Finite Group Databasebeta

← #13140740

Subgroups — #13140740

Order 1728.

600 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 95 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×Q8×C3⋊D9C211yesmaximal, characteristic
2C2×C3×#3674C211yesmaximal, characteristic, complemented
3#11720936C211yesmaximal, characteristic
4C2×#88418C211yesmaximal, characteristic, complemented
5C9×#835C211yesmaximal, characteristic, complemented
6#11720282C211yesmaximal, characteristic
7#11720269C211yesmaximal, characteristic, complemented
8#10591603—31nomaximal
9#10588489—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11Q8×C3⋊D9—22no—
12Q8×C3⋊D9—22no—
13C2×C4×C3⋊D9—21no—
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×#6226—21no—
16C2×#6226C2211yescharacteristic
17C2×C3×Dic18C2211yescharacteristic
18C3×#3674—22no—
19C9×C4⋊Dic3C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21C2×C3×C9⋊C8C2211yescharacteristic
22#88445—21no—
23Dic3⋊Dic9—21no—
24#88418—22no—
25C2×Q8×C3⋊D3—31no—
26C2×C3×#216—31no—
27#63871—31no—
28C2×#3719—31no—
29C2×Q8×D9—31no—
30C2×Q8×D9—61no—
31C3×#835D311yescharacteristic
32#63567—31no—
33#63557—31no—
34C2×#3674D311yescharacteristic, complemented
35C2×Dic36—31no—
36#63459—31no—
37C9×#103—31no—
38(C2×D9)⋊C8—31no—
39#63368—31no—
40C22×C3⋊D9D411yescharacteristic
41C3×Q8×C9—21no—
42C3×Q8×C9D412yes—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3⋊Dic9—21no—
46C2×C3⋊Dic9D411yescharacteristic
47C4×C3⋊D9—24no—
48C4×C3⋊D9—21no—
49C4×C3⋊D9—21no—
50#6226—24no—
51#6226—21no—
52#6226D412yes—
53C2×C9×Dic3—21no—
54C2×C3×Dic9—21no—
55C3×Dic18—21no—
56C3×Dic18—21no—
57C9×C3⋊C8—21no—
58C3×C9⋊C8—21no—
59#5186—91no—
60#4809—91no—
61C2×C32×Q8C2×D311yescharacteristic
62Q8×C3⋊D3—62no—
63Q8×C3⋊D3—62no—
64C2×C4×C3⋊D3—31no—
65C2×C4×C3⋊D3—61no—
66C2×#634—31no—
67C2×#634—61no—
68C2×C3×Dic6—31no—
69C3×#216—62no—
70C3×C4⋊Dic3C2×D311yescharacteristic
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8C2×D311yescharacteristic
73Dic3⋊Dic3—61no—
74#3722—61no—
75#3719—62no—
76C2×Q8×C9—21no—
77C2×Q8×C9C2×D311yescharacteristic
78Q8×D9—62no—
79Q8×D9—64no—
80Q8×D9—64no—
81Q8×D9—62no—
82C2×C4×D9—31no—
83C2×C4×D9—61no—
84C2×C4×D9—61no—
85C2×C4×D9—62no—
86C2×Dic18—31no—
87C2×Dic18—61no—
88C2×Dic18—61no—
89C2×Dic18—62no—
90C2×Dic18C2×D311yescharacteristic
91C2×C8×C9—31no—
92C9×C4⋊C4—31no—
93#3674—22no—
94(C2×D9)⋊C4—61no—
95C4⋊Dic9—61no—
96C2×C9⋊C8C2×D311yescharacteristic
97Dic36—62no—
98C22×C3×C9C2×D411yescharacteristic
99C2×C3⋊D9—21no—
100C2×C3⋊D9—21no—
101C2×C3⋊D9Q1612yescomplemented
102C3×C4×C9—22no—
103C3×C4×C9C2×D411yescharacteristic
104C3×C4×C9C2×D411yescharacteristic
105C3⋊Dic9—21no—
106C3⋊Dic9—22no—
107C9×Dic3—41no—
108C3×Dic9—41no—
109C2×D3×Q8—91no—
110C2×D3×Q8—91no—
111C2×D3×Q8—181no—
112C2×C3×Q16—91no—
113#946—91no—
114C2×#216—31no—
115C2×#216—91no—
116C2×Dic12—91no—
117#897—91no—
118C3×#103—31no—
119#835D911yescharacteristic, complemented
120(C2×D3)⋊C8—91no—
121(C2×D3)⋊C8—91no—
122#816—91no—
123#810—91no—
124C22×C3⋊D3—31no—
125C32×Q8—21no—
126C32×Q8D1212yes—
127C2×C32×C4—21no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—31no—
130C2×C3⋊Dic3—61no—
131C4×C3⋊D3—64no—
132C4×C3⋊D3—61no—
133C4×C3⋊D3—61no—
134#634—32no—
135#634—64no—
136#634—61no—
137C2×C3×Dic3—21no—
138C2×C3×Dic3—61no—
139C3×Dic6—61no—
140C3×Dic6—61no—
141C22×D9—31no—
142C22×D9—61no—
143C3×C3⋊C8—21no—
144C3×C3⋊C8—61no—
145Q8×C9—22no—
146Q8×C9—21no—
147Q8×C9—22no—
148Q8×C9#6712yes—
149C2×C4×C9—22no—
150C2×C4×C9—21no—
151C2×C4×C9—21no—
152C2×C4×C9—61no—
153C2×C4×C9C22×D311yescharacteristic
154C2×Dic9—21no—
155C2×Dic9—31no—
156C2×Dic9—61no—
157C2×Dic9—62no—
158C2×Dic9—61no—
159C4×D9—68no—
160C4×D9—64no—
161C4×D9—62no—
162C4×D9—62no—
163C4×D9—61no—
164C4×D9—61no—
165Dic18—21no—
166Dic18—21no—
167Dic18—32no—
168Dic18—62no—
169Dic18—68no—
170Dic18—64no—
171Dic18—62no—
172Dic18—61no—
173C8×C9—61no—
174C9⋊C8—21no—
175#451—271noSylow
176C2×C3×C9#12011yescharacteristic
177C2×C3×C9C2×Q1611yescharacteristic
178C2×C3×C9#13711yescharacteristic
179C3⋊D9—22no—
180C2×C3×Q8—21no—
181C2×C3×Q8—91no—
182C2×C3×Q8C2×D911yescharacteristic
183C2×C3×Q8D3211yescharacteristic
184D3×Q8—182no—
185D3×Q8—182no—
186D3×Q8—184no—
187D3×Q8—184no—
188D3×Q8—182no—
189D3×Q8—182no—
190C2×C4×D3—91no—
191C2×C4×D3—91no—
192C2×C4×D3—181no—
193C2×C4×D3—181no—
194C2×C4×D3—181no—
195C2×C4×D3—182no—
196C2×Dic6—31no—
197C2×Dic6—91no—
198C2×Dic6—91no—
199C2×Dic6—181no—
200C2×Dic6—181no—
201C2×Dic6—181no—
202C2×Dic6—182no—
203C3×Q16—182no—
204C2×C3×C8—31no—
205C2×C3×C8—91no—
206C3×C4⋊C4—31no—
207#216—62no—
208#216—182no—
209(C2×D3)⋊C4—181no—
210(C2×D3)⋊C4—181no—
211C4⋊Dic3—181no—
212C4⋊Dic3C2×D911yescharacteristic
213Dic3⋊C4—181no—
214C2×C3⋊C8—31no—
215C2×C3⋊C8C2×D911yescharacteristic
216Dic12—182no—
217C22×C32D3×D411yescharacteristic
218C2×C3⋊D3—32no—
219C2×C3⋊D3—61no—
220C2×C3⋊D3—61no—
221C32×C4—22no—
222C32×C4C2×D1211yescharacteristic
223C32×C4D3×D411yescharacteristic
224C3⋊Dic3—62no—
225C3⋊Dic3—61no—
226C3×Dic3—41no—
227C3×Dic3—121no—
228C22×C9—21no—
229C22×C9D3×D411yescharacteristic
230C2×D9—32no—
231C2×D9—62no—
232C2×D9—62no—
233C2×D9—62no—
234C2×D9—61no—
235C2×D9—61no—
236C4×C9—24no—
237C4×C9—22no—
238C4×C9—21no—
239C4×C9—21no—
240C4×C9—121no—
241C4×C9D3×D411yescharacteristic
242C4×C9C2×#6711yescharacteristic
243Dic9—41no—
244Dic9—64no—
245Dic9—62no—
246Dic9—62no—
247Dic9—61no—
248C22×Q8—271no—
249C2×Q16—91no—
250C2×Q16—271no—
251C22⋊Q8—271no—
252#103—31no—
253#103—271no—
254C22⋊C8—271no—
255C3×C9#45111yescharacteristic, complemented, Sylow
256C22×D3—91no—
257C22×D3—91no—
258C22×D3—181no—
259C3×Q8—22no—
260C3×Q8—21no—
261C3×Q8—21no—
262C3×Q8—22no—
263C3×Q8—181no—
264C3×Q8—181no—
265C3×Q8D3612yes—
266C3×Q8#62612yes—
267C2×C3×C4—22no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—61no—
272C2×C3×C4—181no—
273C2×C3×C4C22×D911yescharacteristic
274C2×C3×C4C2×D3211yescharacteristic
275C2×Dic3—21no—
276C2×Dic3—61no—
277C2×Dic3—91no—
278C2×Dic3—91no—
279C2×Dic3—181no—
280C2×Dic3—182no—
281C2×Dic3—181no—
282C2×Dic3—181no—
283C4×D3—184no—
284C4×D3—188no—
285C4×D3—184no—
286C4×D3—182no—
287C4×D3—182no—
288C4×D3—181no—
289C4×D3—181no—
290C4×D3—181no—
291C4×D3—181no—
292Dic6—61no—
293Dic6—61no—
294Dic6—92no—
295Dic6—92no—
296Dic6—182no—
297Dic6—184no—
298Dic6—188no—
299Dic6—184no—
300Dic6—182no—
301Dic6—181no—
302Dic6—181no—
303C3×C8—61no—
304C3×C8—181no—
305C3⋊C8—21no—
306C3⋊C8—61no—
307C2×C32#88211yescharacteristic
308C2×C32#91511yescharacteristic
309C2×C32D3×Q1611yescharacteristic
310C3⋊D3—62no—
311C2×C9—21no—
312C2×C9—21no—
313C2×C9—21no—
314C2×C9D3×Q1611yescharacteristic
315C2×C9#91811yescharacteristic
316C2×C9#93711yescharacteristic
317D9—64no—
318D9—62no—
319C2×Q8—91no—
320C2×Q8—271no—
321C2×Q8—541no—
322C2×Q8—542no—
323C2×Q8—542no—
324C2×Q8D3×D911yescharacteristic
325C22×C4—271no—
326C22×C4—541no—
327Q16—182no—
328Q16—542no—
329C2×C8—31no—
330C2×C8—91no—
331C4⋊C4—31no—
332C4⋊C4—541no—
333C22⋊C4—541no—
334C22×C3—21no—
335C22×C3D4×D911yescharacteristic
336C22×C3#381111yescharacteristic
337C2×D3—92no—
338C2×D3—92no—
339C2×D3—182no—
340C2×D3—182no—
341C2×D3—182no—
342C2×D3—181no—
343C2×D3—181no—
344C2×D3—181no—
345C2×D3—181no—
346C3×C4—24no—
347C3×C4—22no—
348C3×C4—22no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—121no—
352C3×C4—361no—
353C3×C4C2×D3611yescharacteristic
354C3×C4D4×D911yescharacteristic
355C3×C4C2×#62611yescharacteristic
356C3×C4#381111yescharacteristic
357Dic3—41no—
358Dic3—121no—
359Dic3—182no—
360Dic3—184no—
361Dic3—182no—
362Dic3—181no—
363Dic3—182no—
364Dic3—181no—
365C32#480911yescharacteristic
366C9—21no—
367C9#518611yescharacteristic, complemented
368C23—271no—
369Q8—21no—
370Q8—181no—
371Q8—181no—
372Q8—272no—
373Q8—544no—
374Q8—541no—
375Q8#619212yes—
376C2×C4—21no—
377C2×C4—61no—
378C2×C4—181no—
379C2×C4—271no—
380C2×C4—544no—
381C2×C4—541no—
382C2×C4—541no—
383C2×C4—541no—
384C2×C4C2×D3×D911yescharacteristic
385C8—61no—
386C8—181no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3#6344411yescharacteristic
391C2×C3#6347711yescharacteristic
392C2×C3Q16×D911yescharacteristic
393C2×C3#6394411yescharacteristic
394C2×C3#6395111yescharacteristic
395C2×C3#6398111yescharacteristic
396D3—184no—
397D3—182no—
398D3—182no—
399C22—272no—
400C22—541no—
401C22—541no—
402C22#8866811yescharacteristic
403C4—22no—
404C4—121no—
405C4—361no—
406C4—541no—
407C4—542no—
408C4C2×#619211yescharacteristic
409C4#8866811yescharacteristic
410C3—21no—
411C3#1058848911yescharacteristic, complemented
412C3#1059160311yescharacteristic
413C2—542no—
414C2#1172121711yescharacteristic
415C2#1172122011yescharacteristic
416C2#1172127811yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
hypercenterC2×Q8D3×D9
hypocenterC3×C9#451
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. C2×C3×#3674C2semidirect
4. C2×#88418C2semidirect
5. C9×#835C2semidirect
7. #11720269C2semidirect
34. C2×#3674D3semidirect
101. C2×C3⋊D9Q16semidirect
119. #835D9semidirect
255. C3×C9#451semidirect
367. C9#5186semidirect
411. C3#10588489semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊D92
C2×C3×#36742
#117209362
C2×#884182
C9×#8352
#117202822
#117202692
#105916033
#105884893