Finite Group Databasebeta

← #13159218

Subgroups — #13159218

Order 1728.

880 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 142 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#11723933C211yesmaximal, characteristic
2#11723449C211yesmaximal, characteristic
3C3×#63637C211yesmaximal, characteristic, complemented
4C32×#808C211yesmaximal, characteristic, complemented
5#11722057C211yesmaximal, characteristic, complemented
6#11722040C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic
8#10592567—31nomaximal
9#10590527—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13#88859—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C4×C3⋊(C3⋊Dic3)—21no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20#88804—21no—
21#64106—31no—
22#64101—31no—
23#64101—34no—
24#64101—64no—
25#63871—34no—
26C32×#106—31no—
27#63647—31no—
28#63643—31no—
29#63637D311yescharacteristic, complemented
30#63634—31no—
31C3×#810—34no—
32C3×#808D314yescomplemented
33#63575—34no—
34#63566—34no—
35#63557—34no—
36C22×C3⋊(C3⋊D3)D411yescharacteristic
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41C4×C3⋊(C3⋊D3)—21no—
42C4×C3⋊(C3⋊D3)—21no—
43C2×C3×C3⋊Dic3—21no—
44C3×#634—21no—
45C3×#634—21no—
46C2×C32×Dic3—21no—
47C3×C3⋊(C3⋊C8)—21no—
48C32×C3⋊C8—21no—
49#4868—94no—
50#4814—91no—
51C2×C4×C3⋊D3—31no—
52C2×C4×C3⋊D3—34no—
53C2×C4×C3⋊D3—64no—
54C2×#634C2×D311yescharacteristic
55C2×C3×Dic6—34no—
56C2×C32×C8—31no—
57C32×C4⋊C4—24no—
58C32×C4⋊C4—31no—
59C32×C4⋊C4C2×D314yes—
60C32×C4⋊C4C2×D311yescharacteristic
61#3752—61no—
62#3752—61no—
63#3752—64no—
64#3752—68no—
65C4⋊(C3⋊Dic3)—31no—
66C4⋊(C3⋊Dic3)—34no—
67C4⋊(C3⋊Dic3)—61no—
68C4⋊(C3⋊Dic3)—64no—
69C4×C3⋊Dic3—61no—
70C4×C3⋊Dic3—64no—
71C4×C3⋊Dic3—68no—
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×C4⋊Dic3C2×D314yes—
74C2×C3×C3⋊C8—34no—
75C2×C3×C3⋊C8C2×D314yes—
76Dic3⋊Dic3—64no—
77#3722—64no—
78C22×C33C2×D411yescharacteristic
79C2×C3⋊(C3⋊D3)—21no—
80C2×C3⋊(C3⋊D3)—21no—
81C2×C3⋊(C3⋊D3)—21no—
82C33×C4—41no—
83C33×C4#4111yescharacteristic
84C33×C4#4111yescharacteristic
85C3⋊(C3⋊Dic3)—21no—
86C3⋊(C3⋊Dic3)—41no—
87C3×C3⋊Dic3—41no—
88C32×Dic3—41no—
89#946—91no—
90#897—94no—
91#892—94no—
92#892—91no—
93#892—184no—
94C3×#106—34no—
95C3×#103—91no—
96#835—91no—
97(C2×D3)⋊C8—94no—
98(C2×D3)⋊C8—91no—
99#818—91no—
100#816—94no—
101#810—34no—
102#808C3⋊D311yescharacteristic, complemented
103#807—94no—
104C22×C3⋊D3—31no—
105C22×C3⋊D3—34no—
106C22×C3⋊D3—64no—
107C2×C32×C4—28no—
108C2×C32×C4—24no—
109C2×C32×C4—21no—
110C2×C32×C4—24no—
111C2×C32×C4—61no—
112C2×C32×C4C22×D314yes—
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—34no—
117C2×C3⋊Dic3—64no—
118C2×C3⋊Dic3—68no—
119C2×C3⋊Dic3—64no—
120C2×C3⋊Dic3—61no—
121C4×C3⋊D3—68no—
122C4×C3⋊D3—68no—
123C4×C3⋊D3—64no—
124C4×C3⋊D3—61no—
125C4×C3⋊D3—61no—
126C4×C3⋊D3—64no—
127#634—21no—
128#634—21no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—64no—
131C3×Dic6—64no—
132C3×Dic6—64no—
133C32×C8—61no—
134C3⋊(C3⋊C8)—21no—
135C3×C3⋊C8—24no—
136C3×C3⋊C8—64no—
137#485—271noSylow
138C2×C33#12311yescharacteristic
139C2×C33#13511yescharacteristic
140C2×C33#13711yescharacteristic
141C3⋊(C3⋊D3)—41no—
142C2×C3×Q8—91no—
143C2×C4×D3—94no—
144C2×C4×D3—91no—
145C2×C4×D3—184no—
146C2×Dic6—34no—
147C2×C3×C8—34no—
148C2×C3×C8—91no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—34no—
151C3×C4⋊C4D3214yes—
152C3×C4⋊C4C2×C3⋊D311yescharacteristic
153(C2×D3)⋊C4—181no—
154(C2×D3)⋊C4—184no—
155(C2×D3)⋊C4—184no—
156(C2×D3)⋊C4—181no—
157(C2×D3)⋊C4—188no—
158C4⋊Dic3—94no—
159C4⋊Dic3—91no—
160C4⋊Dic3—184no—
161C4⋊Dic3—184no—
162C4⋊Dic3C2×C3⋊D311yescharacteristic
163Dic3⋊C4—181no—
164C4×Dic3—184no—
165C4×Dic3—181no—
166C4×Dic3—188no—
167C2×C3⋊C8—34no—
168C2×C3⋊C8C2×C3⋊D311yescharacteristic
169C22×C32—24no—
170C22×C32D3×D414yes—
171C22×C32D3×D411yescharacteristic
172C2×C3⋊D3—68no—
173C2×C3⋊D3—61no—
174C2×C3⋊D3—64no—
175C2×C3⋊D3—61no—
176C2×C3⋊D3—61no—
177C2×C3⋊D3—64no—
178C2×C3⋊D3—64no—
179C2×C3⋊D3—124no—
180C2×C3⋊D3—124no—
181C32×C4—24no—
182C32×C4—24no—
183C32×C4—48no—
184C32×C4—44no—
185C32×C4—41no—
186C32×C4—121no—
187C32×C4#23514yes—
188C32×C4#23511yescharacteristic
189C32×C4#23711yescharacteristic
190C32×C4#23914yes—
191C3⋊Dic3—41no—
192C3⋊Dic3—68no—
193C3⋊Dic3—61no—
194C3⋊Dic3—64no—
195C3⋊Dic3—128no—
196C3⋊Dic3—124no—
197C3⋊Dic3—121no—
198C3×Dic3—44no—
199C3×Dic3—124no—
200C22⋊Q8—271no—
201#117—271no—
202#107—271no—
203#106—31no—
204#103—91no—
205#103—271no—
206C22⋊C8—271no—
207C33#48511yescharacteristic, complemented, Sylow
208C22×D3—94no—
209C22×D3—91no—
210C22×D3—184no—
211C3×Q8—181no—
212C3×Q8—181no—
213C2×C3×C4—28no—
214C2×C3×C4—24no—
215C2×C3×C4—21no—
216C2×C3×C4—24no—
217C2×C3×C4—64no—
218C2×C3×C4—181no—
219C2×C3×C4C2×D3214yes—
220C2×C3×C4C22×C3⋊D311yescharacteristic
221C2×Dic3—21no—
222C2×Dic3—64no—
223C2×Dic3—94no—
224C2×Dic3—91no—
225C2×Dic3—184no—
226C2×Dic3—188no—
227C2×Dic3—184no—
228C2×Dic3—181no—
229C4×D3—188no—
230C4×D3—188no—
231C4×D3—184no—
232C4×D3—184no—
233C4×D3—181no—
234C4×D3—181no—
235Dic6—64no—
236Dic6—64no—
237C3×C8—64no—
238C3×C8—181no—
239C3⋊C8—21no—
240C3⋊C8—64no—
241C2×C32—24no—
242C2×C32—24no—
243C2×C32—24no—
244C2×C32#88311yescharacteristic
245C2×C32#89414yes—
246C2×C32#91514yes—
247C2×C32#91511yescharacteristic
248C2×C32#91614yes—
249C2×C32#91911yescharacteristic
250C3⋊D3—128no—
251C3⋊D3—124no—
252C3⋊D3—121no—
253C2×Q8—91no—
254C22×C4—271no—
255C2×C8—31no—
256C2×C8—91no—
257C4⋊C4—31no—
258C4⋊C4—271no—
259C4⋊C4—541no—
260C4⋊C4D3×C3⋊D311yescharacteristic
261C22⋊C4—541no—
262C22⋊C4—541no—
263C42—541no—
264C22×C3—24no—
265C22×C3#381114yes—
266C22×C3D4×C3⋊D311yescharacteristic
267C2×D3—188no—
268C2×D3—181no—
269C2×D3—184no—
270C2×D3—181no—
271C2×D3—181no—
272C2×D3—184no—
273C2×D3—184no—
274C2×D3—364no—
275C2×D3—364no—
276C3×C4—24no—
277C3×C4—24no—
278C3×C4—48no—
279C3×C4—41no—
280C3×C4—44no—
281C3×C4—124no—
282C3×C4—361no—
283C3×C4#379614yes—
284C3×C4#379814yes—
285C3×C4#382811yescharacteristic
286C3×C4#383211yescharacteristic
287Dic3—41no—
288Dic3—124no—
289Dic3—188no—
290Dic3—181no—
291Dic3—184no—
292Dic3—368no—
293Dic3—361no—
294Dic3—364no—
295C32—24no—
296C32#481411yescharacteristic, complemented
297C32#486814yescomplemented
298C23—271no—
299Q8—181no—
300Q8—181no—
301C2×C4—21no—
302C2×C4—61no—
303C2×C4—181no—
304C2×C4—271no—
305C2×C4—541no—
306C2×C4—541no—
307C2×C4—541no—
308C2×C4C2×D3×C3⋊D311yescharacteristic
309C8—61no—
310C8—181no—
311C2×C3—24no—
312C2×C3—24no—
313C2×C3—24no—
314C2×C3#6385314yes—
315C2×C3#6393114yes—
316C2×C3#6395014yes—
317C2×C3#6410311yescharacteristic
318C2×C3#6412411yescharacteristic
319C2×C3#6412511yescharacteristic
320D3—368no—
321D3—361no—
322D3—364no—
323C22—541no—
324C22—541no—
325C22—541no—
326C22#8903911yescharacteristic
327C4—41no—
328C4—121no—
329C4—361no—
330C4—541no—
331C4—1081no—
332C4#8901711yescharacteristic
333C4#8902011yescharacteristic
334C3—24no—
335C3#1059052714yescomplemented
336C3#1059256711yescharacteristic, complemented
337C2—1081no—
338C2#1172342111yescharacteristic
339C2#1172355411yescharacteristic
340C2#1172358711yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#485
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C3×#63637C2semidirect
4. C32×#808C2semidirect
5. #11722057C2semidirect
6. #11722040C2semidirect
29. #63637D3semidirect
32. C3×#808D3semidirect
102. #808C3⋊D3semidirect
207. C33#485semidirect
296. C32#4814semidirect
297. C32#4868semidirect
335. C3#10590527semidirect
336. C3#10592567semidirect

Maximal subgroups

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

subgroupindex
#117239332
#117234492
C3×#636372
C32×#8082
#117220572
#117220402
#117220242
#105925673
#105905273