Finite Group Databasebeta

← C2×Q8×#626 #11902680

Subgroups — C2×Q8×#626 #11902680

Order 1152.

3561 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2C22×C3×D3×Q8C211yesmaximal, characteristic, complemented
3C2×C3×Q8×Dic3C211yesmaximal, characteristic, complemented
4Q8×#626C2116yesmaximal, direct product factor
5C2×C4×#626C213yesmaximal
6C2×#63871C213yesmaximal, complemented
7C2×#63866C213yesmaximal, complemented
8C2×#63857C213yesmaximal, complemented
9C2×Q8×#67—31nomaximal
10C2×Q8×D12—31nomaximal
11C22×C32×Q8C2211yescharacteristic
12C2×Q8×C3⋊D3—28no—
13C2×Q8×C3⋊D3C2218yescomplemented
14C22×C4×C3⋊D3C2213yes—
15C22×#634C2213yes—
16C2×C3×D3×Q8—28no—
17C2×C3×D3×Q8C2218yescomplemented
18C22×C3×C4×D3C2213yes—
19C22×C3×Dic6C2213yes—
20C22×#626C2211yescharacteristic
21C3×Q8×Dic3C2218yescomplemented
22C2×C3×C4⋊Dic3C2213yes—
23C2×C3×C4×Dic3C2213yes—
24C2×Dic3⋊Dic3C2216yes—
25C2×Dic3⋊Dic3C2213yes—
26C2×#3722C2213yes—
27C2×(C2×D3)⋊Dic3C2213yes—
28C4×#626C22124yes—
29#63871C22124yescomplemented
30#63866C22124yescomplemented
31#63857C22124yescomplemented
32C23×C3×Q8—31no—
33C22×D3×Q8—31no—
34C22×D3×Q8—31no—
35C22×D3×Q8—61no—
36C22×D3×Q8D311yescharacteristic, complemented
37C2×C3×C4×Q8—31no—
38Q8×#67—316no—
39C2×#946—33no—
40C2×Q8×Dic3D311yescharacteristic, complemented
41C2×#944—33no—
42C2×C4×#67—33no—
43C2×#924—33no—
44Q8×D12—316no—
45C2×#897—33no—
46C2×#897—33no—
47C2×C4×D12—33no—
48C2×#869—33no—
49C23×C3⋊D3C2311yescharacteristic
50C23×C3×D3C2311yescharacteristic
51C2×C32×Q8D418yescomplemented
52C2×C32×Q8C2314yes—
53C22×C32×C4C2313yes—
54C22×C3⋊Dic3C2313yes—
55Q8×C3⋊D3—232no—
56C2×C4×C3⋊D3—212no—
57C2×C4×C3⋊D3C23112yes—
58C2×#634—212no—
59C2×#634C23112yes—
60C22×C3×Dic3C2313yes—
61C22×C3×Dic3C2313yes—
62C22×C3×Dic3C2311yescharacteristic
63C3×D3×Q8—232no—
64C2×C3×C4×D3—212no—
65C2×C3×C4×D3C23112yes—
66C2×C3×Dic6—212no—
67C2×C3×Dic6C23112yes—
68C2×#626Q818yesdirect product factor
69C2×#626C2314yes—
70C3×C4⋊Dic3C23112yes—
71C3×C4×Dic3C23112yes—
72Dic3⋊Dic3C23124yes—
73Dic3⋊Dic3C23112yes—
74#3722C23112yes—
75(C2×D3)⋊Dic3C23112yes—
76C2×D4×Q8—91noSylow
77C22×C3×Q8—21no—
78C22×C3×Q8—38no—
79C22×C3×Q8—33no—
80C22×C3×Q8—68no—
81C22×C3×Q8C2×D311yescharacteristic
82C22×C3×Q8C2×D311yescharacteristic
83C23×C3×C4—33no—
84C22×#67—31no—
85C2×D3×Q8—28no—
86C2×D3×Q8—38no—
87C2×D3×Q8—38no—
88C2×D3×Q8—68no—
89C2×D3×Q8—616no—
90C2×D3×Q8—68no—
91C2×D3×Q8—68no—
92C2×D3×Q8C2×D318yescomplemented
93C22×D12—31no—
94C22×C4×D3—33no—
95C22×C4×D3—33no—
96C22×C4×D3—63no—
97C22×C4×D3C2×D313yes—
98C22×Dic6—33no—
99C22×Dic6—33no—
100C22×Dic6—63no—
101C22×Dic6C2×D313yes—
102C3×C4×Q8—38no—
103C2×C3×C4⋊C4—33no—
104C2×C3×C42—33no—
105C2×#217—33no—
106#946—324no—
107Q8×Dic3C2×D318yescomplemented
108#944—324no—
109C4×#67—324no—
110C2×(C2×D3)⋊C4—33no—
111C2×(C2×D3)⋊C4—33no—
112C2×(C2×D3)⋊C4—33no—
113C2×C4⋊Dic3—33no—
114C2×C4⋊Dic3—36no—
115C2×C4⋊Dic3C2×D313yes—
116#924—324no—
117C2×Dic3⋊C4—36no—
118C2×Dic3⋊C4—33no—
119C2×C4×Dic3C2×D313yes—
120#897—324no—
121#897—324no—
122C4×D12—324no—
123#869—324no—
124C23×C32C2411yescharacteristic
125C22×C3⋊D3—22no—
126C22×C3⋊D3—22no—
127C22×C3⋊D3C2×Q814yescomplemented
128C22×C3⋊D3C2412yes—
129C22×C3×D3—22no—
130C22×C3×D3—22no—
131C22×C3×D3C2×Q814yescomplemented
132C22×C3×D3C2412yes—
133C32×Q8C2×D4116yescomplemented
134C2×C32×C4C2×D4112yes—
135C2×C32×C4C2416yes—
136C2×C3⋊Dic3—26no—
137C2×C3⋊Dic3C2416yes—
138C4×C3⋊D3—224no—
139#634—224no—
140C2×C3×Dic3—26no—
141C2×C3×Dic3—26no—
142C2×C3×Dic3C2×Q814yescomplemented
143C2×C3×Dic3C2416yes—
144C2×C3×Dic3C2416yes—
145C2×C3×Dic3C2412yes—
146C3×C4×D3—224no—
147C3×Dic6—224no—
148#626C2×Q8116yesdirect product factor
149C23×Q8—31no—
150C23×Q8—91no—
151D4×Q8—916no—
152C2×C4⋊Q8—93no—
153C2×C22⋊Q8—93no—
154C2×C22⋊Q8—93no—
155C2×C4×Q8—31no—
156C2×C4×D4—93no—
157C24×C3—31no—
158C23×D3—31no—
159C23×D3—31no—
160C23×D3—61no—
161C23×D3C22×D311yescharacteristic
162C2×C3×Q8—28no—
163C2×C3×Q8—24no—
164C2×C3×Q8—312no—
165C2×C3×Q8—632no—
166C2×C3×Q8—612no—
167C2×C3×Q8D1218yescomplemented
168C2×C3×Q8#6718yescomplemented
169C2×C3×Q8C22×D314yes—
170C2×C3×Q8C22×D314yes—
171C22×C3×C4—23no—
172C22×C3×C4—31no—
173C22×C3×C4—33no—
174C22×C3×C4—33no—
175C22×C3×C4—312no—
176C22×C3×C4—612no—
177C22×C3×C4C22×D313yes—
178C22×C3×C4C22×D313yes—
179C2×#67—38no—
180C2×#67—34no—
181C22×Dic3—33no—
182C22×Dic3—33no—
183C22×Dic3—63no—
184C22×Dic3C22×D313yes—
185C22×Dic3C22×D313yes—
186C22×Dic3C22×D311yescharacteristic
187D3×Q8—232no—
188D3×Q8—664no—
189D3×Q8—632no—
190D3×Q8—632no—
191C2×D12—38no—
192C2×D12—34no—
193C2×C4×D3—212no—
194C2×C4×D3—312no—
195C2×C4×D3—312no—
196C2×C4×D3—612no—
197C2×C4×D3—624no—
198C2×C4×D3—612no—
199C2×C4×D3—612no—
200C2×C4×D3C22×D3112yes—
201C2×Dic6—212no—
202C2×Dic6—312no—
203C2×Dic6—312no—
204C2×Dic6—612no—
205C2×Dic6—624no—
206C2×Dic6—612no—
207C2×Dic6—612no—
208C2×Dic6C22×D3112yes—
209C3×C4⋊C4—312no—
210C3×C42—312no—
211#217—312no—
212(C2×D3)⋊C4—312no—
213(C2×D3)⋊C4—312no—
214(C2×D3)⋊C4—312no—
215C4⋊Dic3—312no—
216C4⋊Dic3—324no—
217C4⋊Dic3C22×D3112yes—
218Dic3⋊C4—324no—
219Dic3⋊C4—312no—
220C4×Dic3C22×D3112yes—
221C22×C32C22×D412yes—
222C22×C32C22×Q812yes—
223C22×C322−1+412yes—
224C22×C32C2511yescharacteristic
225C2×C3⋊D3—28no—
226C2×C3⋊D3—22no—
227C2×C3⋊D3—22no—
228C2×C3⋊D3C22×Q814yescomplemented
229C2×C3×D3—28no—
230C2×C3×D3—22no—
231C2×C3×D3—22no—
232C2×C3×D3C22×Q814yescomplemented
233C32×C4C22×D4112yes—
234C3⋊Dic3—26no—
235C3×Dic3—26no—
236C3×Dic3—26no—
237C3×Dic3C22×Q814yescomplemented
238C22×Q8—38no—
239C22×Q8—33no—
240C22×Q8—68no—
241C22×Q8—98no—
242C22×Q8—93no—
243C22×Q8—188no—
244C22×Q8D3211yescharacteristic
245C22×D4—91no—
246C23×C4—33no—
247C23×C4—93no—
248C4⋊Q8—924no—
249C22⋊Q8—924no—
250C22⋊Q8—924no—
251C4×Q8—38no—
252C4×D4—924no—
253C2×C4⋊C4—33no—
254C2×C4⋊C4—93no—
255C2×C4⋊C4—96no—
256C2×C22⋊C4—93no—
257C2×C22⋊C4—93no—
258C2×C42—33no—
259C23×C3—21no—
260C23×C3—34no—
261C23×C3—32no—
262C23×C3—62no—
263C23×C3—62no—
264C23×C3C23×D311yescharacteristic
265C23×C3C23×D311yescharacteristic
266C22×D3—22no—
267C22×D3—22no—
268C22×D3—34no—
269C22×D3—34no—
270C22×D3—32no—
271C22×D3—32no—
272C22×D3—64no—
273C22×D3—62no—
274C22×D3—64no—
275C22×D3—64no—
276C22×D3—62no—
277C22×D3—62no—
278C22×D3—62no—
279C22×D3—62no—
280C22×D3D3×Q814yescomplemented
281C22×D3C23×D312yes—
282C3×Q8—216no—
283C3×Q8—624no—
284C3×Q8C2×D12116yescomplemented
285C3×Q8C2×#67116yescomplemented
286C2×C3×C4—26no—
287C2×C3×C4—212no—
288C2×C3×C4—34no—
289C2×C3×C4—32no—
290C2×C3×C4—36no—
291C2×C3×C4—36no—
292C2×C3×C4—66no—
293C2×C3×C4—66no—
294C2×C3×C4—624no—
295C2×C3×C4C2×D12112yes—
296C2×C3×C4C2×#67112yes—
297C2×C3×C4C23×D316yes—
298C2×C3×C4C23×D316yes—
299#67—316no—
300C2×Dic3—26no—
301C2×Dic3—26no—
302C2×Dic3—36no—
303C2×Dic3—36no—
304C2×Dic3—66no—
305C2×Dic3—612no—
306C2×Dic3—66no—
307C2×Dic3—66no—
308C2×Dic3D3×Q814yescomplemented
309C2×Dic3C23×D316yes—
310C2×Dic3C23×D316yes—
311C2×Dic3C23×D312yes—
312D12—316no—
313C4×D3—224no—
314C4×D3—648no—
315C4×D3—624no—
316C4×D3—624no—
317Dic6—224no—
318Dic6—648no—
319Dic6—624no—
320Dic6—624no—
321C2×C32D4×Q814yescomplemented
322C2×C32C23×D411yescharacteristic
323C2×C32C23×Q811yescharacteristic
324C2×C32C2×2−1+411yescharacteristic
325C3⋊D3—24no—
326C3×D3—24no—
327C24—31no—
328C24—91no—
329C2×Q8—312no—
330C2×Q8—632no—
331C2×Q8—612no—
332C2×Q8—912no—
333C2×Q8—1832no—
334C2×Q8—1812no—
335C2×Q8#62618yesdirect product factor
336C2×Q8C2×D3214yes—
337C2×D4—94no—
338C2×D4—98no—
339C22×C4—31no—
340C22×C4—33no—
341C22×C4—33no—
342C22×C4—312no—
343C22×C4—612no—
344C22×C4—93no—
345C22×C4—912no—
346C22×C4—1812no—
347C22×C4C2×D3213yes—
348C4⋊C4—312no—
349C4⋊C4—912no—
350C4⋊C4—924no—
351C22⋊C4—912no—
352C22⋊C4—912no—
353C42—312no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—21no—
358C22×C3—34no—
359C22×C3—62no—
360C22×C3—62no—
361C22×C3—68no—
362C22×C3C22×D1212yes—
363C22×C3C2×D3×Q812yes—
364C22×C3C2×D3×Q812yes—
365C22×C3#100712yes—
366C22×C3#101012yes—
367C22×C3C22×#6712yes—
368C22×C3C24×D311yescharacteristic
369C22×C3C24×D311yescharacteristic
370C2×D3—22no—
371C2×D3—22no—
372C2×D3—28no—
373C2×D3—34no—
374C2×D3—34no—
375C2×D3—64no—
376C2×D3—64no—
377C2×D3—64no—
378C2×D3—616no—
379C2×D3—62no—
380C2×D3—62no—
381C2×D3—62no—
382C2×D3—62no—
383C2×D3—68no—
384C2×D3—68no—
385C2×D3C2×D3×Q814yescomplemented
386C3×C4—212no—
387C3×C4—34no—
388C3×C4—66no—
389C3×C4—66no—
390C3×C4C22×D12112yes—
391C3×C4C22×#67112yes—
392Dic3—26no—
393Dic3—26no—
394Dic3—612no—
395Dic3—66no—
396Dic3—66no—
397Dic3C2×D3×Q814yescomplemented
398C32C2×D4×Q811yescharacteristic, complemented, Sylow
399C23—34no—
400C23—32no—
401C23—62no—
402C23—62no—
403C23—94no—
404C23—92no—
405C23—182no—
406C23—182no—
407C23C22×D3211yescharacteristic
408Q8—624no—
409Q8—1824no—
410Q8C2×#626116yesdirect product factor
411D4—916no—
412C2×C4—34no—
413C2×C4—32no—
414C2×C4—36no—
415C2×C4—36no—
416C2×C4—66no—
417C2×C4—66no—
418C2×C4—624no—
419C2×C4—96no—
420C2×C4—186no—
421C2×C4—1824no—
422C2×C4C2×#626112yes—
423C2×C4C22×D3216yes—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—24no—
427C2×C3—21no—
428C2×C3—64no—
429C2×C3Q8×D1214yescomplemented
430C2×C3Q8×#6714yescomplemented
431C2×C3C23×D1211yescharacteristic
432C2×C3C22×D3×Q811yescharacteristic
433C2×C3C22×D3×Q811yescharacteristic
434C2×C3C2×#100711yescharacteristic
435C2×C3C2×#101011yescharacteristic
436C2×C3C23×#6711yescharacteristic
437D3—24no—
438D3—68no—
439D3—64no—
440D3—64no—
441C22—34no—
442C22—62no—
443C22—68no—
444C22—62no—
445C22—94no—
446C22—182no—
447C22—188no—
448C22—182no—
449C22#6431512yes—
450C22D32×Q812yes—
451C22C22×#62612yes—
452C22C23×D3211yescharacteristic
453C4—34no—
454C4—66no—
455C4—66no—
456C4—186no—
457C4C22×#626112yes—
458C3—21no—
459C3C2×Q8×D1211yescharacteristic, complemented
460C3C2×Q8×#6711yescharacteristic, complemented
461C2—64no—
462C2—184no—
463C2Q8×#62614yesdirect product factor
464C2C2×#6431511yescharacteristic
465C2C2×D32×Q811yescharacteristic
466C2C23×#62611yescharacteristic
subgroupG/H
centerC23C22×D32
commutator, cosocleC22×C32C25
FrattiniC22C23×D32
FittingC22×C32×Q8C22
socleC23×C32C24
prime radicalC22×#626C22
hypercenterC22×Q8D32
hypocenterC32C2×D4×Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×Q8×C3⋊D3C2semidirect
2. C22×C3×D3×Q8C2semidirect
3. C2×C3×Q8×Dic3C2semidirect
4. Q8×#626C2direct (N × H)
6. C2×#63871C2semidirect
7. C2×#63866C2semidirect
8. C2×#63857C2semidirect
13. C2×Q8×C3⋊D3C22semidirect
17. C2×C3×D3×Q8C22semidirect
21. C3×Q8×Dic3C22semidirect
29. #63871C22semidirect
30. #63866C22semidirect
31. #63857C22semidirect
36. C22×D3×Q8D3semidirect
40. C2×Q8×Dic3D3semidirect
51. C2×C32×Q8D4semidirect
68. C2×#626Q8direct (N × H)
92. C2×D3×Q8C2×D3semidirect
107. Q8×Dic3C2×D3semidirect
127. C22×C3⋊D3C2×Q8semidirect
131. C22×C3×D3C2×Q8semidirect
133. C32×Q8C2×D4semidirect
142. C2×C3×Dic3C2×Q8semidirect
148. #626C2×Q8direct (N × H)
167. C2×C3×Q8D12semidirect
168. C2×C3×Q8#67semidirect
228. C2×C3⋊D3C22×Q8semidirect
232. C2×C3×D3C22×Q8semidirect
237. C3×Dic3C22×Q8semidirect
280. C22×D3D3×Q8semidirect
284. C3×Q8C2×D12semidirect
285. C3×Q8C2×#67semidirect
308. C2×Dic3D3×Q8semidirect
321. C2×C32D4×Q8semidirect
335. C2×Q8#626direct (N × H)
385. C2×D3C2×D3×Q8semidirect
397. Dic3C2×D3×Q8semidirect
398. C32C2×D4×Q8semidirect
410. Q8C2×#626direct (N × H)
429. C2×C3Q8×D12semidirect
430. C2×C3Q8×#67semidirect
459. C3C2×Q8×D12semidirect
460. C3C2×Q8×#67semidirect
463. C2Q8×#626direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×C3⋊D32
C22×C3×D3×Q82
C2×C3×Q8×Dic32
Q8×#6262
C2×C4×#6262
C2×#638712
C2×#638662
C2×#638572
C2×Q8×#673
C2×Q8×D123