Finite Group Databasebeta

← #13161019

Subgroups — #13161019

Order 1728.

800 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C3×#64145C211yesmaximal, characteristic
2#11723535C211yesmaximal, characteristic, complemented
3#11722202C211yesmaximal, characteristic
4C32×#832C211yesmaximal, characteristic
5#11722040C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7#11722020C211yesmaximal, characteristic, complemented
8#10592464—31nomaximal
9#10591488—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×#6293—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3757—21no—
16C3×#3757—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88807—21no—
21#64145D311yescharacteristic
22#64097—31no—
23C3×#936—34no—
24#63918—34no—
25C32×#102—31no—
26#63659—31no—
27#63659—34no—
28#63659—64no—
29#63643—31no—
30#63642—31no—
31#63637—31no—
32C3×#832D314yes—
33#63566—34no—
34#63565—34no—
35#63554—34no—
36C23×C33—21no—
37C33×D4—21no—
38C33×D4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×#634—21no—
45C3×#634—21no—
46C2×C32×Dic3—21no—
47#6293—42no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C32×C3⋊C8—21no—
50#5055—91no—
51#4765—94no—
52C2×C32×D4—24no—
53C2×C32×D4C2×D314yes—
54C2×C32×D4C2×D311yescharacteristic
55C2×#634—61no—
56C2×#634C2×D311yescharacteristic
57C2×C3×Dic6—34no—
58C2×#627—64no—
59C2×C32×C8—31no—
60C32×C4⋊C4—31no—
61#3757—21no—
62#3757—21no—
63C4⋊(C3⋊Dic3)—31no—
64C4⋊(C3⋊Dic3)—34no—
65C4⋊(C3⋊Dic3)—64no—
66#3750—61no—
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—64no—
71C3×#217—64no—
72C3×#217—64no—
73C3×C4⋊Dic3C2×D314yes—
74C3×C4×Dic3—34no—
75C2×C3×C3⋊C8C2×D314yes—
76Dic3⋊Dic3—64no—
77C22×C33—41no—
78C22×C33—41no—
79C22×C33—41no—
80C22×C33C2×D411yescharacteristic
81C33×C4#4111yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—41no—
84C3×C3⋊Dic3—21no—
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3—41no—
87C32×Dic3—41no—
88C3×#124—91no—
89#936—34no—
90#888—94no—
91#869—91no—
92C3×#102—34no—
93#835—91no—
94#835—91no—
95#832—94no—
96#832—91no—
97#832—184no—
98#832C3⋊D311yescharacteristic
99#816—94no—
100#815—94no—
101#810—94no—
102#803—91no—
103C23×C32—21no—
104C23×C32—24no—
105C23×C32—44no—
106C32×D4—28no—
107C32×D4—24no—
108C32×D4—21no—
109C32×D4—21no—
110C32×D4—24no—
111C32×D4—44no—
112C2×C32×C4—24no—
113C2×C32×C4—61no—
114C2×C32×C4C22×D314yes—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—61no—
118C2×C3⋊Dic3—64no—
119C2×C3⋊Dic3—124no—
120C2×C3⋊Dic3#6711yescharacteristic
121C2×C3⋊Dic3#6711yescharacteristic
122#634—21no—
123#634—21no—
124#634—122no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—64no—
129C3×Dic6—64no—
130C3×Dic6—64no—
131#627—128no—
132C32×C8—61no—
133C3⋊(C3⋊C8)—61no—
134C3⋊(C3⋊C8)—64no—
135C3⋊(C3⋊C8)—124no—
136C3×C3⋊C8—24no—
137#489—271noSylow
138C2×C33—81no—
139C2×C33#12411yescharacteristic
140C2×C33AGL(1,ℤ/8)11yescharacteristic
141C2×C33#13711yescharacteristic
142C2×C3×Q8—91no—
143C2×C3×D4—24no—
144C2×C3×D4D3214yes—
145C2×C3×D4C2×C3⋊D311yescharacteristic
146C2×Dic6—34no—
147C2×Dic6—181no—
148C2×Dic6—184no—
149C2×C3×C8—34no—
150C3×C4⋊C4—34no—
151C3×C22⋊C4—181no—
152C3×C22⋊C4—181no—
153C3×C42—91no—
154#217—64no—
155#217—64no—
156C4⋊Dic3—94no—
157C4⋊Dic3—91no—
158C4⋊Dic3—184no—
159C4⋊Dic3—181no—
160C4⋊Dic3C2×C3⋊D311yescharacteristic
161Dic3⋊C4—184no—
162C4×Dic3—34no—
163C2×C3⋊C8—94no—
164C2×C3⋊C8—91no—
165C2×C3⋊C8—184no—
166C2×C3⋊C8C2×C3⋊D311yescharacteristic
167C22×C32—24no—
168C22×C32—48no—
169C22×C32—44no—
170C22×C32—41no—
171C22×C32—41no—
172C22×C32—41no—
173C22×C32—44no—
174C22×C32—44no—
175C22×C32—84no—
176C22×C32—84no—
177C22×C32D3×D414yes—
178C22×C32C2×#6711yescharacteristic
179C32×C4—24no—
180C32×C4—24no—
181C32×C4—121no—
182C32×C4#23514yes—
183C32×C4#23714yes—
184C32×C4#23711yescharacteristic
185C32×C4#23711yescharacteristic
186C3⋊Dic3—21no—
187C3⋊Dic3—21no—
188C3⋊Dic3—41no—
189C3⋊Dic3—121no—
190C3⋊Dic3—124no—
191C3⋊Dic3—244no—
192C3×Dic3—44no—
193C3×Dic3—64no—
194C3×Dic3—64no—
195C3×Dic3—124no—
196C4⋊Q8—271no—
197#124—91no—
198#103—271no—
199#103—271no—
200#102—31no—
201#102—271no—
202#97—271no—
203C33#48911yescharacteristic, complemented, Sylow
204C23×C3—24no—
205C23×C3—21no—
206C23×C3—44no—
207C3×Q8—181no—
208C3×Q8—181no—
209C3×D4—28no—
210C3×D4—24no—
211C3×D4—24no—
212C3×D4—21no—
213C3×D4—21no—
214C3×D4—44no—
215C2×C3×C4—24no—
216C2×C3×C4—64no—
217C2×C3×C4—91no—
218C2×C3×C4—91no—
219C2×C3×C4—181no—
220C2×C3×C4C2×D3214yes—
221C2×C3×C4C22×C3⋊D311yescharacteristic
222C2×Dic3—21no—
223C2×Dic3—34no—
224C2×Dic3—34no—
225C2×Dic3—64no—
226C2×Dic3—184no—
227C2×Dic3—181no—
228C2×Dic3—364no—
229Dic6—64no—
230Dic6—64no—
231Dic6—362no—
232Dic6—368no—
233C3×C8—64no—
234C3⋊C8—21no—
235C3⋊C8—184no—
236C3⋊C8—181no—
237C3⋊C8—364no—
238C2×C32—24no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32—88no—
242C2×C32—84no—
243C2×C32—81no—
244C2×C32#88514yes—
245C2×C32#91114yes—
246C2×C32#91514yes—
247C2×C32#93611yescharacteristic
248C2×C32#94911yescharacteristic
249C2×C32#95111yescharacteristic
250C2×Q8—91no—
251C2×Q8—541no—
252C2×D4D3×C3⋊D311yescharacteristic
253C2×C8—31no—
254C2×C8—271no—
255C4⋊C4—31no—
256C4⋊C4—271no—
257C4⋊C4—541no—
258C22⋊C4—181no—
259C22⋊C4—181no—
260C42—91no—
261C22×C3—24no—
262C22×C3—48no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—44no—
267C22×C3—44no—
268C22×C3—44no—
269C22×C3—84no—
270C22×C3—84no—
271C22×C3D3×#6714yes—
272C22×C3D4×C3⋊D311yescharacteristic
273C3×C4—24no—
274C3×C4—24no—
275C3×C4—124no—
276C3×C4—181no—
277C3×C4—181no—
278C3×C4—361no—
279C3×C4#380414yes—
280C3×C4#380514yes—
281C3×C4#382811yescharacteristic
282C3×C4#383011yescharacteristic
283Dic3—41no—
284Dic3—64no—
285Dic3—64no—
286Dic3—124no—
287Dic3—361no—
288Dic3—364no—
289Dic3—724no—
290C32—24no—
291C32#476514yescomplemented
292C32#505511yescharacteristic, complemented
293C23—21no—
294Q8—181no—
295Q8—181no—
296Q8—1082no—
297D4—21no—
298D4—21no—
299C2×C4—61no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—181no—
303C2×C4—541no—
304C2×C4C2×D3×C3⋊D311yescharacteristic
305C8—61no—
306C8—541no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—24no—
310C2×C3—88no—
311C2×C3—84no—
312C2×C3—81no—
313C2×C3#6392514yes—
314C2×C3#6392914yes—
315C2×C3#6395914yes—
316C2×C3#6409411yescharacteristic
317C2×C3#6412011yescharacteristic
318C2×C3#6412411yescharacteristic
319C22—41no—
320C22—41no—
321C22—41no—
322C22C3⋊D3×#6711yescharacteristic
323C4—121no—
324C4—181no—
325C4—181no—
326C4—361no—
327C4—1081no—
328C4#8902911yescharacteristic
329C4#8903111yescharacteristic
330C3—24no—
331C3#1059148814yescomplemented
332C3#1059246411yescharacteristic, complemented
333C2—81no—
334C2#1172354611yescharacteristic
335C2#1172355211yescharacteristic
336C2#1172359111yescharacteristic
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#489
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. #11723535C2semidirect
5. #11722040C2semidirect
6. #11722039C2semidirect
7. #11722020C2semidirect
203. C33#489semidirect
291. C32#4765semidirect
292. C32#5055semidirect
331. C3#10591488semidirect
332. C3#10592464semidirect

Maximal subgroups

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

subgroupindex
C3×#641452
#117235352
#117222022
C32×#8322
#117220402
#117220392
#117220202
#105924643
#105914883