Finite Group Databasebeta

← C2×#10596106 #11903227

Subgroups — C2×#10596106 #11903227

Order 1152.

3600 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#64155C211yesmaximal, characteristic, complemented
3C2×Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C2×C4×#638C211yesmaximal, characteristic
5C22×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
6#10596106C2116yesmaximal, direct product factor
7C2×#64106C212yesmaximal, complemented
8C2×C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
9C2×#64098C212yesmaximal, complemented
10C2×#64097C211yesmaximal, characteristic, complemented
11C2×#64090C212yesmaximal
12C2×#64087C212yesmaximal, complemented
13C2×#5635—34nomaximal
14C22×C32×Q8C2211yescharacteristic
15C23×C32×C4C2211yescharacteristic
16C22×#638C2211yescharacteristic
17C23×C3⋊Dic3C2212yes—
18C22×C4×C3⋊D3C2211yescharacteristic
19C22×C4×C3⋊D3C2212yes—
20C22×#634C2212yes—
21C32×C22⋊Q8C2218yescomplemented
22C2×C32×C4⋊C4C2211yescharacteristic
23C2×C32×C4⋊C4C2212yes—
24C2×C32×C22⋊C4C2212yes—
25C2×#3757C2211yescharacteristic
26#64155C2218yescomplemented
27Q8×C3⋊Dic3C2218yescomplemented
28C4×#638C2218yes—
29C2×#3752C2212yes—
30C2×#3752C2211yescharacteristic
31C2×C4⋊(C3⋊Dic3)C2212yes—
32C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
33C2×C4⋊(C3⋊Dic3)C2212yes—
34C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
35C2×C4⋊(C3⋊Dic3)C2218yescomplemented
36C2×#3750C2212yes—
37C2×#3750C2212yes—
38C2×#3750C2212yes—
39C2×#3750C2211yescharacteristic
40C2×C4×C3⋊Dic3C2212yes—
41C2×C4×C3⋊Dic3C2211yescharacteristic
42#64106C22116yescomplemented
43C4⋊C4×C3⋊D3C2218yescomplemented
44#64098C22116yescomplemented
45#64097C2218yescomplemented
46#64090C22116yes—
47#64087C22116yescomplemented
48C2×C3×C22⋊Q8D314yescomplemented
49C2×#946—34no—
50C2×Q8×Dic3—34no—
51C2×C4×#67—34no—
52C22×C4⋊Dic3—34no—
53#5635—364no—
54C2×#897—38no—
55C2×D3×C4⋊C4—34no—
56C2×#889—38no—
57C2×#888—34no—
58C2×#67⋊C4—38no—
59C2×#878—38no—
60C24×C32C2311yescharacteristic
61C23×C3⋊D3C2311yescharacteristic
62C2×C32×Q8—24no—
63C2×C32×Q8C2314yes—
64C22×C32×C4—24no—
65C22×C32×C4C2312yes—
66C22×C32×C4C2311yescharacteristic
67C22×C32×C4C2314yes—
68C22×C32×C4C2312yes—
69C22×C32×C4C2311yescharacteristic
70C2×#638Q818yescomplemented
71C2×#638C2314yes—
72C22×C3⋊Dic3—44no—
73C22×C3⋊Dic3C2311yescharacteristic
74C22×C3⋊Dic3C2312yes—
75C22×C3⋊Dic3C2312yes—
76C22×C3⋊Dic3C2311yescharacteristic
77C22×C3⋊Dic3C2318yes—
78C22×C3⋊Dic3C2311yescharacteristic
79C2×C4×C3⋊D3—24no—
80C2×C4×C3⋊D3—44no—
81C2×C4×C3⋊D3C2314yes—
82C2×C4×C3⋊D3C2318yes—
83C2×#634—44no—
84C2×#634C2318yes—
85C32×C4⋊C4C2314yes—
86C32×C4⋊C4C2318yes—
87C32×C22⋊C4C2318yes—
88#3757C2314yes—
89#3752C2318yes—
90#3752C2314yes—
91C4⋊(C3⋊Dic3)C2318yes—
92C4⋊(C3⋊Dic3)C2314yes—
93C4⋊(C3⋊Dic3)C2318yes—
94C4⋊(C3⋊Dic3)C2314yes—
95#3750C2318yes—
96#3750C2318yes—
97#3750C2318yes—
98#3750C2314yes—
99C4×C3⋊Dic3C2318yes—
100C4×C3⋊Dic3C2314yes—
101C2×#554—91noSylow
102C22×C3×Q8C2×D314yes—
103C23×C3×C4C2×D314yes—
104C22×#67—34no—
105C23×Dic3—38no—
106C22×C4×D3—34no—
107C22×C4×D3—38no—
108C22×Dic6—38no—
109C3×C22⋊Q8C2×D3132yescomplemented
110C2×C3×C4⋊C4C2×D314yes—
111C2×C3×C4⋊C4C2×D318yes—
112C2×C3×C22⋊C4C2×D318yes—
113C2×#217—34no—
114#946—332no—
115Q8×Dic3—332no—
116C4×#67—332no—
117C2×(C2×D3)⋊C4—38no—
118C2×(C2×D3)⋊C4—34no—
119C2×C4⋊Dic3—332no—
120C2×C4⋊Dic3—38no—
121C2×C4⋊Dic3—34no—
122C2×C4⋊Dic3—38no—
123C2×C4⋊Dic3—34no—
124C2×Dic3⋊C4—38no—
125C2×Dic3⋊C4—38no—
126C2×Dic3⋊C4—38no—
127C2×Dic3⋊C4—34no—
128C2×C4×Dic3—38no—
129C2×C4×Dic3—34no—
130#897—364no—
131D3×C4⋊C4—332no—
132#889—364no—
133#888—332no—
134#67⋊C4—364no—
135#878—364no—
136C23×C32—22no—
137C23×C32—22no—
138C23×C32C2×Q814yes—
139C23×C32C2412yes—
140C23×C32C2411yescharacteristic
141C22×C3⋊D3—22no—
142C22×C3⋊D3—22no—
143C22×C3⋊D3C2×Q814yes—
144C22×C3⋊D3C2412yes—
145C32×Q8—28no—
146C2×C32×C4—22no—
147C2×C32×C4—24no—
148C2×C32×C4—24no—
149C2×C32×C4—28no—
150C2×C32×C4#4114yes—
151C2×C32×C4C2414yes—
152C2×C32×C4C2414yes—
153C2×C32×C4C2412yes—
154C2×C32×C4C2412yes—
155#638C2×Q8116yescomplemented
156C2×C3⋊Dic3—22no—
157C2×C3⋊Dic3—22no—
158C2×C3⋊Dic3—24no—
159C2×C3⋊Dic3—24no—
160C2×C3⋊Dic3—48no—
161C2×C3⋊Dic3C2×Q814yes—
162C2×C3⋊Dic3C2414yes—
163C2×C3⋊Dic3C2412yes—
164C2×C3⋊Dic3C2412yes—
165C2×C3⋊Dic3C2414yes—
166C2×C3⋊Dic3C2412yes—
167C4×C3⋊D3—28no—
168C4×C3⋊D3—48no—
169#634—48no—
170#554—916no—
171C2×C4⋊Q8—91no—
172C2×#125—92no—
173C2×C22⋊Q8—91no—
174C2×C22⋊Q8—92no—
175C2×C22⋊Q8—92no—
176C2×C22⋊Q8C3⋊D311yescharacteristic, complemented
177C2×C4×Q8—91no—
178C2×C4×D4—91no—
179C2×C4×D4—92no—
180C22×C4⋊C4—91no—
181C22×C4⋊C4—91no—
182C24×C3C22×D314yes—
183C23×D3—34no—
184C2×C3×Q8—216no—
185C2×C3×Q8C22×D3116yes—
186C22×C3×C4—216no—
187C22×C3×C4C22×D318yes—
188C22×C3×C4C22×D318yes—
189C22×C3×C4C22×D314yes—
190C22×C3×C4C22×D3116yes—
191C22×C3×C4C22×D314yes—
192C2×#67—332no—
193C2×#67—316no—
194C22×Dic3—34no—
195C22×Dic3—38no—
196C22×Dic3—34no—
197C22×Dic3—34no—
198C22×Dic3—38no—
199C22×Dic3—332no—
200C22×Dic3—1216no—
201C2×C4×D3—316no—
202C2×C4×D3—332no—
203C2×C4×D3—616no—
204C2×C4×D3—1216no—
205C2×Dic6—332no—
206C2×Dic6—1216no—
207C3×C4⋊C4C22×D3116yes—
208C3×C4⋊C4C22×D3132yes—
209C3×C22⋊C4C22×D3132yes—
210#217—316no—
211(C2×D3)⋊C4—332no—
212(C2×D3)⋊C4—316no—
213C4⋊Dic3—332no—
214C4⋊Dic3—316no—
215C4⋊Dic3—332no—
216C4⋊Dic3—316no—
217Dic3⋊C4—332no—
218Dic3⋊C4—332no—
219Dic3⋊C4—332no—
220Dic3⋊C4—316no—
221C4×Dic3—332no—
222C4×Dic3—316no—
223C22×C32—28no—
224C22×C32—22no—
225C22×C32—22no—
226C22×C32C22×Q814yes—
227C22×C32C22×Q812yes—
228C22×C32C2×#4112yes—
229C22×C32H(5,𝔽2)12yes—
230C22×C32C2511yescharacteristic
231C2×C3⋊D3—28no—
232C2×C3⋊D3—22no—
233C2×C3⋊D3—22no—
234C2×C3⋊D3C22×Q814yes—
235C32×C4—22no—
236C32×C4—24no—
237C32×C4—24no—
238C32×C4C2×#4114yes—
239C3⋊Dic3—22no—
240C3⋊Dic3—22no—
241C3⋊Dic3—24no—
242C3⋊Dic3—24no—
243C3⋊Dic3C22×Q814yes—
244C22×Q8—92no—
245C22×Q8C2×C3⋊D311yescharacteristic
246C22×D4—91no—
247C23×C4—92no—
248C23×C4—91no—
249C23×C4—92no—
250C23×C4C2×C3⋊D311yescharacteristic
251C4⋊Q8—98no—
252#125—916no—
253C22⋊Q8—98no—
254C22⋊Q8—916no—
255C22⋊Q8—916no—
256C22⋊Q8C2×C3⋊D318yescomplemented
257C4×Q8—98no—
258C4×D4—98no—
259C4×D4—916no—
260C2×C4⋊C4—91no—
261C2×C4⋊C4—92no—
262C2×C4⋊C4—92no—
263C2×C4⋊C4—91no—
264C2×C4⋊C4—98no—
265C2×C4⋊C4—91no—
266C2×C4⋊C4—92no—
267C2×C4⋊C4—92no—
268C2×C4⋊C4—92no—
269C2×C4⋊C4—98no—
270C2×C4⋊C4C2×C3⋊D311yescharacteristic
271C2×C4⋊C4C2×C3⋊D312yes—
272C2×C22⋊C4—91no—
273C2×C22⋊C4—91no—
274C2×C22⋊C4—92no—
275C2×C22⋊C4C2×C3⋊D312yes—
276C2×C42—92no—
277C2×C42—91no—
278C23×C3—28no—
279C23×C3—28no—
280C23×C3D3×Q8116yes—
281C23×C3C23×D318yes—
282C23×C3C23×D314yes—
283C22×D3—316no—
284C22×D3—38no—
285C22×D3—68no—
286C22×D3—68no—
287C3×Q8—232no—
288C2×C3×C4—28no—
289C2×C3×C4—216no—
290C2×C3×C4—216no—
291C2×C3×C4—232no—
292C2×C3×C4#237116yes—
293C2×C3×C4C23×D3116yes—
294C2×C3×C4C23×D3116yes—
295C2×C3×C4C23×D318yes—
296C2×C3×C4C23×D318yes—
297#67—364no—
298C2×Dic3—316no—
299C2×Dic3—38no—
300C2×Dic3—316no—
301C2×Dic3—38no—
302C2×Dic3—38no—
303C2×Dic3—316no—
304C2×Dic3—68no—
305C2×Dic3—68no—
306C2×Dic3—616no—
307C2×Dic3—616no—
308C2×Dic3—1232no—
309C4×D3—632no—
310C4×D3—1232no—
311Dic6—1232no—
312C2×C32—24no—
313C2×C32#55414yescomplemented
314C2×C32C23×Q811yescharacteristic
315C2×C32C22×#4111yescharacteristic
316C2×C32C2×H(5,𝔽2)11yescharacteristic
317C3⋊D3—24no—
318C24—91no—
319C24C22×C3⋊D311yescharacteristic
320C2×Q8—24no—
321C2×Q8—98no—
322C2×Q8—364no—
323C2×Q8C22×C3⋊D314yes—
324C2×D4—94no—
325C2×D4—98no—
326C22×C4—24no—
327C22×C4—91no—
328C22×C4—91no—
329C22×C4—92no—
330C22×C4—91no—
331C22×C4—92no—
332C22×C4—98no—
333C22×C4—94no—
334C22×C4—98no—
335C22×C4—184no—
336C22×C4—364no—
337C22×C4—364no—
338C22×C4C22×C3⋊D314yes—
339C22×C4C22×C3⋊D312yes—
340C22×C4C22×C3⋊D311yescharacteristic
341C22×C4C22×C3⋊D312yes—
342C22×C4C22×C3⋊D311yescharacteristic
343C4⋊C4—94no—
344C4⋊C4—98no—
345C4⋊C4—98no—
346C4⋊C4—94no—
347C4⋊C4—94no—
348C4⋊C4—98no—
349C4⋊C4—98no—
350C4⋊C4—98no—
351C4⋊C4C22×C3⋊D314yes—
352C4⋊C4C22×C3⋊D318yes—
353C22⋊C4—94no—
354C22⋊C4—94no—
355C22⋊C4—98no—
356C22⋊C4C22×C3⋊D318yes—
357C42—98no—
358C42—94no—
359C22×C3—232no—
360C22×C3—28no—
361C22×C3—28no—
362C22×C3C2×#23718yes—
363C22×C3C2×D3×Q8116yes—
364C22×C3C2×D3×Q818yes—
365C22×C3#100918yes—
366C22×C3C24×D314yes—
367C2×D3—316no—
368C2×D3—632no—
369C2×D3—68no—
370C2×D3—68no—
371C3×C4—28no—
372C3×C4—216no—
373C3×C4—216no—
374C3×C4C2×#237116yes—
375Dic3—316no—
376Dic3—68no—
377Dic3—68no—
378Dic3—616no—
379Dic3—616no—
380C32C2×#55411yescharacteristic, complemented, Sylow
381C23—22no—
382C23—22no—
383C23—94no—
384C23—92no—
385C23—182no—
386C23—182no—
387C23Q8×C3⋊D314yes—
388C23C23×C3⋊D312yes—
389C23C23×C3⋊D311yescharacteristic
390Q8—28no—
391Q8—368no—
392D4—916no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—28no—
397C2×C4—94no—
398C2×C4—92no—
399C2×C4—92no—
400C2×C4—94no—
401C2×C4—92no—
402C2×C4—94no—
403C2×C4—182no—
404C2×C4—184no—
405C2×C4—184no—
406C2×C4—182no—
407C2×C4—188no—
408C2×C4—368no—
409C2×C4—368no—
410C2×C4#383014yes—
411C2×C4C23×C3⋊D314yes—
412C2×C4C23×C3⋊D312yes—
413C2×C4C23×C3⋊D314yes—
414C2×C4C23×C3⋊D312yes—
415C2×C3—216no—
416C2×C3#5635116yescomplemented
417C2×C3C22×#23714yes—
418C2×C3C22×D3×Q814yes—
419C2×C3C2×#100914yes—
420D3—616no—
421C22—28no—
422C22—22no—
423C22—22no—
424C22—94no—
425C22—188no—
426C22—182no—
427C22—182no—
428C22C2×#383012yes—
429C22C2×Q8×C3⋊D314yes—
430C22C2×Q8×C3⋊D312yes—
431C22#6436612yes—
432C22C24×C3⋊D311yescharacteristic
433C4—22no—
434C4—24no—
435C4—24no—
436C4—94no—
437C4—182no—
438C4—184no—
439C4—184no—
440C4—182no—
441C4C2×#383014yes—
442C3C2×#563514yescomplemented
443C2—24no—
444C2—184no—
445C2#1059610614yesdirect product factor
446C2C22×#383011yescharacteristic
447C2C22×Q8×C3⋊D311yescharacteristic
448C2C2×#6436611yescharacteristic
subgroupG/H
centerC23C23×C3⋊D3
commutator, cosocleC22×C32C25
FrattiniC22C24×C3⋊D3
FittingC2×C32×C22⋊Q8C2
socleC23×C32C24
prime radicalC22×#638C22
hypercenterC2×C22⋊Q8C3⋊D3
hypocenterC32C2×#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C32×C22⋊Q8C2semidirect
2. C2×#64155C2semidirect
3. C2×Q8×C3⋊Dic3C2semidirect
5. C22×C4⋊(C3⋊Dic3)C2semidirect
6. #10596106C2direct (N × H)
7. C2×#64106C2semidirect
8. C2×C4⋊C4×C3⋊D3C2semidirect
9. C2×#64098C2semidirect
10. C2×#64097C2semidirect
12. C2×#64087C2semidirect
21. C32×C22⋊Q8C22semidirect
26. #64155C22semidirect
27. Q8×C3⋊Dic3C22semidirect
35. C2×C4⋊(C3⋊Dic3)C22semidirect
42. #64106C22semidirect
43. C4⋊C4×C3⋊D3C22semidirect
44. #64098C22semidirect
45. #64097C22semidirect
47. #64087C22semidirect
48. C2×C3×C22⋊Q8D3semidirect
70. C2×#638Q8semidirect
109. C3×C22⋊Q8C2×D3semidirect
155. #638C2×Q8semidirect
176. C2×C22⋊Q8C3⋊D3semidirect
256. C22⋊Q8C2×C3⋊D3semidirect
313. C2×C32#554semidirect
380. C32C2×#554semidirect
416. C2×C3#5635semidirect
442. C3C2×#5635semidirect
445. C2#10596106direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C22⋊Q82
C2×#641552
C2×Q8×C3⋊Dic32
C2×C4×#6382
C22×C4⋊(C3⋊Dic3)2
#105961062
C2×#641062
C2×C4⋊C4×C3⋊D32
C2×#640982
C2×#640972
C2×#640902
C2×#640872
C2×#56353