Finite Group Databasebeta

← C2×#10592899 #11867846

Subgroups — C2×#10592899 #11867846

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×C32×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C2×#64160C211yesmaximal, characteristic, complemented
3C2×#64157C211yesmaximal, characteristic, complemented
4C22×#3754C211yesmaximal, characteristic, complemented
5C2×#63660C211yesmaximal, characteristic, complemented
6#10592899C218yesmaximal, direct product factor
7C2×#63651C211yesmaximal, characteristic, complemented
8C2×#63649C211yesmaximal, characteristic, complemented
9C2×#5200—34nomaximal
10C2×C32×#41C2211yescharacteristic
11C22×C32×D4C2211yescharacteristic
12C22×#634C2211yescharacteristic
13C32×AGL(1,ℤ/8)—22no—
14C32×AGL(1,ℤ/8)C2214yes—
15C2×C32×SD16—21no—
16C2×C32×D8—21no—
17C2×C32×M16C2211yescharacteristic
18#64160—22no—
19#64160C2214yescomplemented
20#64157—22no—
21#64157C2214yescomplemented
22C2×#3756—21no—
23C2×#3754—21no—
24C2×#3754—21no—
25C2×#3754—21no—
26C2×#3754—22no—
27C2×#3754C2214yescomplemented
28C2×#3748—21no—
29C2×#3748C2211yescharacteristic
30C2×#3748C2211yescharacteristic
31C22×C3⋊(C3⋊C8)—21no—
32C22×C3⋊(C3⋊C8)C2211yescharacteristic
33#63660C2214yescomplemented
34#63651C2214yescomplemented
35#63649C2214yescomplemented
36C2×C3×AGL(1,ℤ/8)D314yes—
37C2×#951—34no—
38C2×#948—34no—
39C22×#214—34no—
40C2×#833—34no—
41#5200—332no—
42C2×#824—34no—
43C2×#822—34no—
44C24×C32—21no—
45C32×#41—21no—
46C32×#41—21no—
47C32×#41D412yes—
48C32×#41C2312yes—
49C2×C32×Q8D411yescharacteristic
50C2×C32×D4—21no—
51C2×C32×D4—21no—
52C2×C32×D4—21no—
53C2×C32×D4—21no—
54C2×C32×D4—21no—
55C2×C32×D4—21no—
56C2×C32×D4D411yescharacteristic
57C2×C32×D4C2312yes—
58C22×C32×C4—21no—
59C22×C32×C4C2311yescharacteristic
60C22×C3⋊Dic3—21no—
61C2×#634—21no—
62C2×#634—21no—
63C2×#634—21no—
64C2×#634—21no—
65C2×#634—21no—
66C2×#634C2312yes—
67C32×SD16—24no—
68C32×D8—24no—
69C32×M16—21no—
70C32×M16C2312yes—
71C2×C32×C8—21no—
72#3756—24no—
73#3754—24no—
74#3754—24no—
75#3754—24no—
76#3748—21no—
77#3748—21no—
78#3748—24no—
79#3748C2312yes—
80#3748C2312yes—
81C2×C3⋊(C3⋊C8)—21no—
82C2×C3⋊(C3⋊C8)—21no—
83C2×C3⋊(C3⋊C8)—24no—
84C2×C3⋊(C3⋊C8)D411yescharacteristic
85C2×C3⋊(C3⋊C8)D411yescharacteristic
86C2×C3⋊(C3⋊C8)D412yes—
87C2×C3⋊(C3⋊C8)C2312yes—
88C2×#471—91noSylow
89C2×C3×#41C2×D314yes—
90C22×C3×D4C2×D314yes—
91C22×Dic6—34no—
92C3×AGL(1,ℤ/8)—28no—
93C3×AGL(1,ℤ/8)C2×D3116yes—
94C2×C3×SD16—24no—
95C2×C3×D8—24no—
96C2×C3×M16C2×D314yes—
97#951—316no—
98#951—68no—
99#948—316no—
100#948—68no—
101C2×#216—64no—
102C2×#214—316no—
103C2×#214—64no—
104C2×#214—64no—
105C2×#214—68no—
106C2×#214—64no—
107C2×#208—34no—
108C2×#208—34no—
109C2×#208—64no—
110C22×C3⋊C8—34no—
111C22×C3⋊C8—64no—
112#833—316no—
113#824—316no—
114#822—316no—
115C23×C32—21no—
116C23×C32—22no—
117C23×C32—41no—
118C23×C32—41no—
119C23×C32—82no—
120C23×C32#4111yescharacteristic
121C32×Q8—21no—
122C32×Q8C2×D412yes—
123C32×D4—22no—
124C32×D4—22no—
125C32×D4—21no—
126C32×D4—22no—
127C32×D4—22no—
128C32×D4—41no—
129C32×D4—41no—
130C32×D4C2×D412yes—
131C2×C32×C4—21no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4C2×D411yescharacteristic
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4C2×D411yescharacteristic
137C2×C32×C4C2×D411yescharacteristic
138C2×C32×C4#4111yescharacteristic
139C2×C32×C4C2411yescharacteristic
140C2×C3⋊Dic3—22no—
141C2×C3⋊Dic3—41no—
142C2×C3⋊Dic3—41no—
143#634—22no—
144#634—22no—
145#634—41no—
146#634—41no—
147C32×C8—22no—
148C3⋊(C3⋊C8)—22no—
149C3⋊(C3⋊C8)—22no—
150C3⋊(C3⋊C8)C2×D412yes—
151C3⋊(C3⋊C8)C2×D412yes—
152C2×#137—91no—
153C2×AGL(1,ℤ/8)C3⋊D311yescharacteristic
154C22×SD16—91no—
155C2×#131—91no—
156#471—98no—
157C2×#108—91no—
158C2×#101—91no—
159C2×#100—91no—
160C24×C3—24no—
161C3×#41—24no—
162C3×#41—24no—
163C3×#41#6718yes—
164C3×#41C22×D318yes—
165C2×C3×Q8#6714yes—
166C2×C3×D4—24no—
167C2×C3×D4—24no—
168C2×C3×D4—24no—
169C2×C3×D4—24no—
170C2×C3×D4—24no—
171C2×C3×D4—24no—
172C2×C3×D4#6714yes—
173C2×C3×D4C22×D318yes—
174C22×C3×C4—24no—
175C22×C3×C4C22×D314yes—
176C22×Dic3—64no—
177C2×Dic6—38no—
178C2×Dic6—64no—
179C2×Dic6—64no—
180C2×Dic6—64no—
181C2×Dic6—64no—
182C2×Dic6—64no—
183C3×SD16—216no—
184C3×D8—216no—
185C3×M16—24no—
186C3×M16C22×D318yes—
187C2×C3×C8—24no—
188#216—616no—
189#214—616no—
190#214—616no—
191#214—616no—
192#208—38no—
193#208—38no—
194#208—616no—
195#208—64no—
196#208—64no—
197C2×C3⋊C8—34no—
198C2×C3⋊C8—34no—
199C2×C3⋊C8—38no—
200C2×C3⋊C8—38no—
201C2×C3⋊C8—616no—
202C2×C3⋊C8—64no—
203C2×C3⋊C8—64no—
204C22×C32—22no—
205C22×C32—22no—
206C22×C32—42no—
207C22×C32—42no—
208C22×C32—82no—
209C22×C32—82no—
210C22×C32—82no—
211C22×C32#12111yescharacteristic
212C22×C32#12111yescharacteristic
213C22×C32C2×#4111yescharacteristic
214C32×C4—22no—
215C32×C4#12111yescharacteristic
216C32×C4#12111yescharacteristic
217C32×C4C22×D411yescharacteristic
218C32×C4C22×D411yescharacteristic
219C3⋊Dic3—42no—
220C2×#41C2×C3⋊D311yescharacteristic
221C22×Q8—91no—
222C22×D4C2×C3⋊D311yescharacteristic
223#137—94no—
224#137—182no—
225AGL(1,ℤ/8)—22no—
226AGL(1,ℤ/8)C2×C3⋊D314yes—
227C2×Q16—181no—
228C2×SD16—21no—
229C2×SD16—94no—
230C2×SD16—181no—
231C2×SD16—181no—
232C2×SD16—182no—
233C2×SD16—181no—
234C2×D8—21no—
235#131—94no—
236#131—182no—
237C2×M16—91no—
238C2×M16—91no—
239C2×M16—181no—
240C2×M16C2×C3⋊D311yescharacteristic
241C22×C8—91no—
242C22×C8—181no—
243#108—94no—
244#101—94no—
245#100—94no—
246C23×C3—24no—
247C23×C3—28no—
248C23×C3—44no—
249C23×C3—44no—
250C23×C3—88no—
251C23×C3#23714yes—
252C3×Q8—24no—
253C3×Q8C2×#6718yes—
254C3×D4—28no—
255C3×D4—28no—
256C3×D4—24no—
257C3×D4—28no—
258C3×D4—28no—
259C3×D4—44no—
260C3×D4—44no—
261C3×D4C2×#6718yes—
262C2×C3×C4—24no—
263C2×C3×C4—28no—
264C2×C3×C4—28no—
265C2×C3×C4D3×D414yes—
266C2×C3×C4D3×D414yes—
267C2×C3×C4#23714yes—
268C2×C3×C4C2×#6714yes—
269C2×C3×C4C2×#6714yes—
270C2×C3×C4C23×D314yes—
271C2×Dic3—68no—
272C2×Dic3—124no—
273C2×Dic3—124no—
274Dic6—68no—
275Dic6—68no—
276Dic6—124no—
277Dic6—124no—
278C3×C8—28no—
279C3⋊C8—38no—
280C3⋊C8—38no—
281C3⋊C8—68no—
282C3⋊C8—68no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—42no—
286C2×C32—82no—
287C2×C32#47112yescomplemented
288C2×C32C2×#12111yescharacteristic
289C24—21no—
290#41—21no—
291#41—21no—
292#41#63812yes—
293#41C22×C3⋊D312yes—
294C2×Q8—92no—
295C2×Q8—181no—
296C2×Q8—181no—
297C2×Q8—181no—
298C2×Q8—181no—
299C2×Q8—181no—
300C2×Q8#63811yescharacteristic
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4#63811yescharacteristic
308C2×D4C22×C3⋊D312yes—
309C22×C4—21no—
310C22×C4—181no—
311C22×C4C22×C3⋊D311yescharacteristic
312Q16—184no—
313SD16—24no—
314SD16—184no—
315SD16—184no—
316SD16—184no—
317D8—24no—
318M16—21no—
319M16—92no—
320M16—92no—
321M16—184no—
322M16—181no—
323M16—181no—
324M16C22×C3⋊D312yes—
325C2×C8—21no—
326C2×C8—92no—
327C2×C8—91no—
328C2×C8—91no—
329C2×C8—92no—
330C2×C8—184no—
331C2×C8—181no—
332C2×C8—181no—
333C22×C3—28no—
334C22×C3—28no—
335C22×C3—48no—
336C22×C3—48no—
337C22×C3—88no—
338C22×C3—88no—
339C22×C3—88no—
340C22×C3#93914yes—
341C22×C3#93914yes—
342C22×C3C2×#23714yes—
343C3×C4—28no—
344C3×C4#93914yes—
345C3×C4#93914yes—
346C3×C4C2×D3×D414yes—
347C3×C4C22×#6714yes—
348Dic3—128no—
349C32C2×#47111yescharacteristic, complemented, Sylow
350C23—21no—
351C23—22no—
352C23—41no—
353C23—41no—
354C23—82no—
355C23#383011yescharacteristic
356Q8—21no—
357Q8—182no—
358Q8—182no—
359Q8—361no—
360Q8—361no—
361Q8C2×#63812yes—
362D4—22no—
363D4—22no—
364D4—22no—
365D4—22no—
366D4—21no—
367D4—41no—
368D4—41no—
369D4C2×#63812yes—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—182no—
374C2×C4—361no—
375C2×C4—361no—
376C2×C4D4×C3⋊D311yescharacteristic
377C2×C4D4×C3⋊D311yescharacteristic
378C2×C4#383011yescharacteristic
379C2×C4C2×#63811yescharacteristic
380C2×C4C2×#63811yescharacteristic
381C2×C4C23×C3⋊D311yescharacteristic
382C8—22no—
383C8—92no—
384C8—92no—
385C8—182no—
386C8—182no—
387C2×C3—24no—
388C2×C3—24no—
389C2×C3—48no—
390C2×C3—88no—
391C2×C3#520018yescomplemented
392C2×C3C2×#93914yes—
393C22—22no—
394C22—22no—
395C22—42no—
396C22—42no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22#6414811yescharacteristic
401C22#6414811yescharacteristic
402C22C2×#383011yescharacteristic
403C4—22no—
404C4—362no—
405C4#6414811yescharacteristic
406C4#6414811yescharacteristic
407C4C2×D4×C3⋊D311yescharacteristic
408C4C22×#63811yescharacteristic
409C3C2×#520014yescomplemented
410C2—21no—
411C2—21no—
412C2—42no—
413C2—82no—
414C2#1059289912yesdirect product factor
415C2C2×#6414811yescharacteristic
subgroupG/H
centerC22#64148
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
Fitting, prime radicalC2×C32×AGL(1,ℤ/8)C2
socleC22×C32#121
hypercenterC2×AGL(1,ℤ/8)C3⋊D3
hypocenterC32C2×#471
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
2. C2×#64160C2semidirect
3. C2×#64157C2semidirect
4. C22×#3754C2semidirect
5. C2×#63660C2semidirect
6. #10592899C2direct (N × H)
7. C2×#63651C2semidirect
8. C2×#63649C2semidirect
19. #64160C22semidirect
21. #64157C22semidirect
27. C2×#3754C22semidirect
33. #63660C22semidirect
34. #63651C22semidirect
35. #63649C22semidirect
287. C2×C32#471semidirect
349. C32C2×#471semidirect
391. C2×C3#5200semidirect
409. C3C2×#5200semidirect
414. C2#10592899direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×AGL(1,ℤ/8)2
C2×#641602
C2×#641572
C22×#37542
C2×#636602
#105928992
C2×#636512
C2×#636492
C2×#52003