Finite Group Databasebeta

← #13140700

Subgroups — #13140700

Order 1728.

376 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 79 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#11721857C211yesmaximal, characteristic
2#11720971C211yesmaximal, characteristic, complemented
3C9×#835C211yesmaximal, characteristic, complemented
4C3×#63395C211yesmaximal, characteristic, complemented
5#11720284C211yesmaximal, characteristic
6#11720275C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic
8#10591632—31nomaximal
9#10588474—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×#6189—21no—
13#88536—21no—
14#88536—21no—
15C4×C3⋊Dic9C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20#88442—21no—
21#64156—31no—
22#63919—31no—
23C3×#835—31no—
24C3×#835D311yescharacteristic
25#63564—31no—
26#63564—31no—
27#63555—31no—
28#63509—31no—
29#63509—61no—
30#63450—31no—
31C9×#103—31no—
32#63395D311yescharacteristic, complemented
33#63380—31no—
34#63375—31no—
35#63369—31no—
36C22×C3⋊D9—21no—
37C3×Q8×C9—21no—
38C3×Q8×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9D411yescharacteristic
42C2×C3⋊Dic9D411yescharacteristic
43C4⋊(C3⋊D9)—21no—
44C4⋊(C3⋊D9)—21no—
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47#6189—42no—
48C9×C3⋊C8—21no—
49C3×C9⋊C8—21no—
50#4794—91no—
51#4794—91no—
52C2×C32×Q8C2×D311yescharacteristic
53C2×C4⋊(C3⋊D3)—31no—
54C2×#627—61no—
55#3752—61no—
56#3752—61no—
57C4×C3⋊Dic3—31no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3C2×D311yescharacteristic
60C2×C3×C3⋊C8—31no—
61C2×C3×C3⋊C8C2×D311yescharacteristic
62#3724—61no—
63C2×Q8×C9—21no—
64C2×Q8×C9C2×D311yescharacteristic
65C2×D36—31no—
66C2×D36—61no—
67C2×Dic18—61no—
68C2×C8×C9—31no—
69C9×C4⋊C4—31no—
70(C2×D9)⋊C4—61no—
71(C2×D9)⋊C4—61no—
72(C2×D9)⋊C4—62no—
73(C2×D9)⋊C4—62no—
74C4⋊Dic9C2×D311yescharacteristic
75Dic9⋊C4—61no—
76C4×Dic9—31no—
77C4×Dic9—61no—
78C2×C9⋊C8C2×D311yescharacteristic
79C22×C3×C9C2×D411yescharacteristic
80C2×C3⋊D9—41no—
81C2×C3⋊D9—41no—
82C2×C3⋊D9—41no—
83C3×C4×C9—41no—
84C3×C4×C9#4111yescharacteristic
85C3×C4×C9#4111yescharacteristic
86C3⋊Dic9—21no—
87C3⋊Dic9—21no—
88C9×Dic3—41no—
89C3×Dic9—41no—
90#947—91no—
91#947—91no—
92#947—181no—
93#888—91no—
94#888—91no—
95C3×#103—31no—
96C3×#103—91no—
97#835—31no—
98#835D911yescharacteristic, complemented
99#821—91no—
100#821—91no—
101#815—91no—
102#815—91no—
103#809—91no—
104#809—91no—
105C22×C3⋊D3—61no—
106C32×Q8—21no—
107C32×Q8—21no—
108C2×C32×C4—21no—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—31no—
111C2×C3⋊Dic3—31no—
112C4⋊(C3⋊D3)—61no—
113C4⋊(C3⋊D3)—61no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—61no—
116#627—122no—
117C22×D9—62no—
118C22×D9—61no—
119C3×C3⋊C8—21no—
120C3×C3⋊C8—61no—
121Q8×C9—21no—
122Q8×C9—21no—
123Q8×C9—41no—
124Q8×C9—41no—
125C2×C4×C9—22no—
126C2×C4×C9—21no—
127C2×C4×C9—21no—
128C2×C4×C9—61no—
129C2×C4×C9C22×D311yescharacteristic
130C2×Dic9—21no—
131C2×Dic9—31no—
132C2×Dic9—31no—
133C2×Dic9—61no—
134C2×Dic9—61no—
135D36—61no—
136D36—61no—
137D36—121no—
138D36—121no—
139Dic18—122no—
140C8×C9—61no—
141C9⋊C8—21no—
142#489—271noSylow
143C2×C3×C9#12411yescharacteristic
144C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
145C2×C3×C9#13711yescharacteristic
146C3⋊D9—81no—
147C2×C3×Q8—21no—
148C2×C3×Q8C2×D911yescharacteristic
149C2×C3×Q8D3211yescharacteristic
150C2×D12—91no—
151C2×D12—91no—
152C2×D12—181no—
153C2×Dic6—181no—
154C2×Dic6—181no—
155C2×C3×C8—31no—
156C2×C3×C8—91no—
157C3×C4⋊C4—31no—
158C3×C4⋊C4—91no—
159(C2×D3)⋊C4—181no—
160(C2×D3)⋊C4—181no—
161(C2×D3)⋊C4—181no—
162(C2×D3)⋊C4—181no—
163(C2×D3)⋊C4—182no—
164(C2×D3)⋊C4—182no—
165C4⋊Dic3—31no—
166C4⋊Dic3C2×D911yescharacteristic
167Dic3⋊C4—181no—
168Dic3⋊C4—181no—
169C4×Dic3—91no—
170C4×Dic3—91no—
171C4×Dic3—181no—
172C2×C3⋊C8—31no—
173C2×C3⋊C8C2×D911yescharacteristic
174C22×C32D3×D411yescharacteristic
175C2×C3⋊D3—121no—
176C2×C3⋊D3—121no—
177C2×C3⋊D3—121no—
178C32×C4—41no—
179C32×C4#23511yescharacteristic
180C32×C4#23711yescharacteristic
181C3⋊Dic3—61no—
182C3⋊Dic3—61no—
183C3×Dic3—41no—
184C3×Dic3—121no—
185C22×C9—21no—
186C22×C9D3×D411yescharacteristic
187C2×D9—122no—
188C2×D9—122no—
189C2×D9—122no—
190C2×D9—121no—
191C2×D9—121no—
192C2×D9—121no—
193C4×C9—21no—
194C4×C9—21no—
195C4×C9—42no—
196C4×C9—41no—
197C4×C9—121no—
198C4×C9#23511yescharacteristic
199C4×C9#23711yescharacteristic
200Dic9—41no—
201Dic9—61no—
202Dic9—61no—
203Dic9—121no—
204Dic9—121no—
205C4⋊Q8—271no—
206#124—271no—
207#103—31no—
208#103—91no—
209#102—271no—
210#102—271no—
211#97—271no—
212C3×C9#48911yescharacteristic, complemented, Sylow
213C22×D3—182no—
214C22×D3—181no—
215C22×D3—181no—
216C3×Q8—21no—
217C3×Q8—21no—
218C3×Q8—21no—
219C3×Q8—21no—
220C3×Q8—41no—
221C3×Q8—41no—
222C2×C3×C4—22no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—61no—
227C2×C3×C4—181no—
228C2×C3×C4C22×D911yescharacteristic
229C2×C3×C4C2×D3211yescharacteristic
230C2×Dic3—21no—
231C2×Dic3—61no—
232C2×Dic3—91no—
233C2×Dic3—91no—
234C2×Dic3—91no—
235C2×Dic3—91no—
236C2×Dic3—181no—
237C2×Dic3—181no—
238D12—181no—
239D12—181no—
240D12—181no—
241D12—181no—
242D12—361no—
243D12—361no—
244Dic6—362no—
245Dic6—362no—
246C3×C8—61no—
247C3×C8—181no—
248C3⋊C8—21no—
249C3⋊C8—61no—
250C2×C32#88511yescharacteristic
251C2×C32#91411yescharacteristic
252C2×C32#91811yescharacteristic
253C3⋊D3—241no—
254C2×C9—21no—
255C2×C9—21no—
256C2×C9—21no—
257C2×C9#88511yescharacteristic
258C2×C9#91411yescharacteristic
259C2×C9#91811yescharacteristic
260D9—242no—
261D9—241no—
262C2×Q8—541no—
263C2×Q8D3×D911yescharacteristic
264C2×D4—271no—
265C2×C8—31no—
266C2×C8—91no—
267C4⋊C4—31no—
268C4⋊C4—91no—
269C4⋊C4—541no—
270C22⋊C4—541no—
271C22⋊C4—541no—
272C42—271no—
273C22×C3—21no—
274C22×C3D4×D911yescharacteristic
275C22×C3#381111yescharacteristic
276C2×D3—362no—
277C2×D3—362no—
278C2×D3—362no—
279C2×D3—361no—
280C2×D3—361no—
281C2×D3—361no—
282C2×D3—361no—
283C2×D3—361no—
284C2×D3—361no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—42no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4—121no—
291C3×C4—361no—
292C3×C4#369711yescharacteristic
293C3×C4#369911yescharacteristic
294C3×C4#380411yescharacteristic
295C3×C4#380411yescharacteristic
296Dic3—41no—
297Dic3—121no—
298Dic3—181no—
299Dic3—181no—
300Dic3—181no—
301Dic3—181no—
302Dic3—361no—
303Dic3—361no—
304C32#479411yescharacteristic
305C9—21no—
306C9#479411yescharacteristic, complemented
307C23—541no—
308Q8—21no—
309Q8—21no—
310Q8—1082no—
311D4—541no—
312D4—541no—
313C2×C4—21no—
314C2×C4—61no—
315C2×C4—181no—
316C2×C4—271no—
317C2×C4—271no—
318C2×C4C2×D3×D911yescharacteristic
319C8—61no—
320C8—181no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3#6344711yescharacteristic
325C2×C3#6347611yescharacteristic
326C2×C3#6348011yescharacteristic
327C2×C3#6394111yescharacteristic
328C2×C3#6394511yescharacteristic
329C2×C3#6395311yescharacteristic
330D3—722no—
331D3—721no—
332D3—721no—
333C22—1081no—
334C22—1081no—
335C22—1081no—
336C22#8866811yescharacteristic
337C4—41no—
338C4—121no—
339C4—361no—
340C4—541no—
341C4—541no—
342C4#8865811yescharacteristic
343C4#8866211yescharacteristic
344C3—21no—
345C3#1058847411yescharacteristic, complemented
346C3#1059163211yescharacteristic
347C2—2161no—
348C2#1172121311yescharacteristic
349C2#1172122111yescharacteristic
350C2#1172124211yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC2×Q8D3×D9
hypocenterC3×C9#489
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11720971C2semidirect
3. C9×#835C2semidirect
4. C3×#63395C2semidirect
6. #11720275C2semidirect
32. #63395D3semidirect
98. #835D9semidirect
212. C3×C9#489semidirect
306. C9#4794semidirect
345. C3#10588474semidirect

Maximal subgroups

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

subgroupindex
#117218572
#117209712
C9×#8352
C3×#633952
#117202842
#117202752
#117202672
#105916323
#105884743