Finite Group Databasebeta

← C2×#10592597 #11867855

Subgroups — C2×#10592597 #11867855

Order 1152.

1404 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64118C211yesmaximal, characteristic, complemented
2C2×#64117C211yesmaximal, characteristic
3C22×#3744C211yesmaximal, characteristic, complemented
4C2×#64110C211yesmaximal, characteristic, complemented
5C2×C32×#108C211yesmaximal, characteristic, complemented
6C2×#63651C211yesmaximal, characteristic, complemented
7C2×#63650C211yesmaximal, characteristic, complemented
8#10592597C218yesmaximal, direct product factor
9C2×#4898—34nomaximal
10C2×#3828C2211yescharacteristic
11C22×C4⋊(C3⋊D3)C2211yescharacteristic
12C22×#634C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14C2×C32×M16C2211yescharacteristic
15C22×C32×C8C2211yescharacteristic
16C2×#3748C2211yescharacteristic
17#64118—22no—
18#64118C2214yescomplemented
19#64117—22no—
20#64117C2214yes—
21C2×#3746—21no—
22C2×#3745—21no—
23C2×#3744—21no—
24C2×#3744—21no—
25C2×#3744—21no—
26C2×#3744—21no—
27C2×#3744—22no—
28C2×#3744C2214yescomplemented
29#64110—22no—
30#64110C2214yescomplemented
31C2×#3743—21no—
32C2×C8×C3⋊D3—21no—
33C32×#108C2214yescomplemented
34#63651C2214yescomplemented
35#63650C2214yescomplemented
36C2×#909—34no—
37C2×#908—34no—
38C22×#204—34no—
39C2×#901—34no—
40C2×C3×#108D314yescomplemented
41C2×#824—34no—
42C2×#823—34no—
43#4898—332no—
44C23×C3⋊D3—21no—
45C22×C32×C4C2311yescharacteristic
46C2×#638—21no—
47C22×C3⋊Dic3—21no—
48#3828—21no—
49#3828—21no—
50#3828D412yes—
51#3828C2312yes—
52C2×C4⋊(C3⋊D3)—21no—
53C2×C4⋊(C3⋊D3)—21no—
54C2×C4⋊(C3⋊D3)—21no—
55C2×C4⋊(C3⋊D3)—21no—
56C2×C4⋊(C3⋊D3)—21no—
57C2×C4⋊(C3⋊D3)D411yescharacteristic
58C2×C4⋊(C3⋊D3)C2312yes—
59C2×C4×C3⋊D3—21no—
60C2×#634—21no—
61C2×#634—21no—
62C2×#634—21no—
63C2×#634—21no—
64C2×#634—21no—
65C2×#634D411yescharacteristic
66C2×#634C2312yes—
67C32×M16—21no—
68C32×M16—21no—
69C32×M16C2312yes—
70C32×M16C2312yes—
71C2×C32×C8—21no—
72C2×C32×C8—21no—
73C2×C32×C8D411yescharacteristic
74C2×C32×C8D411yescharacteristic
75C2×C32×C8D412yes—
76C2×C32×C8C2312yes—
77#3748—21no—
78#3748C2312yes—
79C2×C3⋊(C3⋊C8)—21no—
80#3746—24no—
81#3745—24no—
82#3744—24no—
83#3744—24no—
84#3744—24no—
85#3744—24no—
86#3743—24no—
87C8×C3⋊D3—24no—
88C2×#471—91noSylow
89C2×#235—34no—
90C22×D12—34no—
91C22×Dic6—34no—
92C2×C3×M16C2×D314yes—
93C2×C3×M16C2×D314yes—
94C22×C3×C8C2×D314yes—
95C2×#208—34no—
96#909—316no—
97#909—68no—
98#908—316no—
99#908—68no—
100C2×Dic12—64no—
101C2×D24—64no—
102C2×#204—316no—
103C2×#204—64no—
104C2×#204—64no—
105C2×#204—64no—
106C2×#204—64no—
107C2×#204—68no—
108#901—316no—
109#901—68no—
110C2×ΓL(1,𝔽25)—64no—
111C2×D3×C8—64no—
112C3×#108C2×D3116yescomplemented
113#824—316no—
114#823—316no—
115C23×C32#4111yescharacteristic
116C22×C3⋊D3—21no—
117C22×C3⋊D3—22no—
118C22×C3⋊D3—41no—
119C22×C3⋊D3—41no—
120C22×C3⋊D3—82no—
121C2×C32×C4C2×D411yescharacteristic
122C2×C32×C4C2×D411yescharacteristic
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C2×C32×C4#4111yescharacteristic
126C2×C32×C4C2411yescharacteristic
127#638—22no—
128#638—22no—
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—22no—
131C2×C3⋊Dic3—41no—
132C2×C3⋊Dic3—41no—
133C4⋊(C3⋊D3)—21no—
134C4⋊(C3⋊D3)—22no—
135C4⋊(C3⋊D3)—22no—
136C4⋊(C3⋊D3)—41no—
137C4⋊(C3⋊D3)—41no—
138C4⋊(C3⋊D3)C2×D412yes—
139C4×C3⋊D3—22no—
140C4×C3⋊D3—22no—
141#634—21no—
142#634—22no—
143#634—22no—
144#634—41no—
145#634—41no—
146#634C2×D412yes—
147C32×C8—22no—
148C32×C8—22no—
149C32×C8C2×D412yes—
150C32×C8C2×D412yes—
151C3⋊(C3⋊C8)—22no—
152C2×#137—91no—
153C2×AGL(1,ℤ/8)—91no—
154C22×SD16—91no—
155C2×#131—91no—
156#471—98no—
157C2×#108C3⋊D311yescharacteristic, complemented
158C2×#101—91no—
159C2×#100—91no—
160C23×D3—64no—
161C22×C3×C4C22×D314yes—
162C2×#67—64no—
163C22×Dic3—64no—
164#235—38no—
165#235—38no—
166#235—64no—
167#235—64no—
168C2×D12—34no—
169C2×D12—38no—
170C2×D12—64no—
171C2×D12—64no—
172C2×D12—64no—
173C2×D12—64no—
174C2×D12—64no—
175C2×C4×D3—64no—
176C2×Dic6—34no—
177C2×Dic6—38no—
178C2×Dic6—64no—
179C2×Dic6—64no—
180C2×Dic6—64no—
181C2×Dic6—64no—
182C2×Dic6—64no—
183C3×M16—24no—
184C3×M16—24no—
185C3×M16C22×D318yes—
186C3×M16C22×D318yes—
187C2×C3×C8—24no—
188C2×C3×C8—24no—
189C2×C3×C8D1214yes—
190C2×C3×C8D1214yes—
191C2×C3×C8D1218yes—
192C2×C3×C8C22×D318yes—
193#208—38no—
194#208—64no—
195C2×C3⋊C8—64no—
196Dic12—616no—
197D24—616no—
198#204—616no—
199#204—616no—
200#204—616no—
201#204—616no—
202ΓL(1,𝔽25)—616no—
203D3×C8—616no—
204C22×C32—22no—
205C22×C32#12111yescharacteristic
206C22×C32#12111yescharacteristic
207C22×C32C2×#4111yescharacteristic
208C2×C3⋊D3—22no—
209C2×C3⋊D3—42no—
210C2×C3⋊D3—42no—
211C2×C3⋊D3—82no—
212C2×C3⋊D3—82no—
213C2×C3⋊D3—82no—
214C32×C4#12111yescharacteristic
215C32×C4#12111yescharacteristic
216C32×C4C22×D411yescharacteristic
217C32×C4C22×D411yescharacteristic
218C3⋊Dic3—22no—
219C3⋊Dic3—42no—
220C2×#41—91no—
221C22×Q8—91no—
222C22×D4—91no—
223#137—94no—
224#137—182no—
225AGL(1,ℤ/8)—94no—
226AGL(1,ℤ/8)—182no—
227C2×Q16—181no—
228C2×SD16—94no—
229C2×SD16—181no—
230C2×SD16—181no—
231C2×SD16—181no—
232C2×SD16—181no—
233C2×SD16—182no—
234C2×D8—181no—
235#131—94no—
236#131—182no—
237C2×M16—91no—
238C2×M16—181no—
239C2×M16C2×C3⋊D311yescharacteristic
240C2×M16C2×C3⋊D311yescharacteristic
241C22×C8—181no—
242C22×C8C2×C3⋊D311yescharacteristic
243#108C2×C3⋊D314yescomplemented
244#101—94no—
245#100—94no—
246C23×C3#23914yes—
247C22×D3—64no—
248C22×D3—68no—
249C22×D3—124no—
250C22×D3—124no—
251C22×D3—248no—
252C2×C3×C4C2×D1214yes—
253C2×C3×C4C2×D1214yes—
254C2×C3×C4D3×D414yes—
255C2×C3×C4D3×D414yes—
256C2×C3×C4#23914yes—
257C2×C3×C4C23×D314yes—
258#67—68no—
259#67—68no—
260C2×Dic3—64no—
261C2×Dic3—68no—
262C2×Dic3—124no—
263C2×Dic3—124no—
264D12—38no—
265D12—64no—
266D12—68no—
267D12—68no—
268D12—124no—
269D12—124no—
270C4×D3—68no—
271C4×D3—68no—
272Dic6—38no—
273Dic6—64no—
274Dic6—68no—
275Dic6—68no—
276Dic6—124no—
277Dic6—124no—
278C3×C8—28no—
279C3×C8—28no—
280C3×C8C2×D1218yes—
281C3×C8C2×D1218yes—
282C3⋊C8—68no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32#47112yescomplemented
286C2×C32C2×#12111yescharacteristic
287C3⋊D3—42no—
288C3⋊D3—82no—
289C24—181no—
290#41—92no—
291#41—92no—
292#41—181no—
293#41—181no—
294C2×Q8—92no—
295C2×Q8—91no—
296C2×Q8—181no—
297C2×Q8—181no—
298C2×Q8—181no—
299C2×Q8—181no—
300C2×Q8—181no—
301C2×D4—92no—
302C2×D4—91no—
303C2×D4—181no—
304C2×D4—181no—
305C2×D4—181no—
306C2×D4—181no—
307C2×D4—181no—
308C2×D4—181no—
309C22×C4—181no—
310C22×C4—181no—
311C22×C4C22×C3⋊D311yescharacteristic
312Q16—184no—
313SD16—184no—
314SD16—184no—
315SD16—184no—
316SD16—184no—
317D8—184no—
318M16—21no—
319M16—21no—
320M16—92no—
321M16—181no—
322M16—184no—
323M16C22×C3⋊D312yes—
324M16C22×C3⋊D312yes—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—181no—
328C2×C8—184no—
329C2×C8C4⋊(C3⋊D3)11yescharacteristic
330C2×C8C4⋊(C3⋊D3)11yescharacteristic
331C2×C8C4⋊(C3⋊D3)12yes—
332C2×C8C22×C3⋊D312yes—
333C22×C3—28no—
334C22×C3#89514yes—
335C22×C3#89514yes—
336C22×C3C2×#23914yes—
337C2×D3—68no—
338C2×D3—128no—
339C2×D3—128no—
340C2×D3—248no—
341C2×D3—248no—
342C2×D3—248no—
343C3×C4#89514yes—
344C3×C4#89514yes—
345C3×C4C22×D1214yes—
346C3×C4C2×D3×D414yes—
347Dic3—68no—
348Dic3—128no—
349C32C2×#47111yescharacteristic, complemented, Sylow
350C23—181no—
351C23—182no—
352C23—361no—
353C23—361no—
354C23—722no—
355C23#383211yescharacteristic
356Q8—92no—
357Q8—181no—
358Q8—182no—
359Q8—182no—
360Q8—361no—
361Q8—361no—
362D4—92no—
363D4—182no—
364D4—182no—
365D4—182no—
366D4—182no—
367D4—181no—
368D4—361no—
369D4—361no—
370C2×C4—181no—
371C2×C4—182no—
372C2×C4—182no—
373C2×C4—182no—
374C2×C4—361no—
375C2×C4—361no—
376C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
377C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
378C2×C4D4×C3⋊D311yescharacteristic
379C2×C4D4×C3⋊D311yescharacteristic
380C2×C4#383211yescharacteristic
381C2×C4C23×C3⋊D311yescharacteristic
382C8—22no—
383C8—22no—
384C8—182no—
385C8C2×C4⋊(C3⋊D3)12yes—
386C8C2×C4⋊(C3⋊D3)12yes—
387C2×C3—24no—
388C2×C3—24no—
389C2×C3#489818yescomplemented
390C2×C3C2×#89514yes—
391D3—128no—
392D3—248no—
393C22—22no—
394C22—182no—
395C22—362no—
396C22—362no—
397C22—722no—
398C22—722no—
399C22—722no—
400C22#6410411yescharacteristic
401C22#6410411yescharacteristic
402C22C2×#383211yescharacteristic
403C4—182no—
404C4—362no—
405C4#6410411yescharacteristic
406C4#6410411yescharacteristic
407C4C22×C4⋊(C3⋊D3)11yescharacteristic
408C4C2×D4×C3⋊D311yescharacteristic
409C3C2×#489814yescomplemented
410C2—21no—
411C2—21no—
412C2—362no—
413C2—722no—
414C2#1059259712yesdirect product factor
415C2C2×#6410411yescharacteristic
subgroupG/H
centerC22#64104
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC2×C32×#108C2
socleC22×C32#121
prime radicalC2×#64117C2
hypercenterC2×#108C3⋊D3
hypocenterC32C2×#471
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#108

Complements

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

NH ≅ G/Ntype
1. C2×#64118C2semidirect
3. C22×#3744C2semidirect
4. C2×#64110C2semidirect
5. C2×C32×#108C2semidirect
6. C2×#63651C2semidirect
7. C2×#63650C2semidirect
8. #10592597C2direct (N × H)
18. #64118C22semidirect
28. C2×#3744C22semidirect
30. #64110C22semidirect
33. C32×#108C22semidirect
34. #63651C22semidirect
35. #63650C22semidirect
40. C2×C3×#108D3semidirect
112. C3×#108C2×D3semidirect
157. C2×#108C3⋊D3semidirect
243. #108C2×C3⋊D3semidirect
285. C2×C32#471semidirect
349. C32C2×#471semidirect
389. C2×C3#4898semidirect
409. C3C2×#4898semidirect
414. C2#10592597direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#641182
C2×#641172
C22×#37442
C2×#641102
C2×C32×#1082
C2×#636512
C2×#636502
#105925972
C2×#48983