Finite Group Databasebeta

← #13161020

Subgroups — #13161020

Order 1728.

992 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 172 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×#88861C211yesmaximal, characteristic
2C3×D4×C3⋊Dic3C211yesmaximal, characteristic
3C2×C32×#214C211yesmaximal, characteristic
4#11723455C211yesmaximal, characteristic, complemented
5#11722051C211yesmaximal, characteristic, complemented
6#11722042C211yesmaximal, characteristic, complemented
7#11722037C211yesmaximal, characteristic, complemented
8#10592861—31nomaximal
9#10591489—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C22×C3×C3⋊Dic3—21no—
13C2×C32×Dic6C2211yescharacteristic
14#88861—22no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3757—21no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C32×#214—22no—
20C2×C32×C3⋊C8C2211yescharacteristic
21#88807—21no—
22#88807—21no—
23C2×C32×SD16—31no—
24#64153—31no—
25D4×C3⋊Dic3D311yescharacteristic
26C2×#3754—31no—
27C2×#3754—34no—
28C2×#3754—64no—
29C3×D4×Dic3—34no—
30C2×C3×#214D314yes—
31#63866—34no—
32#63662—31no—
33#63643—31no—
34#63641—31no—
35Dic3⋊(C3⋊C8)—34no—
36#63567—34no—
37#63565—34no—
38C23×C33—21no—
39C33×D4—21no—
40C33×D4D412yes—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43#6308—21no—
44#6308—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3—42no—
47C2×C3×C3⋊Dic3D411yescharacteristic
48C2×C3×C3⋊Dic3D411yescharacteristic
49C2×C32×Dic3—21no—
50C32×Dic6—21no—
51C32×Dic6—21no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C32×C3⋊C8—21no—
54#5162—94no—
55#5019—91no—
56C2×C32×Q8—31no—
57C2×C32×D4—24no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C32×D4C2×D314yes—
60C22×C3⋊Dic3—21no—
61C2×#634—31no—
62C2×#634—34no—
63C2×#634—64no—
64C22×C3×Dic3—64no—
65C2×C3×Dic6C2×D314yes—
66C32×SD16—62no—
67C2×C32×C8—31no—
68#3757—21no—
69#3754—616no—
70#3754—62no—
71#3754—68no—
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73#3750—61no—
74#3750—61no—
75C4×C3⋊Dic3C2×D311yescharacteristic
76C2×C3⋊(C3⋊C8)—31no—
77C2×C3⋊(C3⋊C8)—34no—
78C2×C3⋊(C3⋊C8)—64no—
79C3×#217—64no—
80C3×#214—28no—
81C3×C4⋊Dic3—34no—
82C3×C4×Dic3—34no—
83C2×C3×C3⋊C8C2×D314yes—
84Dic3⋊Dic3—64no—
85Dic3⋊Dic3—64no—
86C22×C33—22no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33C2×D411yescharacteristic
90C33×C4C2×D411yescharacteristic
91C33×C4#4111yescharacteristic
92C3⋊(C3⋊Dic3)—41no—
93C3×C3⋊Dic3—21no—
94C3×C3⋊Dic3—41no—
95C3×C3⋊Dic3SD1612yescomplemented
96C32×Dic3—41no—
97C2×C3×SD16—34no—
98C3×C4×D4—91no—
99#944—94no—
100D4×Dic3—34no—
101C2×#214—94no—
102C2×#214—91no—
103C2×#214—184no—
104C2×#214C3⋊D311yescharacteristic
105#869—91no—
106#835—94no—
107#816—94no—
108Dic3⋊C8—94no—
109#810—91no—
110#810—91no—
111C4⋊(C3⋊C8)—91no—
112C23×C32—21no—
113C23×C32—24no—
114C23×C32—44no—
115C32×Q8—61no—
116C32×Q8—61no—
117C32×D4—21no—
118C32×D4—28no—
119C32×D4—24no—
120C32×D4—44no—
121C32×D4D1212yes—
122C32×D4#6718yes—
123C2×C32×C4—24no—
124C2×C32×C4—61no—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—42no—
129C2×C3⋊Dic3—61no—
130C2×C3⋊Dic3—64no—
131C2×C3⋊Dic3—124no—
132C2×C3⋊Dic3#6711yescharacteristic
133C2×C3⋊Dic3#6711yescharacteristic
134#634—68no—
135#634—61no—
136#634—61no—
137#634—64no—
138#634—64no—
139#634—124no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3—34no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3—128no—
145C3×Dic6—24no—
146C3×Dic6—24no—
147C32×C8—61no—
148C3⋊(C3⋊C8)—68no—
149C3⋊(C3⋊C8)—61no—
150C3⋊(C3⋊C8)—64no—
151C3×C3⋊C8—24no—
152#461—271noSylow
153C2×C33—42no—
154C2×C33#12111yescharacteristic
155C2×C33C2×SD1611yescharacteristic
156C2×C33#13711yescharacteristic
157C2×C3×Q8—34no—
158C2×C3×D4—24no—
159C2×C3×D4D3214yes—
160C2×C3×D4C2×C3⋊D311yescharacteristic
161C22×C3×C4—181no—
162C22×Dic3—64no—
163C2×Dic6—94no—
164C2×Dic6—91no—
165C2×Dic6—184no—
166C2×Dic6C2×C3⋊D311yescharacteristic
167C3×SD16—68no—
168C2×C3×C8—34no—
169C3×C4⋊C4—91no—
170C3×C22⋊C4—181no—
171C3×C42—91no—
172#217—64no—
173#214—22no—
174#214—1816no—
175#214—188no—
176#214—182no—
177C4⋊Dic3—34no—
178C4⋊Dic3—181no—
179C4⋊Dic3—181no—
180Dic3⋊C4—184no—
181Dic3⋊C4—184no—
182C4×Dic3—34no—
183C2×C3⋊C8—94no—
184C2×C3⋊C8—91no—
185C2×C3⋊C8—184no—
186C2×C3⋊C8C2×C3⋊D311yescharacteristic
187C22×C32—22no—
188C22×C32—24no—
189C22×C32—28no—
190C22×C32—48no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32—44no—
194C22×C32—44no—
195C22×C32—84no—
196C22×C32—84no—
197C22×C32D3×D414yes—
198C22×C32C2×#6711yescharacteristic
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—121no—
202C32×C4C2×D1211yescharacteristic
203C32×C4#23511yescharacteristic
204C32×C4#23714yes—
205C32×C4C2×#6714yes—
206C3⋊Dic3—21no—
207C3⋊Dic3—41no—
208C3⋊Dic3—128no—
209C3⋊Dic3—121no—
210C3⋊Dic3—124no—
211C3⋊Dic3#21412yescomplemented
212C3×Dic3—38no—
213C3×Dic3—44no—
214C3×Dic3—64no—
215C3×Dic3—124no—
216C2×SD16—31no—
217C2×SD16—271no—
218C4⋊Q8—271no—
219C4×D4—91no—
220C4⋊C8—271no—
221#103—271no—
222#103—271no—
223C33#46111yescharacteristic, complemented, Sylow
224C23×C3—24no—
225C23×C3—21no—
226C23×C3—44no—
227C3×Q8—64no—
228C3×Q8—64no—
229C3×D4—24no—
230C3×D4—21no—
231C3×D4—28no—
232C3×D4—44no—
233C3×D4#62618yes—
234C3×D4#63812yes—
235C2×C3×C4—24no—
236C2×C3×C4—64no—
237C2×C3×C4—91no—
238C2×C3×C4—91no—
239C2×C3×C4—181no—
240C2×C3×C4—362no—
241C2×C3×C4C2×D3214yes—
242C2×C3×C4C22×C3⋊D311yescharacteristic
243C2×Dic3—21no—
244C2×Dic3—34no—
245C2×Dic3—34no—
246C2×Dic3—64no—
247C2×Dic3—128no—
248C2×Dic3—184no—
249C2×Dic3—181no—
250C2×Dic3—364no—
251Dic6—21no—
252Dic6—21no—
253Dic6—188no—
254Dic6—184no—
255Dic6—184no—
256Dic6—181no—
257Dic6—181no—
258Dic6—364no—
259C3×C8—64no—
260C3⋊C8—21no—
261C3⋊C8—188no—
262C3⋊C8—184no—
263C3⋊C8—181no—
264C2×C32—24no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—42no—
268C2×C32—48no—
269C2×C32—88no—
270C2×C32D3×SD1614yes—
271C2×C32#91514yes—
272C2×C32#93011yescharacteristic
273C2×C32C2×#21411yescharacteristic
274C2×C32#93914yes—
275C2×C32#95111yescharacteristic
276C2×Q8—31no—
277C2×Q8—271no—
278C2×D4D3×C3⋊D311yescharacteristic
279C22×C4—181no—
280SD16—62no—
281SD16—542no—
282C2×C8—31no—
283C2×C8—271no—
284C4⋊C4—91no—
285C4⋊C4—541no—
286C4⋊C4—541no—
287C22⋊C4—181no—
288C42—91no—
289C22×C3—22no—
290C22×C3—28no—
291C22×C3—24no—
292C22×C3—48no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—44no—
296C22×C3—44no—
297C22×C3—84no—
298C22×C3—84no—
299C22×C3D3×#6714yes—
300C22×C3D4×C3⋊D311yescharacteristic
301C3×C4—24no—
302C3×C4—24no—
303C3×C4—92no—
304C3×C4—124no—
305C3×C4—181no—
306C3×C4—361no—
307C3×C4#380414yes—
308C3×C4C2×#62614yes—
309C3×C4#383011yescharacteristic
310C3×C4C2×#63811yescharacteristic
311Dic3—38no—
312Dic3—41no—
313Dic3—64no—
314Dic3—124no—
315Dic3—368no—
316Dic3—361no—
317Dic3—364no—
318C32—24no—
319C32#501911yescharacteristic, complemented
320C32#516214yescomplemented
321C23—21no—
322Q8—61no—
323Q8—61no—
324Q8—541no—
325Q8—541no—
326D4—21no—
327D4#629112yes—
328C2×C4—61no—
329C2×C4—91no—
330C2×C4—91no—
331C2×C4—181no—
332C2×C4—362no—
333C2×C4—541no—
334C2×C4C2×D3×C3⋊D311yescharacteristic
335C8—61no—
336C8—541no—
337C2×C3—24no—
338C2×C3—24no—
339C2×C3—24no—
340C2×C3—48no—
341C2×C3—42no—
342C2×C3—88no—
343C2×C3D3×#21414yes—
344C2×C3#6392914yes—
345C2×C3#6397814yes—
346C2×C3SD16×C3⋊D311yescharacteristic
347C2×C3#6412411yescharacteristic
348C2×C3#6414811yescharacteristic
349C22—22no—
350C22—41no—
351C22—41no—
352C22C3⋊D3×#6711yescharacteristic
353C4—92no—
354C4—121no—
355C4—181no—
356C4—361no—
357C4—1081no—
358C4#8902811yescharacteristic
359C4C2×#629111yescharacteristic
360C3—24no—
361C3#1059148914yescomplemented
362C3#1059286111yescharacteristic, complemented
363C2—42no—
364C2C3⋊D3×#21411yescharacteristic
365C2#1172355211yescharacteristic
366C2#1172363511yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#461
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. #11723455C2semidirect
5. #11722051C2semidirect
6. #11722042C2semidirect
7. #11722037C2semidirect
95. C3×C3⋊Dic3SD16semidirect
211. C3⋊Dic3#214semidirect
223. C33#461semidirect
319. C32#5019semidirect
320. C32#5162semidirect
361. C3#10591489semidirect
362. C3#10592861semidirect

Maximal subgroups

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

subgroupindex
C2×#888612
C3×D4×C3⋊Dic32
C2×C32×#2142
#117234552
#117220512
#117220422
#117220372
#105928613
#105914893