Finite Group Databasebeta

← C2×D3×Q8×D9 #13164753

Subgroups — C2×D3×Q8×D9 #13164753

Order 1728.

2706 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C3⋊D9C211yesmaximal, characteristic, complemented
2C2×D3×Q8×C9C211yesmaximal, characteristic, complemented
3C2×C3×Q8×D9C211yesmaximal, characteristic, complemented
4D3×Q8×D9C2116yesmaximal, direct product factor
5C2×C4×D3×D9C213yesmaximal
6C2×D3×Dic18C213yesmaximal, complemented
7C2×(C3⋊D9)⋊Q8C213yesmaximal, complemented
8C2×D9×Dic6C213yesmaximal, complemented
9C2×D32×Q8—31nomaximal
10C22×Q8×D9—31nomaximal
11C22×D3×D9C2211yescharacteristic
12C2×C3×Q8×C9C2211yescharacteristic, complemented
13Q8×C3⋊D9C2218yescomplemented
14C2×C4×C3⋊D9C2213yes—
15C2×#6226C2213yescomplemented
16D3×Q8×C9C2218yescomplemented
17C3×Q8×D9C2218yescomplemented
18C2×C4×D3×C9C2213yes—
19C2×C3×C4×D9C2213yes—
20C2×C9×Dic6C2213yescomplemented
21C2×C3×Dic18C2213yescomplemented
22C2×D3×Dic9C2213yes—
23C2×(C3⋊D9)⋊C4C2213yes—
24C2×Dic3×D9C2213yes—
25C2×#6189C2216yescomplemented
26C4×D3×D9C22124yes—
27D3×Dic18C22124yescomplemented
28(C3⋊D9)⋊Q8C22124yescomplemented
29D9×Dic6C22124yescomplemented
30C2×Q8×C3⋊D3—31no—
31C2×C3×D3×Q8—31no—
32C2×C3×D3×Q8D311yescharacteristic
33D32×Q8—316no—
34C2×C4×D32—33no—
35C2×#3797—33no—
36C2×D3×Dic6—33no—
37C2×D3×Dic6—33no—
38C22×Q8×C9—31no—
39C2×Q8×D9—31no—
40C2×Q8×D9—316no—
41C2×Q8×D9—33no—
42C2×Q8×D9—33no—
43C2×Q8×D9—61no—
44C2×Q8×D9D311yescharacteristic, direct product factor
45C22×C4×D9—33no—
46C22×Dic18—33no—
47C22×C3⋊D9C2311yescharacteristic
48C22×D3×C9C2311yescharacteristic
49C22×C3×D9C2311yescharacteristic
50C2×D3×D9Q818yesdirect product factor
51C2×D3×D9C2314yes—
52C3×Q8×C9C2314yescomplemented
53C2×C3×C4×C9C2313yes—
54C2×C3⋊Dic9C2313yes—
55C4×C3⋊D9C23112yes—
56#6226C23112yescomplemented
57C2×C9×Dic3C2313yes—
58C2×C3×Dic9C2313yes—
59C4×D3×C9C23112yes—
60C3×C4×D9C23112yes—
61C9×Dic6C23112yescomplemented
62C3×Dic18C23112yescomplemented
63D3×Dic9C23112yes—
64(C3⋊D9)⋊C4C23112yes—
65Dic3×D9C23112yes—
66#6189C23124yescomplemented
67C22×D3×Q8—91no—
68C22×D3×Q8—91no—
69C22×D32—31no—
70C2×C32×Q8C2×D311yescharacteristic
71Q8×C3⋊D3—38no—
72C2×C4×C3⋊D3—33no—
73C2×#634—33no—
74C3×D3×Q8—38no—
75C3×D3×Q8C2×D318yes—
76C2×C3×C4×D3—33no—
77C2×C3×C4×D3C2×D313yes—
78C2×C3×Dic6—33no—
79C2×C3×Dic6C2×D313yes—
80C2×#627—36no—
81C2×#624—33no—
82C2×D3×Dic3—33no—
83C2×D3×Dic3—33no—
84C4×D32—324no—
85#3797—324no—
86D3×Dic6—324no—
87D3×Dic6—324no—
88C23×D9—31no—
89C2×Q8×C9—21no—
90C2×Q8×C9—38no—
91C2×Q8×C9—33no—
92C2×Q8×C9C2×D311yescharacteristic, complemented
93C22×C4×C9—33no—
94C22×Dic9—33no—
95Q8×D9—38no—
96Q8×D9—324no—
97Q8×D9—324no—
98Q8×D9—68no—
99Q8×D9C2×D318yesdirect product factor
100C2×C4×D9—33no—
101C2×C4×D9—33no—
102C2×C4×D9—33no—
103C2×C4×D9—324no—
104C2×C4×D9—63no—
105C2×C4×D9C2×D313yes—
106C2×Dic18—33no—
107C2×Dic18—36no—
108C2×Dic18—324no—
109C2×Dic18—63no—
110C2×Dic18C2×D313yescomplemented
111C22×C3×C9C2411yescharacteristic
112C2×C3⋊D9C2×Q814yescomplemented
113C2×C3⋊D9C2412yes—
114C2×D3×C9C2×Q814yescomplemented
115C2×D3×C9C2412yes—
116C2×C3×D9C2×Q814yescomplemented
117C2×C3×D9C2412yes—
118D3×D9C2×Q8116yesdirect product factor
119C3×C4×C9C2416yes—
120C3⋊Dic9C2416yes—
121C9×Dic3C2416yes—
122C3×Dic9C2416yes—
123C22×C3×Q8—31no—
124C22×C3×Q8—91no—
125C2×D3×Q8—31no—
126C2×D3×Q8—93no—
127C2×D3×Q8—93no—
128C2×D3×Q8—916no—
129C2×D3×Q8—91no—
130C2×D3×Q8—91no—
131C2×D3×Q8—916no—
132C2×D3×Q8—93no—
133C2×D3×Q8—93no—
134C2×D3×Q8—181no—
135C2×D3×Q8D911yescharacteristic, direct product factor
136C22×C4×D3—93no—
137C22×C4×D3—93no—
138C22×Dic6—93no—
139C22×Dic6—93no—
140C22×C3⋊D3—31no—
141C22×C3×D3—31no—
142C22×C3×D3C22×D311yescharacteristic
143C2×D32—38no—
144C2×D32—34no—
145C32×Q8C22×D314yes—
146C2×C32×C4C22×D313yes—
147C2×C3⋊Dic3—33no—
148C4×C3⋊D3—312no—
149#634—312no—
150C2×C3×Dic3—33no—
151C2×C3×Dic3C22×D313yes—
152C3×C4×D3—312no—
153C3×C4×D3C22×D3112yes—
154C3×Dic6—312no—
155C3×Dic6C22×D3112yes—
156#627—324no—
157#624—312no—
158D3×Dic3—312no—
159D3×Dic3—312no—
160C23×C9—31no—
161C22×D9—31no—
162C22×D9—38no—
163C22×D9—34no—
164C22×D9—61no—
165C22×D9C22×D311yescharacteristic
166Q8×C9—24no—
167Q8×C9—312no—
168Q8×C9C22×D314yescomplemented
169C2×C4×C9—23no—
170C2×C4×C9—33no—
171C2×C4×C9—312no—
172C2×C4×C9C22×D313yes—
173C2×Dic9—33no—
174C2×Dic9—312no—
175C2×Dic9—63no—
176C2×Dic9C22×D313yes—
177C4×D9—312no—
178C4×D9—312no—
179C4×D9—312no—
180C4×D9—612no—
181C4×D9C22×D3112yes—
182Dic18—312no—
183Dic18—324no—
184Dic18—612no—
185Dic18C22×D3112yescomplemented
186C23×Q8—271noSylow
187C2×C3×C9C22×Q812yescomplemented
188C2×C3×C9C2511yescharacteristic
189C3⋊D9C22×Q814yescomplemented
190D3×C9C22×Q814yescomplemented
191C3×D9C22×Q814yescomplemented
192C23×D3—91no—
193C23×D3—91no—
194C2×C3×Q8—21no—
195C2×C3×Q8—38no—
196C2×C3×Q8—33no—
197C2×C3×Q8—98no—
198C2×C3×Q8—93no—
199C2×C3×Q8C2×D911yescharacteristic, complemented
200C2×C3×Q8D3211yescharacteristic
201C22×C3×C4—33no—
202C22×C3×C4—93no—
203C22×Dic3—93no—
204C22×Dic3—93no—
205D3×Q8—38no—
206D3×Q8—924no—
207D3×Q8—924no—
208D3×Q8—98no—
209D3×Q8—98no—
210D3×Q8—924no—
211D3×Q8—924no—
212D3×Q8—188no—
213D3×Q8C2×D918yesdirect product factor
214C2×C4×D3—33no—
215C2×C4×D3—924no—
216C2×C4×D3—93no—
217C2×C4×D3—93no—
218C2×C4×D3—93no—
219C2×C4×D3—93no—
220C2×C4×D3—93no—
221C2×C4×D3—93no—
222C2×C4×D3—924no—
223C2×C4×D3—183no—
224C2×C4×D3C2×D913yes—
225C2×Dic6—33no—
226C2×Dic6—924no—
227C2×Dic6—96no—
228C2×Dic6—93no—
229C2×Dic6—93no—
230C2×Dic6—96no—
231C2×Dic6—924no—
232C2×Dic6—183no—
233C2×Dic6C2×D913yescomplemented
234C22×C32C23×D311yescharacteristic
235C2×C3⋊D3—34no—
236C2×C3⋊D3—32no—
237C2×C3×D3—34no—
238C2×C3×D3—32no—
239C2×C3×D3D3×Q814yes—
240C2×C3×D3C23×D312yes—
241D32—316no—
242C32×C4C23×D316yes—
243C3⋊Dic3—36no—
244C3×Dic3—36no—
245C3×Dic3C23×D316yes—
246C22×C9—21no—
247C22×C9—34no—
248C22×C9—32no—
249C22×C9C23×D311yescharacteristic
250C2×D9—34no—
251C2×D9—32no—
252C2×D9—316no—
253C2×D9—64no—
254C2×D9—62no—
255C2×D9D3×Q814yesdirect product factor
256C2×D9C23×D312yes—
257C4×C9—26no—
258C4×C9—36no—
259C4×C9C23×D316yes—
260Dic9—36no—
261Dic9—66no—
262Dic9C23×D316yes—
263C22×Q8—31no—
264C22×Q8—91no—
265C22×Q8—273no—
266C22×Q8—273no—
267C22×Q8—2716no—
268C22×Q8—273no—
269C22×Q8—271no—
270C23×C4—273no—
271C3×C9C23×Q811yescharacteristic, complemented, Sylow
272C23×C3—31no—
273C23×C3—91no—
274C22×D3—31no—
275C22×D3—94no—
276C22×D3—98no—
277C22×D3—91no—
278C22×D3—91no—
279C22×D3—98no—
280C22×D3—94no—
281C22×D3—181no—
282C22×D3C22×D911yescharacteristic
283C3×Q8—24no—
284C3×Q8—312no—
285C3×Q8—912no—
286C3×Q8C22×D914yescomplemented
287C3×Q8C2×D3214yes—
288C2×C3×C4—23no—
289C2×C3×C4—33no—
290C2×C3×C4—312no—
291C2×C3×C4—93no—
292C2×C3×C4—912no—
293C2×C3×C4C22×D913yes—
294C2×C3×C4C2×D3213yes—
295C2×Dic3—33no—
296C2×Dic3—912no—
297C2×Dic3—93no—
298C2×Dic3—93no—
299C2×Dic3—912no—
300C2×Dic3—183no—
301C2×Dic3C22×D913yes—
302C4×D3—312no—
303C4×D3—912no—
304C4×D3—912no—
305C4×D3—912no—
306C4×D3—912no—
307C4×D3—912no—
308C4×D3—912no—
309C4×D3—1812no—
310C4×D3C22×D9112yes—
311Dic6—312no—
312Dic6—924no—
313Dic6—912no—
314Dic6—912no—
315Dic6—924no—
316Dic6—1812no—
317Dic6C22×D9112yescomplemented
318C2×C32C2×D3×Q812yes—
319C2×C32C24×D311yescharacteristic
320C3⋊D3—34no—
321C3×D3—34no—
322C3×D3C2×D3×Q814yes—
323C2×C9—22no—
324C2×C9—21no—
325C2×C9—34no—
326C2×C9C2×D3×Q812yescomplemented
327C2×C9C24×D311yescharacteristic
328D9—34no—
329D9—64no—
330D9C2×D3×Q814yesdirect product factor
331C24—271no—
332C2×Q8—38no—
333C2×Q8—33no—
334C2×Q8—98no—
335C2×Q8—93no—
336C2×Q8—276no—
337C2×Q8—2724no—
338C2×Q8—2724no—
339C2×Q8—2724no—
340C2×Q8—278no—
341C2×Q8—273no—
342C2×Q8D3×D911yescharacteristic, direct product factor
343C22×C4—33no—
344C22×C4—93no—
345C22×C4—273no—
346C22×C4—2724no—
347C22×C4—273no—
348C22×C4—273no—
349C22×C4—273no—
350C22×C3—21no—
351C22×C3—34no—
352C22×C3—32no—
353C22×C3—92no—
354C22×C3—94no—
355C22×C3C23×D911yescharacteristic
356C22×C3C22×D3211yescharacteristic
357C2×D3—34no—
358C2×D3—32no—
359C2×D3—916no—
360C2×D3—94no—
361C2×D3—92no—
362C2×D3—92no—
363C2×D3—94no—
364C2×D3—916no—
365C2×D3—182no—
366C2×D3—184no—
367C2×D3Q8×D914yesdirect product factor
368C2×D3C23×D912yes—
369C3×C4—26no—
370C3×C4—36no—
371C3×C4—96no—
372C3×C4C23×D916yes—
373C3×C4C22×D3216yes—
374Dic3—36no—
375Dic3—96no—
376Dic3—96no—
377Dic3—186no—
378Dic3C23×D916yes—
379C32C22×D3×Q811yescharacteristic
380C9—21no—
381C9C22×D3×Q811yescharacteristic, complemented
382C23—31no—
383C23—91no—
384C23—274no—
385C23—278no—
386C23—271no—
387Q8—312no—
388Q8—912no—
389Q8—2724no—
390Q8—2712no—
391Q8C2×D3×D914yesdirect product factor
392C2×C4—33no—
393C2×C4—312no—
394C2×C4—93no—
395C2×C4—912no—
396C2×C4—2712no—
397C2×C4—2712no—
398C2×C4—2712no—
399C2×C4—273no—
400C2×C4—2712no—
401C2×C4C2×D3×D913yes—
402C2×C3—22no—
403C2×C3—21no—
404C2×C3—34no—
405C2×C3—94no—
406C2×C3C2×Q8×D912yescomplemented
407C2×C3C24×D911yescharacteristic
408C2×C3D32×Q812yes—
409C2×C3C23×D3211yescharacteristic
410D3—34no—
411D3—94no—
412D3—94no—
413D3—184no—
414D3C2×Q8×D914yesdirect product factor
415C22—34no—
416C22—32no—
417C22—94no—
418C22—92no—
419C22—2716no—
420C22—274no—
421C22—272no—
422C22C22×D3×D911yescharacteristic
423C4—36no—
424C4—96no—
425C4—276no—
426C4C22×D3×D916yes—
427C3—21no—
428C3C22×Q8×D911yescharacteristic, complemented
429C3C2×D32×Q811yescharacteristic
430C2—34no—
431C2—94no—
432C2—274no—
433C2D3×Q8×D912yesdirect product factor
434C2C23×D3×D911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC2×C3×C9C25
FrattiniC2×C3C23×D32
FittingC2×C3×Q8×C9C22
socleC22×C32C23×D3
prime radicalC22×D3×D9C22
hypercenterC2×Q8D3×D9
hypocenterC3×C9C23×Q8
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×C3⋊D9C2semidirect
2. C2×D3×Q8×C9C2semidirect
3. C2×C3×Q8×D9C2semidirect
4. D3×Q8×D9C2direct (N × H)
6. C2×D3×Dic18C2semidirect
7. C2×(C3⋊D9)⋊Q8C2semidirect
8. C2×D9×Dic6C2semidirect
12. C2×C3×Q8×C9C22semidirect
13. Q8×C3⋊D9C22semidirect
15. C2×#6226C22semidirect
16. D3×Q8×C9C22semidirect
17. C3×Q8×D9C22semidirect
20. C2×C9×Dic6C22semidirect
21. C2×C3×Dic18C22semidirect
25. C2×#6189C22semidirect
27. D3×Dic18C22semidirect
28. (C3⋊D9)⋊Q8C22semidirect
29. D9×Dic6C22semidirect
44. C2×Q8×D9D3direct (N × H)
50. C2×D3×D9Q8direct (N × H)
52. C3×Q8×C9C23semidirect
56. #6226C23semidirect
61. C9×Dic6C23semidirect
62. C3×Dic18C23semidirect
66. #6189C23semidirect
92. C2×Q8×C9C2×D3semidirect
99. Q8×D9C2×D3direct (N × H)
110. C2×Dic18C2×D3semidirect
112. C2×C3⋊D9C2×Q8semidirect
114. C2×D3×C9C2×Q8semidirect
116. C2×C3×D9C2×Q8semidirect
118. D3×D9C2×Q8direct (N × H)
135. C2×D3×Q8D9direct (N × H)
168. Q8×C9C22×D3semidirect
185. Dic18C22×D3semidirect
187. C2×C3×C9C22×Q8semidirect
189. C3⋊D9C22×Q8semidirect
190. D3×C9C22×Q8semidirect
191. C3×D9C22×Q8semidirect
199. C2×C3×Q8C2×D9semidirect
213. D3×Q8C2×D9direct (N × H)
233. C2×Dic6C2×D9semidirect
255. C2×D9D3×Q8direct (N × H)
271. C3×C9C23×Q8semidirect
286. C3×Q8C22×D9semidirect
317. Dic6C22×D9semidirect
326. C2×C9C2×D3×Q8semidirect
330. D9C2×D3×Q8direct (N × H)
342. C2×Q8D3×D9direct (N × H)
367. C2×D3Q8×D9direct (N × H)
381. C9C22×D3×Q8semidirect
391. Q8C2×D3×D9direct (N × H)
406. C2×C3C2×Q8×D9semidirect
414. D3C2×Q8×D9direct (N × H)
428. C3C22×Q8×D9semidirect
433. C2D3×Q8×D9direct (N × H)

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×D3×Q8×C92
C2×C3×Q8×D92
D3×Q8×D92
C2×C4×D3×D92
C2×D3×Dic182
C2×(C3⋊D9)⋊Q82
C2×D9×Dic62
C2×D32×Q83
C22×Q8×D93