Finite Group Databasebeta

← #13161027

Subgroups — #13161027

Order 1728.

992 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 188 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
2C2×C3×#3754C211yesmaximal, characteristic
3C32×D4×Dic3C211yesmaximal, characteristic
4#11723443C211yesmaximal, characteristic, complemented
5Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6#11722041C211yesmaximal, characteristic, complemented
7#11722039C211yesmaximal, characteristic, complemented
8#10592718—31nomaximal
9#10591489—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C22×C32×Dic3—21no—
14#88861—22no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3754—22no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#217—21no—
19C32×C4⋊Dic3C2211yescharacteristic
20C32×C4×Dic3C2211yescharacteristic
21Dic3⋊(C3⋊Dic3)—21no—
22Dic3⋊(C3⋊Dic3)—21no—
23C32×C4×D4—31no—
24C2×#3754—31no—
25C2×#3754—34no—
26C2×#3754—64no—
27C2×#3754D311yescharacteristic
28#64078—31no—
29C3×D4×Dic3D314yes—
30C2×C3×#214—34no—
31#63866—34no—
32#63637—31no—
33#63637—31no—
34C4⋊(C3⋊(C3⋊C8))—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—
46C3×#634—21no—
47C3×#634—21no—
48C2×C32×Dic3—21no—
49C2×C32×Dic3—42no—
50C2×C32×Dic3D411yescharacteristic
51C2×C32×Dic3D411yescharacteristic
52C3⋊(C3⋊(C3⋊C8))—21no—
53C3×C3⋊(C3⋊C8)—21no—
54#5162—91no—
55#5019—94no—
56C2×C32×D4—24no—
57C2×C32×D4C2×D314yes—
58C2×C32×D4C2×D311yescharacteristic
59C22×C32×C4—61no—
60C2×#634—31no—
61C2×#634—34no—
62C2×#634—64no—
63C2×#634C2×D311yescharacteristic
64C22×C3×Dic3—24no—
65C2×C3×Dic6—34no—
66C32×C4⋊C4—31no—
67C32×C22⋊C4—61no—
68C32×C42—31no—
69#3754—22no—
70#3754—616no—
71#3754—68no—
72#3754—62no—
73C4⋊(C3⋊Dic3)—61no—
74C4⋊(C3⋊Dic3)—61no—
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—64no—
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C3×#217—24no—
80C3×#214—68no—
81C3×C4⋊Dic3C2×D314yes—
82C3×C4×Dic3C2×D314yes—
83C2×C3×C3⋊C8—34no—
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—41no—
94C32×Dic3—21no—
95C32×Dic3—41no—
96C32×Dic3SD1612yescomplemented
97C2×C3×SD16—91no—
98C3×C4×D4—34no—
99#944—91no—
100D4×Dic3C3⋊D311yescharacteristic
101C2×#214—34no—
102C2×#214—94no—
103C2×#214—91no—
104C2×#214—184no—
105#869—94no—
106#835—91no—
107#816—91no—
108Dic3⋊C8—91no—
109#810—94no—
110#810—94no—
111C4⋊(C3⋊C8)—94no—
112C23×C32—21no—
113C23×C32—24no—
114C23×C32—44no—
115C32×D4—21no—
116C32×D4—28no—
117C32×D4—24no—
118C32×D4—44no—
119C32×D4D1218yes—
120C32×D4#6712yes—
121C2×C32×C4—24no—
122C2×C32×C4—31no—
123C2×C32×C4—31no—
124C2×C32×C4—61no—
125C2×C32×C4—122no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—61no—
130C2×C3⋊Dic3—64no—
131C2×C3⋊Dic3—124no—
132#634—21no—
133#634—21no—
134#634—68no—
135#634—64no—
136#634—61no—
137#634—61no—
138#634—64no—
139#634—124no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3—48no—
142C2×C3×Dic3—64no—
143C2×C3×Dic3#6714yes—
144C2×C3×Dic3#6714yes—
145C3×Dic6—64no—
146C3×Dic6—64no—
147C3⋊(C3⋊C8)—21no—
148C3⋊(C3⋊C8)—68no—
149C3⋊(C3⋊C8)—64no—
150C3⋊(C3⋊C8)—61no—
151C3×C3⋊C8—64no—
152#461—271noSylow
153C2×C33—42no—
154C2×C33#12111yescharacteristic
155C2×C33C2×SD1611yescharacteristic
156C2×C33#13711yescharacteristic
157C2×C3×Q8—91no—
158C2×C3×D4—24no—
159C2×C3×D4D3214yes—
160C2×C3×D4C2×C3⋊D311yescharacteristic
161C22×C3×C4—64no—
162C22×Dic3—21no—
163C2×Dic6—34no—
164C2×Dic6—94no—
165C2×Dic6—91no—
166C2×Dic6—184no—
167C3×SD16—182no—
168C2×C3×C8—91no—
169C3×C4⋊C4—34no—
170C3×C22⋊C4—64no—
171C3×C42—34no—
172#217—21no—
173#214—68no—
174#214—1816no—
175#214—188no—
176#214—182no—
177C4⋊Dic3—184no—
178C4⋊Dic3—184no—
179C4⋊Dic3C2×C3⋊D311yescharacteristic
180Dic3⋊C4—181no—
181Dic3⋊C4—181no—
182C4×Dic3C2×C3⋊D311yescharacteristic
183C2×C3⋊C8—34no—
184C2×C3⋊C8—94no—
185C2×C3⋊C8—91no—
186C2×C3⋊C8—184no—
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×D411yescharacteristic
198C22×C32C2×#6714yes—
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—32no—
202C32×C4—61no—
203C32×C4—121no—
204C32×C4C2×D1214yes—
205C32×C4#23514yes—
206C32×C4#23711yescharacteristic
207C32×C4C2×#6711yescharacteristic
208C3⋊Dic3—41no—
209C3⋊Dic3—128no—
210C3⋊Dic3—124no—
211C3⋊Dic3—121no—
212C3×Dic3—24no—
213C3×Dic3—44no—
214C3×Dic3—124no—
215C3×Dic3#21418yescomplemented
216C2×SD16—91no—
217C2×SD16—271no—
218C4⋊Q8—271no—
219C4×D4—31no—
220C4⋊C8—271no—
221#103—271no—
222#103—271no—
223C33#46111yescharacteristic, complemented, Sylow
224C23×C3—24no—
225C23×C3—21no—
226C23×C3—44no—
227C3×Q8—181no—
228C3×Q8—181no—
229C3×D4—24no—
230C3×D4—21no—
231C3×D4—28no—
232C3×D4—44no—
233C3×D4#62618yes—
234C3×D4C4⋊(C3⋊D3)12yes—
235C2×C3×C4—24no—
236C2×C3×C4—34no—
237C2×C3×C4—34no—
238C2×C3×C4—64no—
239C2×C3×C4—128no—
240C2×C3×C4—181no—
241C2×C3×C4C2×D3214yes—
242C2×C3×C4C22×C3⋊D311yescharacteristic
243C2×Dic3—21no—
244C2×Dic3—42no—
245C2×Dic3—64no—
246C2×Dic3—184no—
247C2×Dic3—181no—
248C2×Dic3—364no—
249C2×Dic3#63811yescharacteristic
250C2×Dic3#63811yescharacteristic
251Dic6—64no—
252Dic6—64no—
253Dic6—188no—
254Dic6—184no—
255Dic6—184no—
256Dic6—181no—
257Dic6—181no—
258Dic6—364no—
259C3×C8—181no—
260C3⋊C8—64no—
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×SD1611yescharacteristic
271C2×C32#91511yescharacteristic
272C2×C32#93014yes—
273C2×C32C2×#21414yes—
274C2×C32#93911yescharacteristic
275C2×C32#95114yes—
276C2×Q8—91no—
277C2×Q8—271no—
278C2×D4D3×C3⋊D311yescharacteristic
279C22×C4—61no—
280SD16—182no—
281SD16—542no—
282C2×C8—91no—
283C2×C8—271no—
284C4⋊C4—31no—
285C4⋊C4—541no—
286C4⋊C4—541no—
287C22⋊C4—61no—
288C42—31no—
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×C3C2×#63811yescharacteristic
301C3×C4—24no—
302C3×C4—24no—
303C3×C4—38no—
304C3×C4—64no—
305C3×C4—124no—
306C3×C4—361no—
307C3×C4#380414yes—
308C3×C4C2×#62614yes—
309C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
310C3×C4#382811yescharacteristic
311Dic3—21no—
312Dic3—41no—
313Dic3—124no—
314Dic3—368no—
315Dic3—361no—
316Dic3—364no—
317Dic3#375412yescomplemented
318C32—24no—
319C32#501914yescomplemented
320C32#516211yescharacteristic, complemented
321C23—21no—
322Q8—181no—
323Q8—181no—
324Q8—541no—
325Q8—541no—
326D4—21no—
327D4#629212yes—
328C2×C4—31no—
329C2×C4—31no—
330C2×C4—61no—
331C2×C4—122no—
332C2×C4—181no—
333C2×C4—541no—
334C2×C4C2×D3×C3⋊D311yescharacteristic
335C8—181no—
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×C3#6413911yescharacteristic
347C2×C3C2×#375411yescharacteristic
348C2×C3#6416011yescharacteristic
349C22—22no—
350C22—41no—
351C22—41no—
352C22D3×#63811yescharacteristic
353C4—32no—
354C4—61no—
355C4—121no—
356C4—361no—
357C4—1081no—
358C4#8903111yescharacteristic
359C4C2×#629211yescharacteristic
360C3—24no—
361C3#1059148914yescomplemented
362C3#1059271811yescharacteristic, complemented
363C2—42no—
364C2D3×#375411yescharacteristic
365C2#1172355011yescharacteristic
366C2#1172363211yescharacteristic
subgroupG/H
centerC22D3×#638
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. #11723443C2semidirect
5. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
6. #11722041C2semidirect
7. #11722039C2semidirect
96. C32×Dic3SD16semidirect
215. C3×Dic3#214semidirect
223. C33#461semidirect
317. Dic3#3754semidirect
319. C32#5019semidirect
320. C32#5162semidirect
361. C3#10591489semidirect
362. C3#10592718semidirect

Maximal subgroups

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

subgroupindex
C2×#888612
C2×C3×#37542
C32×D4×Dic32
#117234432
Dic3⋊(C3⋊(C3⋊C8))2
#117220412
#117220392
#105927183
#105914893