Finite Group Databasebeta

← C2×#10596200 #11901882

Subgroups — C2×#10596200 #11901882

Order 1152.

3240 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#64155C212yesmaximal
3C2×#64153C212yesmaximal, complemented
4#10596200C2116yesmaximal, direct product factor
5C2×#64105C212yesmaximal
6C2×#64101C211yesmaximal, characteristic
7C2×C4⋊C4×C3⋊D3C211yesmaximal, characteristic
8C2×#64099C211yesmaximal, characteristic, complemented
9C2×#64098C212yesmaximal, complemented
10C2×#64097C211yesmaximal, characteristic, complemented
11C2×#64081C211yesmaximal, characteristic
12C2×#64079C211yesmaximal, characteristic, complemented
13C2×#5729—34nomaximal
14C22×C32×Q8C2212yes—
15C22×C4×C3⋊D3C2212yes—
16C22×C4×C3⋊D3C2211yescharacteristic
17C22×#634C2212yes—
18C32×C4⋊Q8C2218yescomplemented
19C2×C32×C4⋊C4C2211yescharacteristic
20C2×C32×C4⋊C4C2212yes—
21C2×C32×C4⋊C4C2211yescharacteristic
22C2×C32×C42C2211yescharacteristic
23#64155C22116yes—
24#64153C22116yescomplemented
25C2×#3752C2211yescharacteristic
26C2×#3752C2211yescharacteristic
27C2×#3752C2212yes—
28C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
29C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
30C2×C4⋊(C3⋊Dic3)C2212yes—
31C2×#3750C2212yes—
32C2×#3750C2212yes—
33C2×#3750C2211yescharacteristic
34C2×#3750C2211yescharacteristic
35C2×#3750C2212yes—
36C2×#3750C2212yes—
37C2×#3750C2212yes—
38C2×C4×C3⋊Dic3C2211yescharacteristic
39C2×C4×C3⋊Dic3C2212yes—
40#64105C22116yes—
41#64101C2218yes—
42C4⋊C4×C3⋊D3C2218yes—
43#64099C2218yescomplemented
44#64098C22116yescomplemented
45#64097C2218yescomplemented
46#64081C2218yes—
47#64079C2218yescomplemented
48C2×C3×C4⋊Q8D314yescomplemented
49C2×#946—38no—
50C2×#944—38no—
51#5729—364no—
52C2×#896—38no—
53C2×#892—34no—
54C2×D3×C4⋊C4—34no—
55C2×#890—34no—
56C2×#889—38no—
57C2×#888—34no—
58C2×#872—34no—
59C2×#870—34no—
60C23×C3⋊D3C2311yescharacteristic
61C2×C32×Q8—44no—
62C2×C32×Q8C2318yes—
63C22×C32×C4C2312yes—
64C22×C32×C4C2312yes—
65C22×C32×C4C2311yescharacteristic
66C22×C32×C4C2311yescharacteristic
67C22×C32×C4C2311yescharacteristic
68C22×C3⋊Dic3C2311yescharacteristic
69C22×C3⋊Dic3C2312yes—
70C22×C3⋊Dic3C2311yescharacteristic
71C22×C3⋊Dic3C2312yes—
72C22×C3⋊Dic3C2311yescharacteristic
73C2×C4×C3⋊D3—44no—
74C2×C4×C3⋊D3Q818yescomplemented
75C2×C4×C3⋊D3C2318yes—
76C2×C4×C3⋊D3C2314yes—
77C2×#634—44no—
78C2×#634C2318yes—
79C32×C4⋊C4C2314yes—
80C32×C4⋊C4C2318yes—
81C32×C4⋊C4C2314yes—
82C32×C42C2314yes—
83#3752C2314yes—
84#3752C2314yes—
85#3752C2318yes—
86C4⋊(C3⋊Dic3)C2314yes—
87C4⋊(C3⋊Dic3)C2314yes—
88C4⋊(C3⋊Dic3)C2318yes—
89#3750C2318yes—
90#3750C2318yes—
91#3750C2314yes—
92#3750C2314yes—
93#3750C2318yes—
94#3750C2318yes—
95#3750C2318yes—
96C4×C3⋊Dic3C2314yes—
97C4×C3⋊Dic3C2318yes—
98C2×#544—91noSylow
99C22×C3×Q8C2×D318yes—
100C22×C4×D3—34no—
101C22×C4×D3—38no—
102C22×Dic6—38no—
103C3×C4⋊Q8C2×D3132yescomplemented
104C2×C3×C4⋊C4C2×D314yes—
105C2×C3×C4⋊C4C2×D318yes—
106C2×C3×C4⋊C4C2×D314yes—
107C2×C3×C42C2×D314yes—
108#946—364no—
109#944—364no—
110C2×(C2×D3)⋊C4—34no—
111C2×(C2×D3)⋊C4—34no—
112C2×(C2×D3)⋊C4—38no—
113C2×C4⋊Dic3—34no—
114C2×C4⋊Dic3—34no—
115C2×C4⋊Dic3—38no—
116C2×Dic3⋊C4—38no—
117C2×Dic3⋊C4—38no—
118C2×Dic3⋊C4—34no—
119C2×Dic3⋊C4—34no—
120C2×Dic3⋊C4—38no—
121C2×Dic3⋊C4—38no—
122C2×Dic3⋊C4—38no—
123C2×C4×Dic3—34no—
124C2×C4×Dic3—38no—
125#896—364no—
126#892—332no—
127D3×C4⋊C4—332no—
128#890—332no—
129#889—364no—
130#888—332no—
131#872—332no—
132#870—332no—
133C23×C32C2411yescharacteristic
134C22×C3⋊D3—22no—
135C22×C3⋊D3—22no—
136C22×C3⋊D3C2×Q814yes—
137C22×C3⋊D3C2412yes—
138C32×Q8—48no—
139C2×C32×C4—24no—
140C2×C32×C4—24no—
141C2×C32×C4—22no—
142C2×C32×C4—22no—
143C2×C32×C4C2×Q814yes—
144C2×C32×C4C2414yes—
145C2×C32×C4C2412yes—
146C2×C32×C4C2412yes—
147C2×C32×C4C2414yes—
148C2×C32×C4C2412yes—
149C2×C3⋊Dic3—22no—
150C2×C3⋊Dic3—24no—
151C2×C3⋊Dic3—22no—
152C2×C3⋊Dic3—24no—
153C2×C3⋊Dic3C2×Q814yes—
154C2×C3⋊Dic3C2412yes—
155C2×C3⋊Dic3C2414yes—
156C2×C3⋊Dic3C2412yes—
157C2×C3⋊Dic3C2414yes—
158C2×C3⋊Dic3C2412yes—
159C4×C3⋊D3—48no—
160C4×C3⋊D3C2×Q8116yescomplemented
161#634—48no—
162#544—916no—
163C2×C4⋊Q8—92no—
164C2×C4⋊Q8—91no—
165C2×C4⋊Q8C3⋊D311yescharacteristic, complemented
166C2×#125—92no—
167C2×#125—91no—
168C2×#125—91no—
169C2×C22⋊Q8—92no—
170C2×C22⋊Q8—92no—
171C2×#117—91no—
172C2×#117—91no—
173C22×C4⋊C4—91no—
174C23×D3—34no—
175C2×C3×Q8—416no—
176C2×C3×Q8C22×D3132yes—
177C22×C3×C4C22×D318yes—
178C22×C3×C4C22×D314yes—
179C22×C3×C4C22×D314yes—
180C22×C3×C4C22×D318yes—
181C22×C3×C4C22×D314yes—
182C22×Dic3—34no—
183C22×Dic3—34no—
184C22×Dic3—38no—
185C22×Dic3—34no—
186C22×Dic3—38no—
187C2×C4×D3—332no—
188C2×C4×D3—316no—
189C2×C4×D3—332no—
190C2×C4×D3—1216no—
191C2×Dic6—332no—
192C2×Dic6—1216no—
193C3×C4⋊C4C22×D3116yes—
194C3×C4⋊C4C22×D3132yes—
195C3×C4⋊C4C22×D3116yes—
196C3×C42C22×D3116yes—
197(C2×D3)⋊C4—316no—
198(C2×D3)⋊C4—316no—
199(C2×D3)⋊C4—332no—
200C4⋊Dic3—316no—
201C4⋊Dic3—316no—
202C4⋊Dic3—332no—
203Dic3⋊C4—332no—
204Dic3⋊C4—332no—
205Dic3⋊C4—316no—
206Dic3⋊C4—316no—
207Dic3⋊C4—332no—
208Dic3⋊C4—332no—
209Dic3⋊C4—332no—
210C4×Dic3—316no—
211C4×Dic3—332no—
212C22×C32C22×Q812yes—
213C22×C32H(5,𝔽2)12yes—
214C22×C322−1+412yes—
215C22×C32C2511yescharacteristic
216C2×C3⋊D3—28no—
217C2×C3⋊D3—22no—
218C2×C3⋊D3—22no—
219C2×C3⋊D3C22×Q814yes—
220C32×C4—24no—
221C32×C4—24no—
222C32×C4—22no—
223C32×C4—22no—
224C32×C4C22×Q814yes—
225C3⋊Dic3—22no—
226C3⋊Dic3—24no—
227C3⋊Dic3—22no—
228C3⋊Dic3—24no—
229C3⋊Dic3C22×Q814yes—
230C22×Q8—92no—
231C22×Q8C2×C3⋊D312yes—
232C23×C4—91no—
233C23×C4—92no—
234C4⋊Q8—916no—
235C4⋊Q8—98no—
236C4⋊Q8C2×C3⋊D318yescomplemented
237#125—916no—
238#125—98no—
239#125—98no—
240C22⋊Q8—916no—
241C22⋊Q8—916no—
242#117—98no—
243#117—98no—
244C2×C4⋊C4—92no—
245C2×C4⋊C4—91no—
246C2×C4⋊C4—91no—
247C2×C4⋊C4—91no—
248C2×C4⋊C4—91no—
249C2×C4⋊C4—92no—
250C2×C4⋊C4—92no—
251C2×C4⋊C4—92no—
252C2×C4⋊C4—92no—
253C2×C4⋊C4—92no—
254C2×C4⋊C4—98no—
255C2×C4⋊C4C2×C3⋊D311yescharacteristic
256C2×C4⋊C4C2×C3⋊D311yescharacteristic
257C2×C4⋊C4C2×C3⋊D312yes—
258C2×C22⋊C4—92no—
259C2×C22⋊C4—91no—
260C2×C22⋊C4—91no—
261C2×C42—92no—
262C2×C42—91no—
263C2×C42C2×C3⋊D311yescharacteristic
264C23×C3C23×D314yes—
265C22×D3—316no—
266C22×D3—38no—
267C22×D3—68no—
268C22×D3—68no—
269C3×Q8—432no—
270C2×C3×C4—216no—
271C2×C3×C4—216no—
272C2×C3×C4—28no—
273C2×C3×C4—28no—
274C2×C3×C4D3×Q8116yes—
275C2×C3×C4C23×D3116yes—
276C2×C3×C4C23×D318yes—
277C2×C3×C4C23×D318yes—
278C2×C3×C4C23×D3116yes—
279C2×C3×C4C23×D318yes—
280C2×Dic3—316no—
281C2×Dic3—38no—
282C2×Dic3—38no—
283C2×Dic3—316no—
284C2×Dic3—38no—
285C2×Dic3—316no—
286C2×Dic3—68no—
287C2×Dic3—616no—
288C2×Dic3—68no—
289C2×Dic3—616no—
290C4×D3—364no—
291C4×D3—1232no—
292Dic6—1232no—
293C2×C32#54414yescomplemented
294C2×C32C23×Q811yescharacteristic
295C2×C32C2×H(5,𝔽2)11yescharacteristic
296C2×C32C2×2−1+411yescharacteristic
297C3⋊D3—24no—
298C24—91no—
299C2×Q8—44no—
300C2×Q8—98no—
301C2×Q8—364no—
302C2×Q8C22×C3⋊D318yes—
303C22×C4—91no—
304C22×C4—94no—
305C22×C4—98no—
306C22×C4—91no—
307C22×C4—92no—
308C22×C4—91no—
309C22×C4—92no—
310C22×C4—98no—
311C22×C4—364no—
312C22×C4C22×C3⋊D312yes—
313C22×C4C22×C3⋊D311yescharacteristic
314C22×C4C22×C3⋊D311yescharacteristic
315C22×C4C22×C3⋊D312yes—
316C22×C4C22×C3⋊D311yescharacteristic
317C4⋊C4—98no—
318C4⋊C4—94no—
319C4⋊C4—94no—
320C4⋊C4—94no—
321C4⋊C4—94no—
322C4⋊C4—98no—
323C4⋊C4—98no—
324C4⋊C4—98no—
325C4⋊C4—98no—
326C4⋊C4—98no—
327C4⋊C4C22×C3⋊D314yes—
328C4⋊C4C22×C3⋊D314yes—
329C4⋊C4C22×C3⋊D318yes—
330C22⋊C4—98no—
331C22⋊C4—94no—
332C22⋊C4—94no—
333C42—98no—
334C42—94no—
335C42C22×C3⋊D314yes—
336C22×C3#100418yes—
337C22×C3C2×D3×Q818yes—
338C22×C3#100718yes—
339C22×C3C24×D314yes—
340C2×D3—316no—
341C2×D3—68no—
342C2×D3—68no—
343C2×D3—632no—
344C3×C4—216no—
345C3×C4—216no—
346C3×C4—28no—
347C3×C4—28no—
348C3×C4C2×D3×Q8116yes—
349Dic3—316no—
350Dic3—68no—
351Dic3—616no—
352Dic3—68no—
353Dic3—616no—
354C32C2×#54411yescharacteristic, complemented, Sylow
355C23—94no—
356C23—92no—
357C23—182no—
358C23—182no—
359C23C23×C3⋊D311yescharacteristic
360Q8—48no—
361Q8—368no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—94no—
367C2×C4—92no—
368C2×C4—916no—
369C2×C4—92no—
370C2×C4—94no—
371C2×C4—92no—
372C2×C4—94no—
373C2×C4—184no—
374C2×C4—182no—
375C2×C4—182no—
376C2×C4—184no—
377C2×C4—368no—
378C2×C4Q8×C3⋊D314yes—
379C2×C4C23×C3⋊D314yes—
380C2×C4C23×C3⋊D312yes—
381C2×C4C23×C3⋊D312yes—
382C2×C4C23×C3⋊D314yes—
383C2×C4C23×C3⋊D312yes—
384C2×C3#5729116yescomplemented
385C2×C3C2×#100414yes—
386C2×C3C22×D3×Q814yes—
387C2×C3C2×#100714yes—
388D3—616no—
389C22—94no—
390C22—188no—
391C22—182no—
392C22—182no—
393C22#6436112yes—
394C22C2×Q8×C3⋊D312yes—
395C22#6436412yes—
396C22C24×C3⋊D311yescharacteristic
397C4—24no—
398C4—22no—
399C4—24no—
400C4—22no—
401C4—94no—
402C4—184no—
403C4—182no—
404C4—182no—
405C4—184no—
406C4C2×Q8×C3⋊D314yes—
407C3C2×#572914yescomplemented
408C2—184no—
409C2#1059620014yesdirect product factor
410C2C2×#6436111yescharacteristic
411C2C22×Q8×C3⋊D311yescharacteristic
412C2C2×#6436411yescharacteristic
subgroupG/H
centerC23C23×C3⋊D3
commutator, cosocleC22×C32C25
FrattiniC22C24×C3⋊D3
FittingC2×C32×C4⋊Q8C2
socleC23×C32C24
prime radicalC23×C3⋊D3C23
hypercenterC2×C4⋊Q8C3⋊D3
hypocenterC32C2×#544
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4⋊Q8C2semidirect
3. C2×#64153C2semidirect
4. #10596200C2direct (N × H)
8. C2×#64099C2semidirect
9. C2×#64098C2semidirect
10. C2×#64097C2semidirect
12. C2×#64079C2semidirect
18. C32×C4⋊Q8C22semidirect
24. #64153C22semidirect
43. #64099C22semidirect
44. #64098C22semidirect
45. #64097C22semidirect
47. #64079C22semidirect
48. C2×C3×C4⋊Q8D3semidirect
74. C2×C4×C3⋊D3Q8semidirect
103. C3×C4⋊Q8C2×D3semidirect
160. C4×C3⋊D3C2×Q8semidirect
165. C2×C4⋊Q8C3⋊D3semidirect
236. C4⋊Q8C2×C3⋊D3semidirect
293. C2×C32#544semidirect
354. C32C2×#544semidirect
384. C2×C3#5729semidirect
407. C3C2×#5729semidirect
409. C2#10596200direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C4⋊Q82
C2×#641552
C2×#641532
#105962002
C2×#641052
C2×#641012
C2×C4⋊C4×C3⋊D32
C2×#640992
C2×#640982
C2×#640972
C2×#640812
C2×#640792
C2×#57293