Finite Group Databasebeta

← #13161024

Subgroups — #13161024

Order 1728.

908 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 176 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
1C32×#940C211yesmaximal, characteristic
2#11723534C211yesmaximal, characteristic, complemented
3#11722202C211yesmaximal, characteristic, complemented
4C3×#63659C211yesmaximal, characteristic, complemented
5#11722032C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic, complemented
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10592769—31nomaximal
9#10591487—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C32×#67—21no—
12C2×C32×#67—21no—
13C2×C32×D12C2211yescharacteristic
14C2×#6293—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21C32×#127—31no—
22#64078—31no—
23C3×#940D314yes—
24#63918—34no—
25#63659—31no—
26#63659—31no—
27#63659—31no—
28#63659—34no—
29#63659—64no—
30#63659D311yescharacteristic, complemented
31C4×C3⋊(C3⋊C8)—31no—
32C3×#832—34no—
33#63563—34no—
34#63562—34no—
35Dic3×C3⋊C8—34no—
36C23×C33—21no—
37C22×C32×D3—21no—
38C33×D4—21no—
39C33×D4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3×C3⋊Dic3—21no—
43C32×#67—42no—
44C32×#67—42no—
45C2×C32×Dic3D411yescharacteristic
46C2×C32×Dic3D411yescharacteristic
47C32×D12—21no—
48C32×D12—21no—
49#6293—42no—
50C3⋊(C3⋊(C3⋊C8))—21no—
51C3×C3⋊(C3⋊C8)—21no—
52#5070—94no—
53#4760—91no—
54C2×C32×D4—24no—
55C2×C32×D4—31no—
56C2×C32×D4—61no—
57C2×C32×D4—61no—
58C2×C32×D4C2×D314yes—
59C2×C32×D4C2×D311yescharacteristic
60C2×#634—61no—
61C2×C3×#67—24no—
62C2×C3×#67—24no—
63C2×C3×D12C2×D314yes—
64C2×#627—64no—
65C32×C42—31no—
66C4⋊(C3⋊Dic3)—31no—
67C4⋊(C3⋊Dic3)—34no—
68C4⋊(C3⋊Dic3)—61no—
69C4⋊(C3⋊Dic3)—64no—
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—34no—
73C2×C3⋊(C3⋊C8)—64no—
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C3×C4⋊Dic3—34no—
76C3×C4×Dic3C2×D314yes—
77C2×C3×C3⋊C8—34no—
78Dic3⋊Dic3—64no—
79C22×C33—41no—
80C22×C33—41no—
81C22×C33—41no—
82C22×C33C2×D411yescharacteristic
83C2×C32×D3—41no—
84C2×C32×D3—41no—
85C2×C32×D3—41no—
86C33×C4#4111yescharacteristic
87C33×C4#4111yescharacteristic
88C3⋊(C3⋊Dic3)—41no—
89C3×C3⋊Dic3—41no—
90C32×Dic3D812yes—
91C32×Dic3SD1612yes—
92C3×#127—34no—
93#940C3⋊D311yescharacteristic
94#888—91no—
95#869—94no—
96C3×#102—91no—
97#832—34no—
98#832—94no—
99#832—94no—
100#832—94no—
101#832—91no—
102#832—184no—
103#821—91no—
104C8×Dic3—91no—
105#809—91no—
106C4×C3⋊C8—94no—
107C23×C32—21no—
108C23×C32—24no—
109C23×C32—44no—
110C23×C32—61no—
111C22×C3×D3—24no—
112C32×D4—28no—
113C32×D4—21no—
114C32×D4—24no—
115C32×D4—21no—
116C32×D4—24no—
117C32×D4—44no—
118C32×D4—61no—
119C32×D4—61no—
120C32×D4—122no—
121C32×D4—122no—
122C2×C32×C4—24no—
123C2×C32×C4—31no—
124C2×C32×C4—31no—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—61no—
129C2×C3⋊Dic3—64no—
130C2×C3⋊Dic3—124no—
131#634—122no—
132C3×#67—48no—
133C3×#67—48no—
134C2×C3×Dic3—64no—
135C2×C3×Dic3#6714yes—
136C2×C3×Dic3#6714yes—
137C3×D12—24no—
138C3×D12—24no—
139#627—128no—
140C3⋊(C3⋊C8)—21no—
141C3⋊(C3⋊C8)—61no—
142C3⋊(C3⋊C8)—64no—
143C3⋊(C3⋊C8)—124no—
144C3×C3⋊C8—64no—
145#486—271noSylow
146C2×C33—81no—
147C2×C33#12411yescharacteristic
148C2×C33C2×D811yescharacteristic
149C2×C33C2×SD1611yescharacteristic
150C32×D3—81no—
151C2×C3×D4—24no—
152C2×C3×D4—34no—
153C2×C3×D4—64no—
154C2×C3×D4—64no—
155C2×C3×D4D3214yes—
156C2×C3×D4C2×C3⋊D311yescharacteristic
157C2×#67—21no—
158C2×#67—21no—
159C2×D12C2×C3⋊D311yescharacteristic
160C2×Dic6—184no—
161C2×Dic6—181no—
162C2×C3×C8—91no—
163C3×C4⋊C4—91no—
164C3×C42—34no—
165C4⋊Dic3—34no—
166C4⋊Dic3—94no—
167C4⋊Dic3—91no—
168C4⋊Dic3—184no—
169C4⋊Dic3—184no—
170Dic3⋊C4—181no—
171C4×Dic3C2×C3⋊D311yescharacteristic
172C2×C3⋊C8—34no—
173C2×C3⋊C8—94no—
174C2×C3⋊C8—91no—
175C2×C3⋊C8—184no—
176C22×C32—24no—
177C22×C32—48no—
178C22×C32—41no—
179C22×C32—41no—
180C22×C32—41no—
181C22×C32—44no—
182C22×C32—44no—
183C22×C32—44no—
184C22×C32—84no—
185C22×C32—84no—
186C22×C32—121no—
187C22×C32—121no—
188C22×C32—121no—
189C22×C32D3×D411yescharacteristic
190C22×C32C2×#6714yes—
191C2×C3×D3—44no—
192C2×C3×D3—44no—
193C2×C3×D3—44no—
194C32×C4—24no—
195C32×C4—24no—
196C32×C4—32no—
197C32×C4—32no—
198C32×C4#23511yescharacteristic
199C32×C4#23714yes—
200C32×C4#23714yes—
201C32×C4#23711yescharacteristic
202C3⋊Dic3—41no—
203C3⋊Dic3—121no—
204C3⋊Dic3—124no—
205C3⋊Dic3—244no—
206C3×Dic3—124no—
207C3×Dic3#21318yes—
208C3×Dic3#21418yes—
209C4⋊Q8—271no—
210#127—31no—
211#102—91no—
212#102—271no—
213#102—271no—
214#102—271no—
215C4×C8—271no—
216C33#48611yescharacteristic, complemented, Sylow
217C23×C3—24no—
218C23×C3—21no—
219C23×C3—44no—
220C23×C3—64no—
221C22×D3—21no—
222C3×D4—28no—
223C3×D4—24no—
224C3×D4—24no—
225C3×D4—21no—
226C3×D4—21no—
227C3×D4—44no—
228C3×D4—64no—
229C3×D4—64no—
230C3×D4—128no—
231C3×D4—128no—
232C2×C3×C4—24no—
233C2×C3×C4—34no—
234C2×C3×C4—34no—
235C2×C3×C4—181no—
236C2×C3×C4C2×D3214yes—
237C2×C3×C4C22×C3⋊D311yescharacteristic
238#67—42no—
239#67—42no—
240C2×Dic3—64no—
241C2×Dic3—184no—
242C2×Dic3—181no—
243C2×Dic3—364no—
244C2×Dic3#63811yescharacteristic
245C2×Dic3#63811yescharacteristic
246D12—21no—
247D12—21no—
248Dic6—368no—
249Dic6—362no—
250C3×C8—181no—
251C3⋊C8—64no—
252C3⋊C8—184no—
253C3⋊C8—181no—
254C3⋊C8—364no—
255C2×C32—24no—
256C2×C32—24no—
257C2×C32—24no—
258C2×C32—88no—
259C2×C32—81no—
260C2×C32—84no—
261C2×C32—241no—
262C2×C32#88511yescharacteristic
263C2×C32D3×D811yescharacteristic
264C2×C32D3×SD1611yescharacteristic
265C2×C32C2×#21314yes—
266C2×C32C2×#21414yes—
267C2×C32#93614yes—
268C3×D3—84no—
269C2×Q8—541no—
270C2×D4—31no—
271C2×D4—61no—
272C2×D4—61no—
273C2×D4D3×C3⋊D311yescharacteristic
274C2×C8—91no—
275C2×C8—271no—
276C4⋊C4—91no—
277C4⋊C4—271no—
278C4⋊C4—541no—
279C42—31no—
280C22×C3—24no—
281C22×C3—48no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—44no—
286C22×C3—44no—
287C22×C3—44no—
288C22×C3—84no—
289C22×C3—84no—
290C22×C3—124no—
291C22×C3—124no—
292C22×C3—124no—
293C22×C3D3×#6714yes—
294C22×C3C2×#63811yescharacteristic
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—41no—
298C3×C4—24no—
299C3×C4—24no—
300C3×C4—38no—
301C3×C4—38no—
302C3×C4—361no—
303C3×C4#380414yes—
304C3×C4#380514yes—
305C3×C4#383011yescharacteristic
306C3×C4#383011yescharacteristic
307Dic3—124no—
308Dic3—361no—
309Dic3—364no—
310Dic3—724no—
311Dic3#375312yes—
312Dic3#375412yes—
313C32—24no—
314C32#476011yescharacteristic, complemented
315C32#507014yescomplemented
316C23—21no—
317C23—61no—
318Q8—1082no—
319D4—21no—
320D4—21no—
321D4—61no—
322D4—61no—
323D4—122no—
324D4—122no—
325C2×C4—31no—
326C2×C4—31no—
327C2×C4—181no—
328C2×C4—541no—
329C2×C4C2×D3×C3⋊D311yescharacteristic
330C8—181no—
331C8—541no—
332C2×C3—24no—
333C2×C3—24no—
334C2×C3—24no—
335C2×C3—88no—
336C2×C3—81no—
337C2×C3—84no—
338C2×C3—244no—
339C2×C3D3×#21314yes—
340C2×C3D3×#21414yes—
341C2×C3#6395914yes—
342C2×C3C2×#375311yescharacteristic
343C2×C3C2×#375411yescharacteristic
344C2×C3#6414511yescharacteristic
345D3—81no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—121no—
350C22—121no—
351C22—121no—
352C22D3×#63811yescharacteristic
353C4—32no—
354C4—32no—
355C4—361no—
356C4—1081no—
357C4#8902811yescharacteristic
358C4#8902911yescharacteristic
359C3—24no—
360C3#1059148714yescomplemented
361C3#1059276911yescharacteristic, complemented
362C2—81no—
363C2—241no—
364C2D3×#375311yescharacteristic
365C2D3×#375411yescharacteristic
366C2#1172360011yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC32×#940C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#486
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
2. #11723534C2semidirect
3. #11722202C2semidirect
4. C3×#63659C2semidirect
5. #11722032C2semidirect
6. #11722031C2semidirect
7. Dic3×C3⋊(C3⋊C8)C2semidirect
30. #63659D3semidirect
216. C33#486semidirect
314. C32#4760semidirect
315. C32#5070semidirect
360. C3#10591487semidirect
361. C3#10592769semidirect

Maximal subgroups

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

subgroupindex
C32×#9402
#117235342
#117222022
C3×#636592
#117220322
#117220312
Dic3×C3⋊(C3⋊C8)2
#105927693
#105914873