—64no—
210C3⋊Dic3—64no—
211C3⋊Dic3—124no—
212C3⋊Dic3—124no—
213C3⋊Dic3—128no—
214C3⋊Dic3—121no—
215C3⋊Dic3—124no—
216C3×Dic3—28no—
217C3×Dic3—124no—
218C2×Q16—271no—
219C2×SD16—31no—
220#124—271no—
221C4×Q8—271no—
222C4⋊C8—271no—
223#103—271no—
224#102—91no—
225C33#46411yescharacteristic, complemented, Sylow
226C23×C3—28no—
227C23×C3—24no—
228C23×C3—21no—
229C3×Q8—38no—
230C3×Q8—64no—
231C3×D4—24no—
232C3×D4—24no—
233C3×D4—21no—
234C3×D4—21no—
235C3×D4—44no—
236C3×D4—44no—
237C2×C3×C4—24no—
238C2×C3×C4—64no—
239C2×C3×C4—181no—
240C2×C3×C4C2×D3214yes—
241C2×C3×C4C22×C3⋊D311yescharacteristic
242C2×Dic3—21no—
243C2×Dic3—64no—
244C2×Dic3—94no—
245C2×Dic3—94no—
246C2×Dic3—91no—
247C2×Dic3—91no—
248C2×Dic3—184no—
249C2×Dic3—184no—
250C2×Dic3—188no—
251C2×Dic3—184no—
252C2×Dic3—181no—
253Dic6—21no—
254Dic6—184no—
255Dic6—184no—
256Dic6—181no—
257Dic6—181no—
258Dic6—364no—
259Dic6—364no—
260Dic6#63812yes—
261C3×C8—64no—
262C3×C8—181no—
263C3⋊C8—21no—
264C3⋊C8—64no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—88no—
269C2×C32—84no—
270C2×C32—81no—
271C2×C32#88411yescharacteristic
272C2×C32#91211yescharacteristic
273C2×C32#91514yes—
274C2×C32#91511yescharacteristic
275C2×C32#91614yes—
276C2×C32#93914yes—
277C2×Q8—31no—
278C2×Q8—271no—
279C2×D4D3×C3⋊D311yescharacteristic
280Q16—542no—
281SD16—62no—
282C2×C8—31no—
283C2×C8—91no—
284C4⋊C4—91no—
285C4⋊C4—541no—
286C22⋊C4—541no—
287C22⋊C4—541no—
288C42—271no—
289C42—541no—
290C22×C3—24no—
291C22×C3—48no—
292C22×C3—48no—
293C22×C3—48no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—44no—
298C22×C3—44no—
299C22×C3—44no—
300C22×C3#381114yes—
301C22×C3D4×C3⋊D311yescharacteristic
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—68no—
305C3×C4—361no—
306C3×C4#380414yes—
307C3×C4D3×#6714yes—
308C3×C4#383011yescharacteristic
309C3×C4C2×#63811yescharacteristic
310Dic3—22no—
311Dic3—124no—
312Dic3—181no—
313Dic3—181no—
314Dic3—184no—
315Dic3—184no—
316Dic3—364no—
317Dic3—364no—
318Dic3—368no—
319Dic3—361no—
320Dic3—364no—
321C32—24no—
322C32#476711yescharacteristic, complemented
323C32#516614yescomplemented
324C23—21no—
325Q8—32no—
326Q8—61no—
327Q8—541no—
328Q8—541no—
329D4—21no—
330D4—21no—
331C2×C4—61no—
332C2×C4—181no—
333C2×C4—271no—
334C2×C4—271no—
335C2×C4—541no—
336C2×C4C2×D3×C3⋊D311yescharacteristic
337C8—61no—
338C8—181no—
339C2×C3—24no—
340C2×C3—24no—
341C2×C3—24no—
342C2×C3—88no—
343C2×C3—81no—
344C2×C3—84no—
345C2×C3#6393114yes—
346C2×C3#6393614yes—
347C2×C3#6397114yes—
348C2×C3#6412411yescharacteristic
349C2×C3#6412511yescharacteristic
350C2×C3#6414811yescharacteristic
351C22—41no—
352C22—41no—
353C22—41no—
354C22#8903911yescharacteristic
355C4—62no—
356C4—361no—
357C4—541no—
358C4—541no—
359C4—1081no—
360C4#8902811yescharacteristic
361C4D3×#63811yescharacteristic
362C3—24no—
363C3#1059144314yescomplemented
364C3#1059286511yescharacteristic, complemented
365C2—81no—
366C2#1172355411yescharacteristic
367C2#1172356511yescharacteristic
368C2#1172361611yescharacteristic
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#464
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
3. #11723397C2semidirect
4. C2×#88799C2semidirect
6. #11722050C2semidirect
7. #11722042C2semidirect
225. C33#464semidirect
322. C32#4767semidirect
323. C32#5166semidirect
363. C3#10591443semidirect
364. C3#10592865semidirect

Maximal subgroups

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

subgroupindex
#117239772
C2×C32×#2142
#117233972
C2×#887992
C3×#636592
#117220502
#117220422
#105928653
#105914433