Finite Group Databasebeta

← #13140324

Subgroups — #13140324

Order 1728.

462 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721840C211yesmaximal, characteristic
2#11720971C211yesmaximal, characteristic, complemented
3C9×#832C211yesmaximal, characteristic, complemented
4C3×#63392C211yesmaximal, characteristic, complemented
5#11720284C211yesmaximal, characteristic, complemented
6#11720271C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic, complemented
8#10591402—31nomaximal
9#10588440—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C2×#6233—21no—
13C2×C4⋊(C3⋊D9)C2211yescharacteristic
14C2×#6189—21no—
15C4×C3⋊Dic9C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20#88442—21no—
21#64149—31no—
22#63919—31no—
23C3×#832—31no—
24C3×#832D311yescharacteristic
25#63564—31no—
26#63564—31no—
27#63553—31no—
28#63502—31no—
29#63502—61no—
30#63450—31no—
31C9×#102—31no—
32#63392D311yescharacteristic, complemented
33#63380—31no—
34C8×Dic9—31no—
35#63369—31no—
36C23×C3×C9—21no—
37C22×C3⋊D9—21no—
38C3×D4×C9—21no—
39C3×D4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41#6233—42no—
42#6233—42no—
43C2×C3⋊Dic9D411yescharacteristic
44C2×C3⋊Dic9D411yescharacteristic
45C4⋊(C3⋊D9)—21no—
46C4⋊(C3⋊D9)—21no—
47C2×C9×Dic3—21no—
48C2×C3×Dic9—21no—
49#6189—42no—
50C9×C3⋊C8—21no—
51C3×C9⋊C8—21no—
52#4760—91no—
53#4760—91no—
54C2×C32×D4C2×D311yescharacteristic
55C2×#638—61no—
56C2×#638—61no—
57C2×C4⋊(C3⋊D3)—31no—
58C2×#627—61no—
59C4×C3⋊Dic3—31no—
60C3×C4⋊Dic3—31no—
61C3×C4⋊Dic3C2×D311yescharacteristic
62C2×C3×C3⋊C8—31no—
63C2×C3×C3⋊C8C2×D311yescharacteristic
64#3724—61no—
65C2×D4×C9—21no—
66C2×D4×C9C2×D311yescharacteristic
67C2×#611—61no—
68C2×#611—61no—
69C2×#611—62no—
70C2×#611—62no—
71C2×D36—31no—
72C2×D36—61no—
73C2×Dic18—61no—
74C2×C8×C9—31no—
75C9×C4⋊C4—31no—
76C4⋊Dic9C2×D311yescharacteristic
77Dic9⋊C4—61no—
78C4×Dic9—31no—
79C4×Dic9—61no—
80C2×C9⋊C8C2×D311yescharacteristic
81C22×C3×C9—41no—
82C22×C3×C9—41no—
83C22×C3×C9—41no—
84C22×C3×C9C2×D411yescharacteristic
85C2×C3⋊D9—41no—
86C2×C3⋊D9—41no—
87C2×C3⋊D9—41no—
88C3×C4×C9#4111yescharacteristic
89C3×C4×C9#4111yescharacteristic
90C3⋊Dic9D812yes—
91C3⋊Dic9SD1612yes—
92C9×Dic3—41no—
93C3×Dic9—41no—
94#940—91no—
95#940—91no—
96#940—181no—
97#888—91no—
98#888—91no—
99C3×#102—31no—
100C3×#102—91no—
101#832—31no—
102#832D911yescharacteristic, complemented
103#821—91no—
104#821—91no—
105C8×Dic3—91no—
106C8×Dic3—91no—
107#809—91no—
108#809—91no—
109C23×C32—21no—
110C22×C3⋊D3—61no—
111C32×D4—21no—
112C32×D4—21no—
113C2×C32×C4C22×D311yescharacteristic
114#638—122no—
115#638—122no—
116C2×C3⋊Dic3—31no—
117C2×C3⋊Dic3—31no—
118C4⋊(C3⋊D3)—61no—
119C4⋊(C3⋊D3)—61no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—61no—
122#627—122no—
123C23×C9—22no—
124C23×C9—21no—
125C22×D9—62no—
126C22×D9—61no—
127C3×C3⋊C8—21no—
128C3×C3⋊C8—61no—
129D4×C9—21no—
130D4×C9—21no—
131D4×C9—41no—
132D4×C9—41no—
133C2×C4×C9—21no—
134C2×C4×C9—61no—
135C2×C4×C9C22×D311yescharacteristic
136#611—122no—
137#611—122no—
138#611—124no—
139#611—124no—
140C2×Dic9—21no—
141C2×Dic9—31no—
142C2×Dic9—31no—
143C2×Dic9—61no—
144C2×Dic9—61no—
145D36—61no—
146D36—61no—
147D36—121no—
148D36—121no—
149Dic18—122no—
150C8×C9—61no—
151C9⋊C8—21no—
152#486—271noSylow
153C2×C3×C9—81no—
154C2×C3×C9#12411yescharacteristic
155C2×C3×C9C2×D811yescharacteristic
156C2×C3×C9C2×SD1611yescharacteristic
157C3⋊D9—81no—
158C2×C3×D4—21no—
159C2×C3×D4C2×D911yescharacteristic
160C2×C3×D4D3211yescharacteristic
161C2×#67—181no—
162C2×#67—181no—
163C2×#67—181no—
164C2×#67—181no—
165C2×#67—182no—
166C2×#67—182no—
167C2×D12—91no—
168C2×D12—91no—
169C2×D12—181no—
170C2×Dic6—181no—
171C2×Dic6—181no—
172C2×C3×C8—31no—
173C2×C3×C8—91no—
174C3×C4⋊C4—31no—
175C3×C4⋊C4—91no—
176C4⋊Dic3—31no—
177C4⋊Dic3C2×D911yescharacteristic
178Dic3⋊C4—181no—
179Dic3⋊C4—181no—
180C4×Dic3—91no—
181C4×Dic3—91no—
182C4×Dic3—181no—
183C2×C3⋊C8—31no—
184C2×C3⋊C8C2×D911yescharacteristic
185C22×C32—41no—
186C22×C32—41no—
187C22×C32—41no—
188C22×C32D3×D411yescharacteristic
189C2×C3⋊D3—121no—
190C2×C3⋊D3—121no—
191C2×C3⋊D3—121no—
192C32×C4#23511yescharacteristic
193C32×C4#23711yescharacteristic
194C3⋊Dic3—32no—
195C3⋊Dic3—32no—
196C3×Dic3—41no—
197C3×Dic3—121no—
198C22×C9—21no—
199C22×C9—42no—
200C22×C9—42no—
201C22×C9—42no—
202C22×C9—41no—
203C22×C9—41no—
204C22×C9—41no—
205C22×C9D3×D411yescharacteristic
206C2×D9—122no—
207C2×D9—122no—
208C2×D9—122no—
209C2×D9—121no—
210C2×D9—121no—
211C2×D9—121no—
212C4×C9—21no—
213C4×C9—21no—
214C4×C9—121no—
215C4×C9#23511yescharacteristic
216C4×C9#23711yescharacteristic
217Dic9—32no—
218Dic9—32no—
219Dic9—41no—
220Dic9—62no—
221Dic9—62no—
222C4⋊Q8—271no—
223#127—271no—
224#102—31no—
225#102—91no—
226#102—271no—
227#102—271no—
228C4×C8—271no—
229C3×C9#48611yescharacteristic, complemented, Sylow
230C23×C3—22no—
231C23×C3—21no—
232C23×C3—21no—
233C22×D3—182no—
234C22×D3—181no—
235C22×D3—181no—
236C3×D4—21no—
237C3×D4—21no—
238C3×D4—21no—
239C3×D4—21no—
240C3×D4—41no—
241C3×D4—41no—
242C2×C3×C4—21no—
243C2×C3×C4—61no—
244C2×C3×C4—181no—
245C2×C3×C4C22×D911yescharacteristic
246C2×C3×C4C2×D3211yescharacteristic
247#67—362no—
248#67—362no—
249#67—362no—
250#67—362no—
251#67—364no—
252#67—364no—
253C2×Dic3—21no—
254C2×Dic3—61no—
255C2×Dic3—91no—
256C2×Dic3—91no—
257C2×Dic3—91no—
258C2×Dic3—91no—
259C2×Dic3—181no—
260C2×Dic3—181no—
261D12—181no—
262D12—181no—
263D12—181no—
264D12—181no—
265D12—361no—
266D12—361no—
267Dic6—362no—
268Dic6—362no—
269C3×C8—61no—
270C3×C8—181no—
271C3⋊C8—21no—
272C3⋊C8—61no—
273C2×C32—81no—
274C2×C32#88511yescharacteristic
275C2×C32D3×D811yescharacteristic
276C2×C32D3×SD1611yescharacteristic
277C3⋊D3—241no—
278C2×C9—21no—
279C2×C9—21no—
280C2×C9—21no—
281C2×C9—82no—
282C2×C9—81no—
283C2×C9#88511yescharacteristic
284C2×C9D3×D811yescharacteristic
285C2×C9D3×SD1611yescharacteristic
286D9—242no—
287D9—241no—
288C2×Q8—541no—
289C2×D4—271no—
290C2×D4—541no—
291C2×D4—541no—
292C2×D4D3×D911yescharacteristic
293C2×C8—31no—
294C2×C8—91no—
295C4⋊C4—31no—
296C4⋊C4—91no—
297C4⋊C4—541no—
298C42—271no—
299C22×C3—21no—
300C22×C3—42no—
301C22×C3—42no—
302C22×C3—42no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3D4×D911yescharacteristic
310C22×C3#381111yescharacteristic
311C2×D3—362no—
312C2×D3—362no—
313C2×D3—362no—
314C2×D3—361no—
315C2×D3—361no—
316C2×D3—361no—
317C2×D3—361no—
318C2×D3—361no—
319C2×D3—361no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—121no—
323C3×C4—361no—
324C3×C4#369711yescharacteristic
325C3×C4#369911yescharacteristic
326C3×C4#380411yescharacteristic
327C3×C4#380411yescharacteristic
328Dic3—41no—
329Dic3—92no—
330Dic3—92no—
331Dic3—92no—
332Dic3—92no—
333Dic3—121no—
334Dic3—182no—
335Dic3—182no—
336C32#476011yescharacteristic
337C9—21no—
338C9#476011yescharacteristic, complemented
339C23—21no—
340C23—541no—
341Q8—1082no—
342D4—21no—
343D4—21no—
344D4—541no—
345D4—541no—
346D4—1082no—
347D4—1082no—
348C2×C4—61no—
349C2×C4—181no—
350C2×C4—271no—
351C2×C4—271no—
352C2×C4C2×D3×D911yescharacteristic
353C8—61no—
354C8—181no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—82no—
359C2×C3—81no—
360C2×C3—81no—
361C2×C3#6344711yescharacteristic
362C2×C3D8×D911yescharacteristic
363C2×C3SD16×D911yescharacteristic
364C2×C3#6392611yescharacteristic
365C2×C3#6393011yescharacteristic
366C2×C3#6395311yescharacteristic
367D3—722no—
368D3—721no—
369D3—721no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—1081no—
374C22—1081no—
375C22—1081no—
376C22#8866811yescharacteristic
377C4—121no—
378C4—272no—
379C4—272no—
380C4—361no—
381C4#8865811yescharacteristic
382C4#8866211yescharacteristic
383C3—21no—
384C3#1058844011yescharacteristic, complemented
385C3#1059140211yescharacteristic
386C2—81no—
387C2—2161no—
388C2#1172117411yescharacteristic
389C2#1172118211yescharacteristic
390C2#1172124211yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radical#11721840C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#486
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
3. C9×#832C2semidirect
4. C3×#63392C2semidirect
5. #11720284C2semidirect
6. #11720271C2semidirect
7. #11720267C2semidirect
32. #63392D3semidirect
102. #832D9semidirect
229. C3×C9#486semidirect
338. C9#4760semidirect
384. C3#10588440semidirect

Maximal subgroups

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

subgroupindex
#117218402
#117209712
C9×#8322
C3×#633922
#117202842
#117202712
#117202672
#105914023
#105884403