Finite Group Databasebeta

← C2×D3×Q8×C3⋊D3 #13181625

Subgroups — C2×D3×Q8×C3⋊D3 #13181625

Order 1728.

5724 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⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×C32×D3×Q8C211yesmaximal, characteristic, complemented
4D3×Q8×C3⋊D3C2116yesmaximal, direct product factor
5C2×C4×D3×C3⋊D3C213yesmaximal
6C2×#89019C213yesmaximal, complemented
7C2×C3⋊D3×Dic6C213yesmaximal, complemented
8C2×D3×#634C213yesmaximal, complemented
9C22×Q8×C3⋊D3—31nomaximal
10C2×D32×Q8—34nomaximal
11C22×D3×C3⋊D3C2211yescharacteristic
12C2×C33×Q8C2211yescharacteristic, complemented
13Q8×C3⋊(C3⋊D3)C2218yescomplemented
14C2×C4×C3⋊(C3⋊D3)C2213yes—
15C2×#6308C2213yescomplemented
16C3×Q8×C3⋊D3C2218yescomplemented
17C2×C3×C4×C3⋊D3C2213yes—
18C2×C3×#634C2213yescomplemented
19C32×D3×Q8C2218yescomplemented
20C2×C32×C4×D3C2213yes—
21C2×C32×Dic6C2213yescomplemented
22C2×#6293C2216yescomplemented
23C2×#6289C2213yes—
24C2×Dic3×C3⋊D3C2213yes—
25C2×D3×C3⋊Dic3C2213yes—
26C4×D3×C3⋊D3C22124yes—
27#89019C22124yescomplemented
28C3⋊D3×Dic6C22124yescomplemented
29D3×#634C22124yescomplemented
30C22×C32×Q8—31no—
31C2×Q8×C3⋊D3—31no—
32C2×Q8×C3⋊D3—316no—
33C2×Q8×C3⋊D3—33no—
34C2×Q8×C3⋊D3—33no—
35C2×Q8×C3⋊D3—34no—
36C2×Q8×C3⋊D3—64no—
37C2×Q8×C3⋊D3D311yescharacteristic, direct product factor
38C22×C4×C3⋊D3—33no—
39C22×#634—33no—
40C2×C3×D3×Q8—34no—
41C2×C3×D3×Q8D314yescomplemented
42D32×Q8—364no—
43C2×C4×D32—312no—
44C2×#3797—312no—
45C2×D3×Dic6—312no—
46C2×D3×Dic6—312no—
47C22×C3⋊(C3⋊D3)C2311yescharacteristic
48C22×C3×C3⋊D3C2311yescharacteristic
49C22×C32×D3C2311yescharacteristic
50C2×D3×C3⋊D3Q818yesdirect product factor
51C2×D3×C3⋊D3C2314yes—
52C33×Q8C2314yescomplemented
53C2×C33×C4C2313yes—
54C2×C3⋊(C3⋊Dic3)C2313yes—
55C4×C3⋊(C3⋊D3)C23112yes—
56#6308C23112yescomplemented
57C2×C3×C3⋊Dic3C2313yes—
58C3×C4×C3⋊D3C23112yes—
59C3×#634C23112yescomplemented
60C2×C32×Dic3C2313yes—
61C32×C4×D3C23112yes—
62C32×Dic6C23112yescomplemented
63#6293C23124yescomplemented
64#6289C23112yes—
65Dic3×C3⋊D3C23112yes—
66D3×C3⋊Dic3C23112yes—
67C22×D3×Q8—94no—
68C22×D3×Q8—91no—
69C23×C3⋊D3—31no—
70C22×D32—34no—
71C2×C32×Q8—24no—
72C2×C32×Q8—38no—
73C2×C32×Q8—33no—
74C2×C32×Q8C2×D314yescomplemented
75C2×C32×Q8C2×D311yescharacteristic, complemented
76C22×C32×C4—33no—
77C22×C3⋊Dic3—33no—
78Q8×C3⋊D3—38no—
79Q8×C3⋊D3—332no—
80Q8×C3⋊D3—324no—
81Q8×C3⋊D3—324no—
82Q8×C3⋊D3—632no—
83Q8×C3⋊D3C2×D318yesdirect product factor
84C2×C4×C3⋊D3—33no—
85C2×C4×C3⋊D3—33no—
86C2×C4×C3⋊D3—312no—
87C2×C4×C3⋊D3—33no—
88C2×C4×C3⋊D3—324no—
89C2×C4×C3⋊D3—612no—
90C2×C4×C3⋊D3C2×D313yes—
91C2×#634—33no—
92C2×#634—312no—
93C2×#634—36no—
94C2×#634—324no—
95C2×#634—612no—
96C2×#634C2×D313yescomplemented
97C3×D3×Q8—332no—
98C3×D3×Q8C2×D3132yescomplemented
99C2×C3×C4×D3—312no—
100C2×C3×C4×D3C2×D3112yes—
101C2×C3×Dic6—312no—
102C2×C3×Dic6C2×D3112yescomplemented
103C2×#627—324no—
104C2×#624—312no—
105C2×D3×Dic3—312no—
106C2×D3×Dic3—312no—
107C4×D32—396no—
108#3797—396no—
109D3×Dic6—396no—
110D3×Dic6—396no—
111C22×C33C2411yescharacteristic
112C2×C3⋊(C3⋊D3)C2×Q814yescomplemented
113C2×C3⋊(C3⋊D3)C2412yes—
114C2×C3×C3⋊D3C2×Q814yescomplemented
115C2×C3×C3⋊D3C2412yes—
116C2×C32×D3C2×Q814yescomplemented
117C2×C32×D3C2412yes—
118D3×C3⋊D3C2×Q8116yesdirect product factor
119C33×C4C2416yes—
120C3⋊(C3⋊Dic3)C2416yes—
121C3×C3⋊Dic3C2416yes—
122C32×Dic3C2416yes—
123C22×C3×Q8—34no—
124C22×C3×Q8—91no—
125C2×D3×Q8—34no—
126C2×D3×Q8—912no—
127C2×D3×Q8—912no—
128C2×D3×Q8—964no—
129C2×D3×Q8—94no—
130C2×D3×Q8—91no—
131C2×D3×Q8—916no—
132C2×D3×Q8—93no—
133C2×D3×Q8—93no—
134C2×D3×Q8—184no—
135C2×D3×Q8C3⋊D311yescharacteristic, direct product factor
136C22×C4×D3—912no—
137C22×C4×D3—93no—
138C22×Dic6—912no—
139C22×Dic6—93no—
140C23×C32—31no—
141C22×C3⋊D3—31no—
142C22×C3⋊D3—38no—
143C22×C3⋊D3—34no—
144C22×C3⋊D3—34no—
145C22×C3⋊D3—64no—
146C22×C3⋊D3C22×D311yescharacteristic
147C22×C3×D3—34no—
148C22×C3×D3C22×D314yes—
149C2×D32—332no—
150C2×D32—316no—
151C32×Q8—216no—
152C32×Q8—312no—
153C32×Q8C22×D3116yescomplemented
154C32×Q8C22×D314yescomplemented
155C2×C32×C4—212no—
156C2×C32×C4—33no—
157C2×C32×C4—312no—
158C2×C32×C4C22×D3112yes—
159C2×C32×C4C22×D313yes—
160C2×C3⋊Dic3—33no—
161C2×C3⋊Dic3—312no—
162C2×C3⋊Dic3—312no—
163C2×C3⋊Dic3—612no—
164C2×C3⋊Dic3C22×D313yes—
165C4×C3⋊D3—312no—
166C4×C3⋊D3—312no—
167C4×C3⋊D3—312no—
168C4×C3⋊D3—348no—
169C4×C3⋊D3—648no—
170C4×C3⋊D3C22×D3112yes—
171#634—312no—
172#634—324no—
173#634—348no—
174#634—648no—
175#634C22×D3112yescomplemented
176C2×C3×Dic3—312no—
177C2×C3×Dic3C22×D3112yes—
178C3×C4×D3—348no—
179C3×C4×D3C22×D3148yes—
180C3×Dic6—348no—
181C3×Dic6C22×D3148yescomplemented
182#627—396no—
183#624—348no—
184D3×Dic3—348no—
185D3×Dic3—348no—
186C23×Q8—271noSylow
187C2×C33C22×Q812yescomplemented
188C2×C33C2511yescharacteristic
189C3⋊(C3⋊D3)C22×Q814yescomplemented
190C3×C3⋊D3C22×Q814yescomplemented
191C32×D3C22×Q814yescomplemented
192C23×D3—94no—
193C23×D3—91no—
194C2×C3×Q8—24no—
195C2×C3×Q8—332no—
196C2×C3×Q8—312no—
197C2×C3×Q8—98no—
198C2×C3×Q8—93no—
199C2×C3×Q8D3214yescomplemented
200C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
201C22×C3×C4—312no—
202C22×C3×C4—93no—
203C22×Dic3—912no—
204C22×Dic3—93no—
205D3×Q8—332no—
206D3×Q8—996no—
207D3×Q8—996no—
208D3×Q8—932no—
209D3×Q8—98no—
210D3×Q8—924no—
211D3×Q8—924no—
212D3×Q8—1832no—
213D3×Q8C2×C3⋊D318yesdirect product factor
214C2×C4×D3—312no—
215C2×C4×D3—996no—
216C2×C4×D3—912no—
217C2×C4×D3—912no—
218C2×C4×D3—93no—
219C2×C4×D3—912no—
220C2×C4×D3—93no—
221C2×C4×D3—93no—
222C2×C4×D3—924no—
223C2×C4×D3—1812no—
224C2×C4×D3C2×C3⋊D313yes—
225C2×Dic6—312no—
226C2×Dic6—996no—
227C2×Dic6—924no—
228C2×Dic6—912no—
229C2×Dic6—93no—
230C2×Dic6—96no—
231C2×Dic6—924no—
232C2×Dic6—1812no—
233C2×Dic6C2×C3⋊D313yescomplemented
234C22×C32—24no—
235C22×C32—34no—
236C22×C32—32no—
237C22×C32C23×D314yes—
238C22×C32C23×D311yescharacteristic
239C2×C3⋊D3—34no—
240C2×C3⋊D3—32no—
241C2×C3⋊D3—316no—
242C2×C3⋊D3—316no—
243C2×C3⋊D3—38no—
244C2×C3⋊D3—616no—
245C2×C3⋊D3—68no—
246C2×C3⋊D3D3×Q814yesdirect product factor
247C2×C3⋊D3C23×D312yes—
248C2×C3×D3—316no—
249C2×C3×D3—38no—
250C2×C3×D3D3×Q8116yescomplemented
251C2×C3×D3C23×D318yes—
252D32—364no—
253C32×C4—224no—
254C32×C4—36no—
255C32×C4C23×D3124yes—
256C32×C4C23×D316yes—
257C3⋊Dic3—36no—
258C3⋊Dic3—324no—
259C3⋊Dic3—624no—
260C3⋊Dic3C23×D316yes—
261C3×Dic3—324no—
262C3×Dic3C23×D3124yes—
263C22×Q8—31no—
264C22×Q8—91no—
265C22×Q8—273no—
266C22×Q8—273no—
267C22×Q8—2716no—
268C22×Q8—273no—
269C22×Q8—271no—
270C23×C4—273no—
271C33C23×Q811yescharacteristic, complemented, Sylow
272C23×C3—34no—
273C23×C3—91no—
274C22×D3—34no—
275C22×D3—916no—
276C22×D3—932no—
277C22×D3—94no—
278C22×D3—91no—
279C22×D3—94no—
280C22×D3—98no—
281C22×D3—184no—
282C22×D3C22×C3⋊D311yescharacteristic
283C3×Q8—216no—
284C3×Q8—348no—
285C3×Q8—912no—
286C3×Q8C2×D32116yescomplemented
287C3×Q8C22×C3⋊D314yescomplemented
288C2×C3×C4—212no—
289C2×C3×C4—312no—
290C2×C3×C4—348no—
291C2×C3×C4—93no—
292C2×C3×C4—912no—
293C2×C3×C4C2×D32112yes—
294C2×C3×C4C22×C3⋊D313yes—
295C2×Dic3—312no—
296C2×Dic3—948no—
297C2×Dic3—912no—
298C2×Dic3—93no—
299C2×Dic3—912no—
300C2×Dic3—1812no—
301C2×Dic3C22×C3⋊D313yes—
302C4×D3—348no—
303C4×D3—948no—
304C4×D3—948no—
305C4×D3—912no—
306C4×D3—948no—
307C4×D3—912no—
308C4×D3—912no—
309C4×D3—1848no—
310C4×D3C22×C3⋊D3112yes—
311Dic6—348no—
312Dic6—996no—
313Dic6—948no—
314Dic6—912no—
315Dic6—924no—
316Dic6—1848no—
317Dic6C22×C3⋊D3112yescomplemented
318C2×C32—28no—
319C2×C32—24no—
320C2×C32—34no—
321C2×C32C2×D3×Q818yescomplemented
322C2×C32C2×D3×Q812yescomplemented
323C2×C32C24×D314yes—
324C2×C32C24×D311yescharacteristic
325C3⋊D3—34no—
326C3⋊D3—316no—
327C3⋊D3—616no—
328C3⋊D3C2×D3×Q814yesdirect product factor
329C3×D3—316no—
330C3×D3C2×D3×Q8116yescomplemented
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×C3⋊D311yescharacteristic, 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—24no—
351C22×C3—316no—
352C22×C3—38no—
353C22×C3—94no—
354C22×C3—92no—
355C22×C3C22×D3214yes—
356C22×C3C23×C3⋊D311yescharacteristic
357C2×D3—316no—
358C2×D3—38no—
359C2×D3—964no—
360C2×D3—916no—
361C2×D3—94no—
362C2×D3—98no—
363C2×D3—92no—
364C2×D3—916no—
365C2×D3—188no—
366C2×D3—1816no—
367C2×D3Q8×C3⋊D314yesdirect product factor
368C2×D3C23×C3⋊D312yes—
369C3×C4—224no—
370C3×C4—324no—
371C3×C4—96no—
372C3×C4C22×D32124yes—
373C3×C4C23×C3⋊D316yes—
374Dic3—324no—
375Dic3—924no—
376Dic3—96no—
377Dic3—1824no—
378Dic3C23×C3⋊D316yes—
379C32—24no—
380C32C22×D3×Q814yescomplemented
381C32C22×D3×Q811yescharacteristic, complemented
382C23—31no—
383C23—91no—
384C23—274no—
385C23—278no—
386C23—271no—
387Q8—312no—
388Q8—912no—
389Q8—2724no—
390Q8—2712no—
391Q8C2×D3×C3⋊D314yesdirect 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×C3⋊D313yes—
402C2×C3—28no—
403C2×C3—24no—
404C2×C3—316no—
405C2×C3—94no—
406C2×C3D32×Q818yescomplemented
407C2×C3C2×Q8×C3⋊D312yescomplemented
408C2×C3C23×D3214yes—
409C2×C3C24×C3⋊D311yescharacteristic
410D3—316no—
411D3—916no—
412D3—94no—
413D3—1816no—
414D3C2×Q8×C3⋊D314yesdirect product factor
415C22—34no—
416C22—32no—
417C22—94no—
418C22—92no—
419C22—2716no—
420C22—274no—
421C22—272no—
422C22C22×D3×C3⋊D311yescharacteristic
423C4—36no—
424C4—96no—
425C4—276no—
426C4C22×D3×C3⋊D316yes—
427C3—24no—
428C3C2×D32×Q814yescomplemented
429C3C22×Q8×C3⋊D311yescharacteristic, complemented
430C2—34no—
431C2—94no—
432C2—274no—
433C2D3×Q8×C3⋊D312yesdirect product factor
434C2C23×D3×C3⋊D311yescharacteristic
subgroupG/H
centerC22C22×D3×C3⋊D3
commutator, cosocleC2×C33C25
FrattiniC2C23×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C24
prime radicalC22×D3×C3⋊D3C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33C23×Q8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×C3⋊(C3⋊D3)C2semidirect
2. C2×C3×Q8×C3⋊D3C2semidirect
3. C2×C32×D3×Q8C2semidirect
4. D3×Q8×C3⋊D3C2direct (N × H)
6. C2×#89019C2semidirect
7. C2×C3⋊D3×Dic6C2semidirect
8. C2×D3×#634C2semidirect
12. C2×C33×Q8C22semidirect
13. Q8×C3⋊(C3⋊D3)C22semidirect
15. C2×#6308C22semidirect
16. C3×Q8×C3⋊D3C22semidirect
18. C2×C3×#634C22semidirect
19. C32×D3×Q8C22semidirect
21. C2×C32×Dic6C22semidirect
22. C2×#6293C22semidirect
27. #89019C22semidirect
28. C3⋊D3×Dic6C22semidirect
29. D3×#634C22semidirect
37. C2×Q8×C3⋊D3D3direct (N × H)
41. C2×C3×D3×Q8D3semidirect
50. C2×D3×C3⋊D3Q8direct (N × H)
52. C33×Q8C23semidirect
56. #6308C23semidirect
59. C3×#634C23semidirect
62. C32×Dic6C23semidirect
63. #6293C23semidirect
74. C2×C32×Q8C2×D3semidirect
75. C2×C32×Q8C2×D3semidirect
83. Q8×C3⋊D3C2×D3direct (N × H)
96. C2×#634C2×D3semidirect
98. C3×D3×Q8C2×D3semidirect
102. C2×C3×Dic6C2×D3semidirect
112. C2×C3⋊(C3⋊D3)C2×Q8semidirect
114. C2×C3×C3⋊D3C2×Q8semidirect
116. C2×C32×D3C2×Q8semidirect
118. D3×C3⋊D3C2×Q8direct (N × H)
135. C2×D3×Q8C3⋊D3direct (N × H)
153. C32×Q8C22×D3semidirect
154. C32×Q8C22×D3semidirect
175. #634C22×D3semidirect
181. C3×Dic6C22×D3semidirect
187. C2×C33C22×Q8semidirect
189. C3⋊(C3⋊D3)C22×Q8semidirect
190. C3×C3⋊D3C22×Q8semidirect
191. C32×D3C22×Q8semidirect
199. C2×C3×Q8D32semidirect
200. C2×C3×Q8C2×C3⋊D3semidirect
213. D3×Q8C2×C3⋊D3direct (N × H)
233. C2×Dic6C2×C3⋊D3semidirect
246. C2×C3⋊D3D3×Q8direct (N × H)
250. C2×C3×D3D3×Q8semidirect
271. C33C23×Q8semidirect
286. C3×Q8C2×D32semidirect
287. C3×Q8C22×C3⋊D3semidirect
317. Dic6C22×C3⋊D3semidirect
321. C2×C32C2×D3×Q8semidirect
322. C2×C32C2×D3×Q8semidirect
328. C3⋊D3C2×D3×Q8direct (N × H)
330. C3×D3C2×D3×Q8semidirect
342. C2×Q8D3×C3⋊D3direct (N × H)
367. C2×D3Q8×C3⋊D3direct (N × H)
380. C32C22×D3×Q8semidirect
381. C32C22×D3×Q8semidirect
391. Q8C2×D3×C3⋊D3direct (N × H)
406. C2×C3D32×Q8semidirect
407. C2×C3C2×Q8×C3⋊D3semidirect
414. D3C2×Q8×C3⋊D3direct (N × H)
428. C3C2×D32×Q8semidirect
429. C3C22×Q8×C3⋊D3semidirect
433. C2D3×Q8×C3⋊D3direct (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⋊(C3⋊D3)2
C2×C3×Q8×C3⋊D32
C2×C32×D3×Q82
D3×Q8×C3⋊D32
C2×C4×D3×C3⋊D32
C2×#890192
C2×C3⋊D3×Dic62
C2×D3×#6342
C22×Q8×C3⋊D33
C2×D32×Q83