Finite Group Databasebeta

← C2×#11722510 #13164750

Subgroups — C2×#11722510 #13164750

Order 1728.

2538 conjugacy classes of subgroups.

The subgroup lattice has 424 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×#3701C211yesmaximal, characteristic, complemented
4#11722510C2116yesmaximal, direct product factor
5C2×#88641C213yesmaximal, complemented
6C2×#88640C213yesmaximal, complemented
7C2×#88634C213yesmaximal, complemented
8C2×D9×Dic6C213yesmaximal, complemented
9C2×#64315—31nomaximal
10C2×#63713—31nomaximal
11C2×C3×Q8×C9C2211yescharacteristic
12Q8×C3⋊D9C2218yescomplemented
13C2×C4×C3⋊D9C2213yes—
14C2×#6226C2213yes—
15D3×Q8×C9C2218yescomplemented
16C3×#3701C2218yescomplemented
17C2×C4×D3×C9C2213yes—
18C2×C3×D36C2213yes—
19C2×C3×C4×D9C2213yes—
20C2×C9×Dic6C2213yes—
21C2×#6195C2211yescharacteristic
22C2×#6194C2213yes—
23C2×#6192C2213yes—
24C2×Dic3×D9C2216yes—
25C2×#6189C2213yes—
26#88641C22124yescomplemented
27#88640C22124yescomplemented
28#88634C22124yescomplemented
29D9×Dic6C22124yescomplemented
30C2×Q8×C3⋊D3—31no—
31C2×C3×#239—31no—
32C2×C3×D3×Q8D311yescharacteristic
33#64315—316no—
34C2×#3798—33no—
35C2×#3796—33no—
36C2×#3795—33no—
37C2×D3×Dic6—33no—
38C22×Q8×C9—31no—
39#63713—316no—
40C2×#3701—33no—
41C2×#3701D311yescharacteristic, complemented
42C2×Q8×D9—31no—
43C2×Q8×D9—33no—
44C2×Q8×D9—61no—
45C2×#3697—33no—
46C2×#3697—33no—
47C22×C3⋊D9C2311yescharacteristic
48C22×D3×C9C2311yescharacteristic
49C22×C3×D9C2313yes—
50C3×Q8×C9C2314yes—
51C2×C3×C4×C9C2313yes—
52C2×C3⋊Dic9C2313yes—
53C4×C3⋊D9C23112yes—
54#6226C23112yes—
55C2×C9×Dic3C2313yes—
56C2×C3×Dic9C2311yescharacteristic
57C4×D3×C9C23112yes—
58C3×D36C23112yes—
59C3×C4×D9C23112yes—
60C9×Dic6C23112yes—
61#6195C2314yes—
62#6194C23112yes—
63#6192C23112yes—
64Dic3×D9C23124yes—
65#6189C23112yes—
66C2×#1010—91no—
67C2×#1007—91no—
68C2×C32×Q8C2×D311yescharacteristic
69Q8×C3⋊D3—38no—
70C2×C4×C3⋊D3—33no—
71C2×#634—33no—
72C3×#239—38no—
73C3×D3×Q8C2×D318yes—
74C2×C3×D12—33no—
75C2×C3×C4×D3—33no—
76C2×C3×C4×D3C2×D313yes—
77C2×C3×Dic6C2×D313yes—
78C2×#627—33no—
79C2×#626—31no—
80C2×#626—33no—
81C2×#625—33no—
82C2×D3×Dic3—36no—
83#3798—324no—
84#3796—324no—
85#3795—324no—
86D3×Dic6—324no—
87C2×Q8×C9—21no—
88C2×Q8×C9—38no—
89C2×Q8×C9—33no—
90C2×Q8×C9C2×D311yescharacteristic
91C22×C4×C9—33no—
92C2×#611—33no—
93C2×#611—31no—
94#3701—324no—
95#3701C2×D318yescomplemented
96Q8×D9—38no—
97Q8×D9—324no—
98Q8×D9—68no—
99#3697—324no—
100#3697—324no—
101C2×D36—33no—
102C2×D36C2×D313yes—
103C2×C4×D9—36no—
104C2×C4×D9—33no—
105C2×C4×D9—63no—
106C2×C4×D9C2×D313yes—
107C2×Dic18—33no—
108C2×Dic18—33no—
109C2×Dic18—63no—
110C22×C3×C9C2411yescharacteristic
111C2×C3⋊D9—22no—
112C2×C3⋊D9C2412yes—
113C2×D3×C9—22no—
114C2×D3×C9C2412yes—
115C2×C3×D9—26no—
116C2×C3×D9C2416yes—
117C3×C4×C9C2416yes—
118C3⋊Dic9C2416yes—
119C9×Dic3C2416yes—
120C3×Dic9C2412yes—
121C2×C3×#41—91no—
122C22×C3×Q8—31no—
123#1010—916no—
124#1007—916no—
125C2×#239—31no—
126C2×#239—93no—
127C2×D3×Q8—93no—
128C2×D3×Q8—91no—
129C2×D3×Q8—91no—
130C2×D3×Q8—181no—
131C2×D3×Q8D911yescharacteristic, complemented
132C2×#237—93no—
133C2×#237—93no—
134C2×#235—93no—
135C2×#235—93no—
136C2×#235—93no—
137C22×Dic6—93no—
138C22×C3⋊D3—31no—
139C22×C3×D3—33no—
140C22×C3×D3C22×D311yescharacteristic
141C32×Q8C22×D314yes—
142C2×C32×C4C22×D313yes—
143C2×C3⋊Dic3—33no—
144C4×C3⋊D3—312no—
145#634—312no—
146C2×C3×Dic3—31no—
147C2×C3×Dic3C22×D313yes—
148C3×D12—312no—
149C3×C4×D3—312no—
150C3×C4×D3C22×D3112yes—
151C3×Dic6C22×D3112yes—
152#627—312no—
153#626—34no—
154#626—312no—
155#625—312no—
156D3×Dic3—324no—
157C23×C9—31no—
158C22×D9—31no—
159C22×D9—61no—
160C22×D9C22×D313yes—
161Q8×C9—24no—
162Q8×C9—312no—
163Q8×C9C22×D314yes—
164C2×C4×C9—23no—
165C2×C4×C9—33no—
166C2×C4×C9—312no—
167C2×C4×C9C22×D313yes—
168#611—312no—
169#611—34no—
170C2×Dic9—33no—
171C2×Dic9—63no—
172C2×Dic9C22×D311yescharacteristic
173D36—312no—
174D36C22×D3112yes—
175C4×D9—324no—
176C4×D9—312no—
177C4×D9—612no—
178C4×D9C22×D3112yes—
179Dic18—312no—
180Dic18—312no—
181Dic18—612no—
182C2×2−1+4—271noSylow
183C2×C3×C92−1+412yescomplemented
184C2×C3×C9C2511yescharacteristic
185C3⋊D9—22no—
186D3×C9—22no—
187C3×D9—26no—
188C3×#41—98no—
189C2×C3×Q8—21no—
190C2×C3×Q8—38no—
191C2×C3×Q8—33no—
192C2×C3×Q8C2×D911yescharacteristic
193C2×C3×Q8D3211yescharacteristic
194C2×C3×D4—93no—
195C22×C3×C4—33no—
196C22×C3×C4—93no—
197C2×#67—91no—
198C2×#67—93no—
199C2×#67—93no—
200C2×#67—93no—
201C22×Dic3—96no—
202#239—38no—
203#239—924no—
204D3×Q8—924no—
205D3×Q8—98no—
206D3×Q8—98no—
207D3×Q8—188no—
208D3×Q8C2×D918yescomplemented
209#237—924no—
210#237—924no—
211#235—924no—
212#235—924no—
213#235—924no—
214C2×D12—33no—
215C2×D12—93no—
216C2×D12—91no—
217C2×C4×D3—33no—
218C2×C4×D3—96no—
219C2×C4×D3—93no—
220C2×C4×D3—93no—
221C2×C4×D3—183no—
222C2×C4×D3C2×D913yes—
223C2×Dic6—93no—
224C2×Dic6—93no—
225C2×Dic6—93no—
226C2×Dic6—93no—
227C2×Dic6—924no—
228C2×Dic6—183no—
229C2×Dic6C2×D913yes—
230C22×C32C23×D311yescharacteristic
231C2×C3⋊D3—32no—
232C2×C3⋊D3—62no—
233C2×C3×D3—22no—
234C2×C3×D3—36no—
235C2×C3×D3—66no—
236C2×C3×D3C23×D312yes—
237C32×C4C23×D316yes—
238C3⋊Dic3—36no—
239C3×Dic3—32no—
240C3×Dic3C23×D316yes—
241C22×C9—21no—
242C22×C9—32no—
243C22×C9—62no—
244C22×C9C23×D311yescharacteristic
245C2×D9—26no—
246C2×D9—32no—
247C2×D9—62no—
248C2×D9—64no—
249C2×D9—62no—
250C2×D9C23×D316yes—
251C4×C9—26no—
252C4×C9—36no—
253C4×C9C23×D316yes—
254Dic9—36no—
255Dic9—66no—
256Dic9C23×D312yes—
2572−1+4—2716no—
258C2×#41—91no—
259C2×#41—273no—
260C2×#41—273no—
261C2×#41—273no—
262C22×Q8—31no—
263C22×Q8—273no—
264C22×Q8—271no—
265C3×C9C2×2−1+411yescharacteristic, complemented, Sylow
266C23×C3—31no—
267C23×C3—93no—
268C22×D3—33no—
269C22×D3—91no—
270C22×D3—91no—
271C22×D3—181no—
272C22×D3C22×D911yescharacteristic
273C3×Q8—24no—
274C3×Q8—312no—
275C3×Q8C22×D914yes—
276C3×Q8C2×D3214yes—
277C3×D4—912no—
278C2×C3×C4—23no—
279C2×C3×C4—33no—
280C2×C3×C4—312no—
281C2×C3×C4—91no—
282C2×C3×C4—912no—
283C2×C3×C4C22×D913yes—
284C2×C3×C4C2×D3213yes—
285#67—94no—
286#67—912no—
287#67—912no—
288#67—912no—
289C2×Dic3—31no—
290C2×Dic3—93no—
291C2×Dic3—93no—
292C2×Dic3—924no—
293C2×Dic3—183no—
294C2×Dic3C22×D913yes—
295D12—312no—
296D12—912no—
297D12—94no—
298C4×D3—312no—
299C4×D3—924no—
300C4×D3—912no—
301C4×D3—912no—
302C4×D3—1812no—
303C4×D3C22×D9112yes—
304Dic6—912no—
305Dic6—912no—
306Dic6—912no—
307Dic6—912no—
308Dic6—1812no—
309Dic6C22×D9112yes—
310C2×C32#100712yes—
311C2×C32C24×D311yescharacteristic
312C3⋊D3—62no—
313C3×D3—22no—
314C3×D3—66no—
315C2×C9—22no—
316C2×C9—21no—
317C2×C9—62no—
318C2×C9#101012yescomplemented
319C2×C9C24×D311yescharacteristic
320D9—26no—
321D9—64no—
322D9—62no—
323#41—98no—
324#41—2724no—
325#41—2724no—
326#41—2724no—
327C2×Q8—38no—
328C2×Q8—33no—
329C2×Q8—273no—
330C2×Q8—2724no—
331C2×Q8—278no—
332C2×Q8—273no—
333C2×Q8D3×D911yescharacteristic
334C2×D4—93no—
335C2×D4—273no—
336C2×D4—271no—
337C2×D4—273no—
338C22×C4—33no—
339C22×C4—93no—
340C22×C4—276no—
341C22×C4—273no—
342C22×C3—21no—
343C22×C3—32no—
344C22×C3—62no—
345C22×C3—96no—
346C22×C3—186no—
347C22×C3C23×D911yescharacteristic
348C22×C3C22×D3211yescharacteristic
349C2×D3—22no—
350C2×D3—36no—
351C2×D3—66no—
352C2×D3—92no—
353C2×D3—92no—
354C2×D3—182no—
355C2×D3—184no—
356C2×D3—182no—
357C2×D3—182no—
358C2×D3C23×D912yes—
359C3×C4—26no—
360C3×C4—36no—
361C3×C4—92no—
362C3×C4C23×D916yes—
363C3×C4C22×D3216yes—
364Dic3—32no—
365Dic3—96no—
366Dic3—96no—
367Dic3—186no—
368Dic3C23×D916yes—
369C32C2×#100711yescharacteristic
370C9—21no—
371C9C2×#101011yescharacteristic, complemented
372C23—31no—
373C23—93no—
374C23—271no—
375Q8—312no—
376Q8—2712no—
377Q8—2712no—
378Q8C2×D3×D914yes—
379D4—912no—
380D4—2712no—
381D4—274no—
382D4—2712no—
383C2×C4—33no—
384C2×C4—312no—
385C2×C4—91no—
386C2×C4—912no—
387C2×C4—2724no—
388C2×C4—273no—
389C2×C4—2712no—
390C2×C4C2×D3×D913yes—
391C2×C3—22no—
392C2×C3—21no—
393C2×C3—62no—
394C2×C3—186no—
395C2×C3#6371312yescomplemented
396C2×C3C24×D911yescharacteristic
397C2×C3#6431512yes—
398C2×C3C23×D3211yescharacteristic
399D3—22no—
400D3—66no—
401D3—184no—
402D3—182no—
403D3—182no—
404C22—32no—
405C22—62no—
406C22—96no—
407C22—186no—
408C22—272no—
409C22—542no—
410C22C22×D3×D911yescharacteristic
411C4—36no—
412C4—92no—
413C4—276no—
414C4C22×D3×D916yes—
415C3—21no—
416C3C2×#6371311yescharacteristic, complemented
417C3C2×#6431511yescharacteristic
418C2—62no—
419C2—186no—
420C2—542no—
421C2#1172251012yesdirect product factor
422C2C23×D3×D911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC2×C3×C9C25
FrattiniC2×C3C23×D32
FittingC2×C3×Q8×C9C22
socleC22×C32C23×D3
hypercenterC2×Q8D3×D9
hypocenterC3×C9C2×2−1+4
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×#3701C2semidirect
4. #11722510C2direct (N × H)
5. C2×#88641C2semidirect
6. C2×#88640C2semidirect
7. C2×#88634C2semidirect
8. C2×D9×Dic6C2semidirect
12. Q8×C3⋊D9C22semidirect
15. D3×Q8×C9C22semidirect
16. C3×#3701C22semidirect
26. #88641C22semidirect
27. #88640C22semidirect
28. #88634C22semidirect
29. D9×Dic6C22semidirect
41. C2×#3701D3semidirect
95. #3701C2×D3semidirect
131. C2×D3×Q8D9semidirect
183. C2×C3×C92−1+4semidirect
208. D3×Q8C2×D9semidirect
265. C3×C9C2×2−1+4semidirect
318. C2×C9#1010semidirect
371. C9C2×#1010semidirect
395. C2×C3#63713semidirect
416. C3C2×#63713semidirect
421. C2#11722510direct (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×#37012
#117225102
C2×#886412
C2×#886402
C2×#886342
C2×D9×Dic62
C2×#643153
C2×#637133