Finite Group Databasebeta

← #13139803

Subgroups — #13139803

Order 1728.

416 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 105 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#11721813C211yesmaximal, characteristic
2C9×#924C211yesmaximal, characteristic
3C22×C3×C9⋊C8C211yesmaximal, characteristic
4C2×#88428C211yesmaximal, characteristic, complemented
5#11720298C211yesmaximal, characteristic, complemented
6#11720269C211yesmaximal, characteristic, complemented
7#11720262C211yesmaximal, characteristic, complemented
8#10591187—31nomaximal
9#10588727—31nomaximal
10C22×C3×C4×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13#88551—21no—
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15#88534—21no—
16C9×#217—21no—
17C9×C4⋊Dic3C2211yescharacteristic
18C9×Dic3⋊C4—21no—
19C2×C3×C9⋊C8—22no—
20C2×C3×C9⋊C8C2211yescharacteristic
21C2×C3×C9⋊C8C2211yescharacteristic
22#88428—22no—
23#64133—31no—
24C3×#924D311yescharacteristic
25C22×C3×C3⋊C8—31no—
26C2×#3719—31no—
27#63575—31no—
28#63567—31no—
29#63566—31no—
30C9×C22⋊Q8—31no—
31C2×#3674—31no—
32#63486—31no—
33#63486—61no—
34C22×C9⋊C8D311yescharacteristic
35#63395—31no—
36#63368—31no—
37#63366—31no—
38C23×C3×C9D411yescharacteristic
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9D411yescharacteristic
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44C2×C3⋊Dic9—21no—
45#6226—21no—
46#6226—21no—
47C2×C9×Dic3—21no—
48C2×C9×Dic3—21no—
49C9×Dic6—21no—
50C9×Dic6—21no—
51C3×C9⋊C8—21no—
52C3×C9⋊C8D412yes—
53#5116—91no—
54#5051—91no—
55C22×C32×C4C2×D311yescharacteristic
56C2×#634—31no—
57C2×C3×Dic6C2×D311yescharacteristic
58#3757—61no—
59C4⋊(C3⋊Dic3)—31no—
60#3750—61no—
61C3×#217—21no—
62C3×C4⋊Dic3C2×D311yescharacteristic
63C3×Dic3⋊C4—21no—
64C2×C3×C3⋊C8—31no—
65C2×C3×C3⋊C8—31no—
66C2×C3×C3⋊C8—62no—
67#3719—62no—
68C2×Q8×C9—31no—
69C22×C4×C9—21no—
70C22×C4×C9C2×D311yescharacteristic
71C2×Dic18—31no—
72C2×Dic18—61no—
73C9×C4⋊C4—31no—
74C9×C4⋊C4—61no—
75C9×C22⋊C4—61no—
76#3676—62no—
77#3676—61no—
78#3674—62no—
79C4⋊Dic9—31no—
80C4⋊Dic9—61no—
81Dic9⋊C4—62no—
82Dic9⋊C4—61no—
83C2×C9⋊C8—22no—
84C2×C9⋊C8C2×D311yescharacteristic
85C2×C9⋊C8C2×D311yescharacteristic
86C22×C3×C9—21no—
87C22×C3×C9—21no—
88C22×C3×C9Q1612yescomplemented
89C22×C3×C9C2×D411yescharacteristic
90C3×C4×C9—21no—
91C3×C4×C9C2×D411yescharacteristic
92C3×C4×C9#4111yescharacteristic
93C3⋊Dic9—41no—
94C3⋊Dic9—41no—
95C9×Dic3—41no—
96C9×Dic3—41no—
97C22×C3×C8—91no—
98C3×C22⋊Q8—31no—
99C2×#216—91no—
100#924—91no—
101#924—91no—
102#924—181no—
103#924D911yescharacteristic
104C22×C3⋊C8—31no—
105C2×Dic12—91no—
106#835—91no—
107#818—91no—
108#816—91no—
109#816—91no—
110#810—91no—
111#807—91no—
112C23×C32#6711yescharacteristic
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4#6711yescharacteristic
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—61no—
118C2×C3⋊Dic3—61no—
119#634—61no—
120#634—61no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—21no—
123C3×Dic6—21no—
124C3×Dic6—21no—
125C23×C9—21no—
126C23×C9D1211yescharacteristic
127C3×C3⋊C8—32no—
128C3×C3⋊C8—61no—
129Q8×C9—61no—
130Q8×C9—61no—
131C2×C4×C9—21no—
132C2×C4×C9—21no—
133C2×C4×C9—21no—
134C2×C4×C9—21no—
135C2×C4×C9—41no—
136C2×C4×C9—41no—
137C2×C4×C9—61no—
138C2×C4×C9—61no—
139C2×C4×C9D1211yescharacteristic
140C2×C4×C9C22×D311yescharacteristic
141C2×Dic9—62no—
142C2×Dic9—62no—
143C2×Dic9—61no—
144C2×Dic9—61no—
145Dic18—61no—
146Dic18—61no—
147Dic18—121no—
148Dic18—121no—
149C9⋊C8—21no—
150C9⋊C8#6712yes—
151#467—271noSylow
152C2×C3×C9—22no—
153C2×C3×C9#12111yescharacteristic
154C2×C3×C9C2×Q1611yescharacteristic
155C2×C3×C9#13511yescharacteristic
156C2×C3×Q8—31no—
157C22×C3×C4—21no—
158C22×C3×C4C2×D911yescharacteristic
159C22×C3×C4D3211yescharacteristic
160C2×Dic6—91no—
161C2×Dic6—91no—
162C2×Dic6—181no—
163C2×Dic6C2×D911yescharacteristic
164C2×C3×C8—91no—
165C2×C3×C8—91no—
166C2×C3×C8—182no—
167C3×C4⋊C4—31no—
168C3×C4⋊C4—61no—
169C3×C22⋊C4—61no—
170#217—21no—
171#217—182no—
172#217—181no—
173#217—181no—
174#216—182no—
175C4⋊Dic3—91no—
176C4⋊Dic3—91no—
177C4⋊Dic3—181no—
178C4⋊Dic3C2×D911yescharacteristic
179Dic3⋊C4—21no—
180Dic3⋊C4—182no—
181Dic3⋊C4—181no—
182Dic3⋊C4—181no—
183C2×C3⋊C8—31no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—62no—
186Dic12—182no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32#21612yes—
190C22×C32C2×#6711yescharacteristic
191C32×C4—21no—
192C32×C4D3×D411yescharacteristic
193C32×C4#23711yescharacteristic
194C3⋊Dic3—121no—
195C3⋊Dic3—121no—
196C3×Dic3—41no—
197C3×Dic3—41no—
198C22×C9—21no—
199C22×C9—21no—
200C22×C9—22no—
201C22×C9—21no—
202C22×C9—41no—
203C22×C9—41no—
204C22×C9Dic1212yescomplemented
205C22×C9C2×D1211yescharacteristic
206C4×C9—21no—
207C4×C9—21no—
208C4×C9—21no—
209C4×C9—41no—
210C4×C9—121no—
211C4×C9—121no—
212C4×C9#23511yescharacteristic
213C4×C9C2×#6711yescharacteristic
214Dic9—122no—
215Dic9—122no—
216Dic9—121no—
217Dic9—121no—
218C2×Q16—271no—
219C22×C8—91no—
220C22⋊Q8—31no—
221C22⋊Q8—271no—
222#107—271no—
223#103—271no—
224#103—271no—
225C3×C9#46711yescharacteristic, complemented, Sylow
226C23×C3—21no—
227C23×C3#61111yescharacteristic
228C23×C3#62611yescharacteristic
229C3×Q8—61no—
230C3×Q8—61no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4#61111yescharacteristic
242C2×C3×C4C22×D911yescharacteristic
243C2×C3×C4#62611yescharacteristic
244C2×C3×C4C2×D3211yescharacteristic
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—182no—
248C2×Dic3—182no—
249C2×Dic3—181no—
250C2×Dic3—181no—
251C2×Dic3—181no—
252C2×Dic3—181no—
253Dic6—21no—
254Dic6—21no—
255Dic6—181no—
256Dic6—181no—
257Dic6—181no—
258Dic6—181no—
259Dic6—361no—
260Dic6—361no—
261C3×C8—92no—
262C3×C8—181no—
263C3⋊C8—32no—
264C3⋊C8—61no—
265C2×C32—22no—
266C2×C32#93911yescharacteristic
267C2×C32C2×#21611yescharacteristic
268C2×C32#95011yescharacteristic
269C2×C9—22no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9—21no—
273C2×C9—42no—
274C2×C9#90411yescharacteristic
275C2×C9C2×Dic1211yescharacteristic
276C2×C9#93011yescharacteristic
277C2×Q8—31no—
278C2×Q8—271no—
279C22×C4D3×D911yescharacteristic
280Q16—542no—
281C2×C8—91no—
282C2×C8—91no—
283C2×C8—182no—
284C4⋊C4—31no—
285C4⋊C4—61no—
286C4⋊C4—271no—
287C4⋊C4—541no—
288C22⋊C4—61no—
289C22⋊C4—541no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—22no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3#367412yescomplemented
299C22×C3C2×#61111yescharacteristic
300C22×C3#371912yes—
301C22×C3C2×#62611yescharacteristic
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—41no—
307C3×C4—121no—
308C3×C4—121no—
309C3×C4D4×D911yescharacteristic
310C3×C4#369911yescharacteristic
311C3×C4#380411yescharacteristic
312C3×C4D3×#6711yescharacteristic
313Dic3—41no—
314Dic3—41no—
315Dic3—362no—
316Dic3—362no—
317Dic3—361no—
318Dic3—361no—
319Dic3—361no—
320Dic3—361no—
321C32#505111yescharacteristic
322C9—21no—
323C9#511611yescharacteristic, complemented
324C23#619511yescharacteristic
325Q8—61no—
326Q8—61no—
327Q8—541no—
328Q8—541no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—541no—
334C2×C4—541no—
335C2×C4#619511yescharacteristic
336C2×C4C2×D3×D911yescharacteristic
337C8—92no—
338C8—181no—
339C2×C3—22no—
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—42no—
345C2×C3#6350111yescharacteristic
346C2×C3C2×#367411yescharacteristic
347C2×C3#6351311yescharacteristic
348C2×C3#6382911yescharacteristic
349C2×C3C2×#371911yescharacteristic
350C2×C3#6397811yescharacteristic
351C22—21no—
352C22—21no—
353C22#8842812yescomplemented
354C22C2×#619511yescharacteristic
355C4—21no—
356C4—121no—
357C4—121no—
358C4—1081no—
359C4—1081no—
360C4#8866211yescharacteristic
361C4D9×#6711yescharacteristic
362C3—21no—
363C3#1058872711yescharacteristic, complemented
364C3#1059118711yescharacteristic
365C2—22no—
366C2#1172088911yescharacteristic
367C2C2×#8842811yescharacteristic
368C2#1172128811yescharacteristic
subgroupG/H
centerC22C2×#6195
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×#67
prime radicalC23×C32#67
hypercenterC22×C4D3×D9
hypocenterC3×C9#467
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. C2×#88428C2semidirect
5. #11720298C2semidirect
6. #11720269C2semidirect
7. #11720262C2semidirect
88. C22×C3×C9Q16semidirect
204. C22×C9Dic12semidirect
225. C3×C9#467semidirect
298. C22×C3#3674semidirect
323. C9#5116semidirect
353. C22#88428semidirect
363. C3#10588727semidirect

Maximal subgroups

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

subgroupindex
#117218132
C9×#9242
C22×C3×C9⋊C82
C2×#884282
#117202982
#117202692
#117202622
#105911873
#105887273