Finite Group Databasebeta

← #13137848

Subgroups — #13137848

Order 1728.

526 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×(C3⋊D9)⋊Q8C211yesmaximal, characteristic
2#11721748C211yesmaximal, characteristic
3C2×#88414C211yesmaximal, characteristic
4C9×#810C211yesmaximal, characteristic, complemented
5C3×#63368C211yesmaximal, characteristic, complemented
6#11720298C211yesmaximal, characteristic, complemented
7#11720276C211yesmaximal, characteristic, complemented
8#10590602—31nomaximal
9#10588482—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C2×(C3⋊D9)⋊C4—21no—
14C2×#6189—21no—
15(C3⋊D9)⋊Q8—22no—
16(C3⋊D9)⋊Q8C414yescomplemented
17C3×C9×C4⋊C4C2211yescharacteristic
18#88536—21no—
19C4⋊(C3⋊Dic9)C2211yescharacteristic
20C4×C3⋊Dic9—21no—
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23#88414—22no—
24C2×#3797—31no—
25#64101—31no—
26C2×#3712—31no—
27C2×Q8×D9—31no—
28C3×#810—31no—
29C3×#810D311yescharacteristic
30#63566—31no—
31#63566—31no—
32C2×#3663—31no—
33#63454—31no—
34#63454—61no—
35C9×#103—31no—
36#63395—31no—
37#63376—31no—
38#63368D311yescharacteristic, complemented
39C22×C3⋊D9D411yescharacteristic
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9D411yescharacteristic
44C4×C3⋊D9C2×C412yes—
45C4×C3⋊D9D412yes—
46C2×C9×Dic3—21no—
47C2×C3×Dic9—21no—
48C9×Dic6—21no—
49C9×Dic6C2×C412yes—
50C3×Dic18—21no—
51C3×Dic18C2×C412yes—
52(C3⋊D9)⋊C4—24no—
53#6189—24no—
54C9×C3⋊C8—21no—
55C3×C9⋊C8—21no—
56#4802—91no—
57#4802—91no—
58C2×C4×C3⋊D3—31no—
59C2×C3×Dic6—31no—
60C2×C3×Dic6C2×D311yescharacteristic
61C2×#627—61no—
62C2×#624—61no—
63#3797—34no—
64#3797—62no—
65C32×C4⋊C4C2×D311yescharacteristic
66#3752—61no—
67C4⋊(C3⋊Dic3)—31no—
68C4×C3⋊Dic3—61no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8C2×D311yescharacteristic
71#3712—62no—
72C2×Q8×C9—31no—
73Q8×D9—34no—
74Q8×D9—62no—
75C2×C4×D9—31no—
76C2×C4×D9—61no—
77C2×C4×D9—61no—
78C2×Dic18—61no—
79C2×Dic18C2×D311yescharacteristic
80C2×C8×C9—31no—
81C9×C4⋊C4—21no—
82C9×C4⋊C4C2×D311yescharacteristic
83(C2×D9)⋊C4—61no—
84(C2×D9)⋊C4—62no—
85C4⋊Dic9—31no—
86C4⋊Dic9—61no—
87C4×Dic9—61no—
88C4×Dic9—62no—
89C2×C9⋊C8C2×D311yescharacteristic
90#3663—62no—
91C22×C3×C9C2×D411yescharacteristic
92C2×C3⋊D9—21no—
93C2×C3⋊D9—21no—
94C2×C3⋊D9C22⋊C412yes—
95C3×C4×C9—41no—
96C3×C4×C9C22×C411yescharacteristic
97C3×C4×C9C2×D411yescharacteristic
98C3⋊Dic9—41no—
99C3⋊Dic9C22⋊C412yes—
100C9×Dic3—22no—
101C3×Dic9—22no—
102C2×D3×Q8—91no—
103C2×D3×Q8—91no—
104C2×ΓL(1,𝔽25)—91no—
105C2×ΓL(1,𝔽25)—91no—
106#892—91no—
107#892—91no—
108#892—181no—
109C3×#103—31no—
110C3×#103—91no—
111#835—91no—
112#835—91no—
113#816—91no—
114#816—91no—
115#810—31no—
116#810D911yescharacteristic, complemented
117C22×C3⋊D3—31no—
118C2×C32×C4—21no—
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—31no—
121C2×C3⋊Dic3—61no—
122C4×C3⋊D3—32no—
123C4×C3⋊D3—32no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—61no—
126C3×Dic6—21no—
127C3×Dic6—32no—
128C3×Dic6—61no—
129C3×Dic6C4×D312yes—
130#627—64no—
131#624—64no—
132C22×D9—31no—
133C22×D9—61no—
134C3×C3⋊C8—21no—
135C3×C3⋊C8—61no—
136Q8×C9—32no—
137Q8×C9—61no—
138C2×C4×C9—22no—
139C2×C4×C9—21no—
140C2×C4×C9—21no—
141C2×C4×C9—61no—
142C2×C4×C9C22×D311yescharacteristic
143C2×Dic9—21no—
144C2×Dic9—31no—
145C2×Dic9—61no—
146C2×Dic9—62no—
147C2×Dic9—61no—
148C4×D9—32no—
149C4×D9—32no—
150C4×D9—62no—
151C4×D9—62no—
152C4×D9—64no—
153Dic18—21no—
154Dic18—64no—
155Dic18C4×D312yes—
156C8×C9—61no—
157C9⋊C8—21no—
158#419—271noSylow
159C2×C3×C9C2×C22⋊C411yescharacteristic
160C2×C3×C9#13711yescharacteristic
161C2×C3×C9#13711yescharacteristic
162C3⋊D9—22no—
163C2×C3×Q8—31no—
164C2×C3×Q8—91no—
165D3×Q8—94no—
166D3×Q8—94no—
167D3×Q8—182no—
168D3×Q8—182no—
169C2×C4×D3—91no—
170C2×C4×D3—91no—
171C2×C4×D3—181no—
172C2×C4×D3—181no—
173C2×C4×D3—181no—
174C2×Dic6—31no—
175C2×Dic6—181no—
176C2×Dic6—181no—
177C2×Dic6C2×D911yescharacteristic
178C2×C3×C8—31no—
179C2×C3×C8—91no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4C2×D911yescharacteristic
182C3×C4⋊C4D3211yescharacteristic
183(C2×D3)⋊C4—181no—
184(C2×D3)⋊C4—181no—
185(C2×D3)⋊C4—182no—
186C4⋊Dic3—91no—
187C4⋊Dic3—91no—
188C4⋊Dic3—181no—
189C4×Dic3—181no—
190C4×Dic3—181no—
191C4×Dic3—182no—
192C2×C3⋊C8—31no—
193C2×C3⋊C8C2×D911yescharacteristic
194ΓL(1,𝔽25)—182no—
195ΓL(1,𝔽25)—182no—
196C22×C32D3×D411yescharacteristic
197C2×C3⋊D3—32no—
198C2×C3⋊D3—61no—
199C2×C3⋊D3—61no—
200C32×C4—41no—
201C32×C4C2×C4×D311yescharacteristic
202C32×C4D3×D411yescharacteristic
203C3⋊Dic3—32no—
204C3⋊Dic3—121no—
205C3×Dic3—22no—
206C3×Dic3—62no—
207C22×C9—21no—
208C22×C9D3×D411yescharacteristic
209C2×D9—32no—
210C2×D9—62no—
211C2×D9—61no—
212C2×D9—61no—
213C2×D9—121no—
214C2×D9—121no—
215C4×C9—21no—
216C4×C9—21no—
217C4×C9—42no—
218C4×C9—41no—
219C4×C9—62no—
220C4×C9C2×C4×D311yescharacteristic
221C4×C9D3×D411yescharacteristic
222Dic9—22no—
223Dic9—32no—
224Dic9—62no—
225Dic9—122no—
226Dic9—121no—
227C22×Q8—271no—
228C2×M16—271no—
229#117—271no—
230#103—31no—
231#103—91no—
232#103—271no—
233#103—271no—
234C3×C9#41911yescharacteristic, complemented, Sylow
235C22×D3—91no—
236C22×D3—91no—
237C22×D3—181no—
238C3×Q8—32no—
239C3×Q8—61no—
240C3×Q8—92no—
241C3×Q8—181no—
242C2×C3×C4—22no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—61no—
247C2×C3×C4—181no—
248C2×C3×C4C22×D911yescharacteristic
249C2×C3×C4C2×D3211yescharacteristic
250C2×Dic3—21no—
251C2×Dic3—61no—
252C2×Dic3—91no—
253C2×Dic3—91no—
254C2×Dic3—181no—
255C2×Dic3—182no—
256C2×Dic3—181no—
257C2×Dic3—181no—
258C4×D3—92no—
259C4×D3—92no—
260C4×D3—92no—
261C4×D3—92no—
262C4×D3—182no—
263C4×D3—182no—
264C4×D3—184no—
265C4×D3—184no—
266Dic6—21no—
267Dic6—32no—
268Dic6—61no—
269Dic6—184no—
270Dic6—184no—
271Dic6C4×D912yes—
272C3×C8—61no—
273C3×C8—181no—
274C3⋊C8—21no—
275C3⋊C8—61no—
276C2×C32D3×C22⋊C411yescharacteristic
277C2×C32#91511yescharacteristic
278C2×C32#91811yescharacteristic
279C3⋊D3—62no—
280C2×C9—21no—
281C2×C9—21no—
282C2×C9—21no—
283C2×C9D3×C22⋊C411yescharacteristic
284C2×C9#91511yescharacteristic
285C2×C9#91811yescharacteristic
286D9—62no—
287D9—122no—
288C2×Q8—31no—
289C2×Q8—91no—
290C2×Q8—274no—
291C2×Q8—541no—
292C2×Q8—542no—
293C22×C4—271no—
294C22×C4—541no—
295M16—542no—
296C2×C8—31no—
297C2×C8—91no—
298C4⋊C4—271no—
299C4⋊C4D3×D911yescharacteristic
300C22⋊C4—541no—
301C42—541no—
302C22×C3—21no—
303C22×C3D4×D911yescharacteristic
304C22×C3#381111yescharacteristic
305C2×D3—92no—
306C2×D3—92no—
307C2×D3—182no—
308C2×D3—181no—
309C2×D3—181no—
310C2×D3—181no—
311C2×D3—181no—
312C2×D3—361no—
313C2×D3—361no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—42no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4—62no—
320C3×C4—182no—
321C3×C4C2×C4×D911yescharacteristic
322C3×C4D4×D911yescharacteristic
323C3×C4C4×D3211yescharacteristic
324C3×C4#380211yescharacteristic
325Dic3—22no—
326Dic3—62no—
327Dic3—92no—
328Dic3—92no—
329Dic3—182no—
330Dic3—362no—
331Dic3—361no—
332Dic3—361no—
333C32#480211yescharacteristic
334C9—21no—
335C9#480211yescharacteristic, complemented
336C23—271no—
337Q8—32no—
338Q8—61no—
339Q8—92no—
340Q8—181no—
341Q8—544no—
342C2×C4—21no—
343C2×C4—61no—
344C2×C4—181no—
345C2×C4—271no—
346C2×C4—272no—
347C2×C4—272no—
348C2×C4—544no—
349C2×C4—541no—
350C2×C4C2×D3×D911yescharacteristic
351C8—61no—
352C8—181no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3C22⋊C4×D911yescharacteristic
357C2×C3#6347711yescharacteristic
358C2×C3#6348011yescharacteristic
359C2×C3#6392111yescharacteristic
360C2×C3#6393111yescharacteristic
361C2×C3#6394511yescharacteristic
362D3—182no—
363D3—182no—
364D3—362no—
365C22—272no—
366C22—541no—
367C22—541no—
368C22#8866811yescharacteristic
369C4—41no—
370C4—62no—
371C4—182no—
372C4—272no—
373C4—1081no—
374C4C4×D3×D911yescharacteristic
375C4#8864611yescharacteristic
376C3—21no—
377C3#1058848211yescharacteristic, complemented
378C3#1059060211yescharacteristic
379C2—542no—
380C2#1172104811yescharacteristic
381C2#1172118311yescharacteristic
382C2#1172122111yescharacteristic
subgroupG/H
centerC22#88668
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
cosocleC2×C3×C4×C9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#419
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C9×#810C2semidirect
5. C3×#63368C2semidirect
6. #11720298C2semidirect
7. #11720276C2semidirect
16. (C3⋊D9)⋊Q8C4semidirect
38. #63368D3semidirect
116. #810D9semidirect
234. C3×C9#419semidirect
335. C9#4802semidirect
377. C3#10588482semidirect

Maximal subgroups

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

subgroupindex
C2×(C3⋊D9)⋊Q82
#117217482
C2×#884142
C9×#8102
C3×#633682
#117202982
#117202762
#105906023
#105884823