Finite Group Databasebeta

← #13140326

Subgroups — #13140326

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#11721836C211yesmaximal, characteristic
2#11720971C211yesmaximal, characteristic, complemented
3C9×#832C211yesmaximal, characteristic
4C3×#63392C211yesmaximal, characteristic
5#11720277C211yesmaximal, characteristic, complemented
6#11720275C211yesmaximal, characteristic, complemented
7#11720269C211yesmaximal, characteristic, complemented
8#10591496—31nomaximal
9#10588445—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×#6189—21no—
13#88551—21no—
14#88551—21no—
15C4×C3⋊Dic9C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20#88442—21no—
21#64145—31no—
22#63919—31no—
23C3×#832—31no—
24C3×#832D311yescharacteristic
25#63567—31no—
26#63567—31no—
27#63555—31no—
28#63498—31no—
29#63498—61no—
30#63450—31no—
31C9×#102—31no—
32#63392D311yescharacteristic
33#63376—31no—
34#63375—31no—
35#63368—31no—
36C23×C3×C9—21no—
37C3×D4×C9—21no—
38C3×D4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C3⋊Dic9D411yescharacteristic
42C2×C3⋊Dic9D411yescharacteristic
43#6226—21no—
44#6226—21no—
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47#6189—42no—
48C9×C3⋊C8—21no—
49C3×C9⋊C8—21no—
50#4765—91no—
51#4765—91no—
52C2×C32×D4C2×D311yescharacteristic
53C2×#634—31no—
54C2×#627—61no—
55#3757—61no—
56#3757—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×D4×C9—21no—
64C2×D4×C9C2×D311yescharacteristic
65C2×Dic18—31no—
66C2×Dic18—61no—
67C2×Dic18—61no—
68C2×C8×C9—31no—
69C9×C4⋊C4—31no—
70#3676—61no—
71#3676—61no—
72#3676—62no—
73#3676—62no—
74C4⋊Dic9C2×D311yescharacteristic
75Dic9⋊C4—61no—
76C4×Dic9—31no—
77C4×Dic9—61no—
78C2×C9⋊C8C2×D311yescharacteristic
79C22×C3×C9—41no—
80C22×C3×C9—41no—
81C22×C3×C9—41no—
82C22×C3×C9C2×D411yescharacteristic
83C3×C4×C9#4111yescharacteristic
84C3×C4×C9#4111yescharacteristic
85C3⋊Dic9—21no—
86C3⋊Dic9—21no—
87C3⋊Dic9—41no—
88C9×Dic3—41no—
89C3×Dic9—41no—
90#936—91no—
91#936—91no—
92#936—181no—
93#888—91no—
94#888—91no—
95C3×#102—31no—
96C3×#102—91no—
97#832—31no—
98#832D911yescharacteristic
99#816—91no—
100#816—91no—
101#815—91no—
102#815—91no—
103#810—91no—
104#810—91no—
105C23×C32—21no—
106C32×D4—21no—
107C32×D4—21no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—31no—
110C2×C3⋊Dic3—31no—
111C2×C3⋊Dic3—61no—
112#634—61no—
113#634—61no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—61no—
116#627—122no—
117C23×C9—22no—
118C23×C9—21no—
119C3×C3⋊C8—21no—
120C3×C3⋊C8—61no—
121D4×C9—21no—
122D4×C9—21no—
123D4×C9—41no—
124D4×C9—41no—
125C2×C4×C9—21no—
126C2×C4×C9—61no—
127C2×C4×C9C22×D311yescharacteristic
128C2×Dic9—21no—
129C2×Dic9—31no—
130C2×Dic9—31no—
131C2×Dic9—61no—
132C2×Dic9—61no—
133C2×Dic9—62no—
134C2×Dic9—61no—
135Dic18—61no—
136Dic18—61no—
137Dic18—121no—
138Dic18—121no—
139Dic18—122no—
140C8×C9—61no—
141C9⋊C8—21no—
142#489—271noSylow
143C2×C3×C9—81no—
144C2×C3×C9#12411yescharacteristic
145C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
146C2×C3×C9#13711yescharacteristic
147C2×C3×D4—21no—
148C2×C3×D4C2×D911yescharacteristic
149C2×C3×D4D3211yescharacteristic
150C2×Dic6—91no—
151C2×Dic6—91no—
152C2×Dic6—181no—
153C2×Dic6—181no—
154C2×Dic6—181no—
155C2×C3×C8—31no—
156C2×C3×C8—91no—
157C3×C4⋊C4—31no—
158C3×C4⋊C4—91no—
159#217—181no—
160#217—181no—
161#217—181no—
162#217—181no—
163#217—182no—
164#217—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×C32—41no—
175C22×C32—41no—
176C22×C32—41no—
177C22×C32D3×D411yescharacteristic
178C32×C4#23511yescharacteristic
179C32×C4#23711yescharacteristic
180C3⋊Dic3—61no—
181C3⋊Dic3—61no—
182C3⋊Dic3—121no—
183C3×Dic3—41no—
184C3×Dic3—121no—
185C22×C9—21no—
186C22×C9—42no—
187C22×C9—42no—
188C22×C9—42no—
189C22×C9—41no—
190C22×C9—41no—
191C22×C9—41no—
192C22×C9D3×D411yescharacteristic
193C4×C9—21no—
194C4×C9—21no—
195C4×C9—121no—
196C4×C9#23511yescharacteristic
197C4×C9#23711yescharacteristic
198Dic9—41no—
199Dic9—61no—
200Dic9—61no—
201Dic9—121no—
202Dic9—121no—
203Dic9—122no—
204Dic9—121no—
205C4⋊Q8—271no—
206#124—271no—
207#103—271no—
208#103—271no—
209#102—31no—
210#102—91no—
211#97—271no—
212C3×C9#48911yescharacteristic, complemented, Sylow
213C23×C3—22no—
214C23×C3—21no—
215C23×C3—21no—
216C3×D4—21no—
217C3×D4—21no—
218C3×D4—21no—
219C3×D4—21no—
220C3×D4—41no—
221C3×D4—41no—
222C2×C3×C4—21no—
223C2×C3×C4—61no—
224C2×C3×C4—181no—
225C2×C3×C4C22×D911yescharacteristic
226C2×C3×C4C2×D3211yescharacteristic
227C2×Dic3—21no—
228C2×Dic3—61no—
229C2×Dic3—91no—
230C2×Dic3—91no—
231C2×Dic3—91no—
232C2×Dic3—91no—
233C2×Dic3—181no—
234C2×Dic3—181no—
235C2×Dic3—182no—
236C2×Dic3—181no—
237C2×Dic3—181no—
238Dic6—181no—
239Dic6—181no—
240Dic6—181no—
241Dic6—181no—
242Dic6—361no—
243Dic6—361no—
244Dic6—362no—
245Dic6—362no—
246C3×C8—61no—
247C3×C8—181no—
248C3⋊C8—21no—
249C3⋊C8—61no—
250C2×C32—81no—
251C2×C32#88511yescharacteristic
252C2×C32#91111yescharacteristic
253C2×C32#91511yescharacteristic
254C2×C9—21no—
255C2×C9—21no—
256C2×C9—21no—
257C2×C9—82no—
258C2×C9—81no—
259C2×C9#88511yescharacteristic
260C2×C9#91111yescharacteristic
261C2×C9#91511yescharacteristic
262C2×Q8—271no—
263C2×Q8—541no—
264C2×D4D3×D911yescharacteristic
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×C3—42no—
275C22×C3—42no—
276C22×C3—42no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3D4×D911yescharacteristic
284C22×C3#381111yescharacteristic
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—121no—
288C3×C4—361no—
289C3×C4#369711yescharacteristic
290C3×C4#369911yescharacteristic
291C3×C4#380411yescharacteristic
292C3×C4#380411yescharacteristic
293Dic3—41no—
294Dic3—121no—
295Dic3—181no—
296Dic3—181no—
297Dic3—181no—
298Dic3—181no—
299Dic3—361no—
300Dic3—361no—
301Dic3—362no—
302Dic3—361no—
303Dic3—361no—
304C32#476511yescharacteristic
305C9—21no—
306C9#476511yescharacteristic, complemented
307C23—21no—
308Q8—541no—
309Q8—541no—
310Q8—1082no—
311D4—21no—
312D4—21no—
313C2×C4—61no—
314C2×C4—181no—
315C2×C4—271no—
316C2×C4—271no—
317C2×C4—541no—
318C2×C4C2×D3×D911yescharacteristic
319C8—61no—
320C8—181no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—82no—
325C2×C3—81no—
326C2×C3—81no—
327C2×C3#6344711yescharacteristic
328C2×C3#6347311yescharacteristic
329C2×C3#6347711yescharacteristic
330C2×C3#6392711yescharacteristic
331C2×C3#6393111yescharacteristic
332C2×C3#6395311yescharacteristic
333C22—41no—
334C22—41no—
335C22—41no—
336C22#8866811yescharacteristic
337C4—121no—
338C4—361no—
339C4—541no—
340C4—541no—
341C4—1081no—
342C4#8865811yescharacteristic
343C4#8866211yescharacteristic
344C3—21no—
345C3#1058844511yescharacteristic, complemented
346C3#1059149611yescharacteristic
347C2—81no—
348C2#1172117511yescharacteristic
349C2#1172118311yescharacteristic
350C2#1172124211yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC2×C32×D4C2×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#489
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11720971C2semidirect
5. #11720277C2semidirect
6. #11720275C2semidirect
7. #11720269C2semidirect
212. C3×C9#489semidirect
306. C9#4765semidirect
345. C3#10588445semidirect

Maximal subgroups

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

subgroupindex
#117218362
#117209712
C9×#8322
C3×#633922
#117202772
#117202752
#117202692
#105914963
#105884453