Finite Group Databasebeta

← C2×#20242390 #26188494

Subgroups — C2×#20242390 #26188494

Order 12032.

1308 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1308 conjugacy classes of subgroups — too many to draw. Only its 203 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689704C211yesmaximal, characteristic
2C2×#15689436C212yesmaximal, complemented
3#20242390C218yesmaximal, direct product factor
4C2×#15688856C212yesmaximal, complemented
5C2×#15688849C211yesmaximal, characteristic, complemented
6C2×C47×#398C211yesmaximal, characteristic, complemented
7C22×#10625350—21no—
8#15689704—24no—
9#15689704—22no—
10#15689704C2214yes—
11C2×#13094004—22no—
12C2×#13093999—22no—
13C2×#13093998—21no—
14C2×#13093998C2212yes—
15#15689436—24no—
16#15689436C2218yescomplemented
17C2×#13093957—22no—
18C2×#13093944—22no—
19C2×C22⋊(C47⋊C8)C2212yes—
20#15688856C2218yescomplemented
21#15688849C2214yescomplemented
22C22×C4⋊C4×C47—21no—
23C22×C4⋊C4×C47C2211yescharacteristic
24C22×C4⋊C4×C47C2211yescharacteristic
25C47×#398—22no—
26C47×#398C2214yescomplemented
27C2×#95×C47—21no—
28C2×#95×C47C2211yescharacteristic
29C2×#10625350—24no—
30C2×#10625350—24no—
31C2×#10625350—44no—
32C2×#10625350—44no—
33C22×Dic94—21no—
34C22×D188—21no—
35C22×D188—21no—
36C22×D188D412yes—
37C22×C22⋊D47—22no—
38C22×C4×D47—22no—
39C22×C47⋊C8—22no—
40#13094004—28no—
41#13094004—28no—
42#13093999—28no—
43#13093999—28no—
44#13093998—24no—
45#13093998—24no—
46#13093998—22no—
47#13093998—22no—
48#13093998D414yes—
49#13093998C2314yes—
50C2×C4⋊Dic47—21no—
51C2×C4⋊Dic47D412yes—
52C2×(C2×D47)⋊C4—22no—
53C2×(C2×D47)⋊C4—22no—
54#13093957—28no—
55#13093944—28no—
56C22⋊(C47⋊C8)—22no—
57C22⋊(C47⋊C8)C2314yes—
58C23×C4×C47—21no—
59C23×C4×C47—21no—
60C23×C4×C47D411yescharacteristic
61C23×C4×C47D411yescharacteristic
62C23×C4×C47C2311yescharacteristic
63C2×C4⋊C4×C47—22no—
64C2×C4⋊C4×C47—22no—
65C2×C4⋊C4×C47—22no—
66C2×C4⋊C4×C47—22no—
67C2×C4⋊C4×C47—24no—
68C2×C4⋊C4×C47—21no—
69C2×C4⋊C4×C47—44no—
70C2×C4⋊C4×C47D412yes—
71C2×C4⋊C4×C47C2312yes—
72C2×C4⋊C4×C47C2312yes—
73#95×C47—24no—
74#95×C47—21no—
75#95×C47C2312yes—
76C24×C47C2×D411yescharacteristic
77C23×D47—22no—
78C22×C4×C47—21no—
79C22×C4×C47—21no—
80C22×C4×C47—22no—
81C22×C4×C47—24no—
82C22×C4×C47—21no—
83C22×C4×C47—22no—
84C22×C4×C47—21no—
85C22×C4×C47—22no—
86C22×C4×C47—42no—
87C22×C4×C47—42no—
88C22×C4×C47—42no—
89C22×C4×C47—42no—
90C22×C4×C47—41no—
91C22×C4×C47—41no—
92C22×C4×C47—42no—
93C22×C4×C47—42no—
94C22×C4×C47D814yes—
95C22×C4×C47C2×D412yes—
96C22×C4×C47C2×D412yes—
97C22×C4×C47C2×D412yes—
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47C2×D412yes—
100C22×C4×C47C2411yescharacteristic
101C2×C22⋊D47—24no—
102C2×C22⋊D47—24no—
103C2×C22⋊D47—44no—
104C2×C22⋊D47—44no—
105C22×Dic47—22no—
106#10625350—48no—
107#10625350—416no—
108#10625350—48no—
109C2×D188—24no—
110C2×D188—24no—
111C2×D188—22no—
112C2×D188—44no—
113C2×D188—44no—
114C2×D188—82no—
115C2×D188C2×D414yes—
116C2×C4×D47—28no—
117C2×C4×D47—48no—
118C2×Dic94—24no—
119C2×Dic94—44no—
120C4⋊C4×C47—24no—
121C4⋊C4×C47—24no—
122C4⋊C4×C47—24no—
123C4⋊C4×C47—42no—
124C4⋊C4×C47—48no—
125(C2×D47)⋊C4—28no—
126(C2×D47)⋊C4—24no—
127(C2×D47)⋊C4—24no—
128C4⋊Dic47—24no—
129C4⋊Dic47—22no—
130C4⋊Dic47C2×D414yes—
131C2×C47⋊C8—24no—
132C2×C47⋊C8—44no—
133C23×C47—22no—
134C23×C47—22no—
135C23×C47#12011yescharacteristic
136C23×C47#12011yescharacteristic
137C23×C47C2×D812yes—
138C23×C47AGL(1,ℤ/8)12yes—
139C23×C47C22×D411yescharacteristic
140C22×D47—24no—
141C22×D47—44no—
142C22×D47—44no—
143C22×D47—44no—
144C2×C4×C47—22no—
145C2×C4×C47—22no—
146C2×C4×C47—24no—
147C2×C4×C47—24no—
148C2×C4×C47—24no—
149C2×C4×C47—24no—
150C2×C4×C47—22no—
151C2×C4×C47—42no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47—48no—
155C2×C4×C47—42no—
156C2×C4×C47—42no—
157C2×C4×C47—42no—
158C2×C4×C47—42no—
159C2×C4×C47—42no—
160C2×C4×C47—42no—
161C2×C4×C47—42no—
162C2×C4×C47—81no—
163C2×C4×C47—82no—
164C2×C4×C47—82no—
165C2×C4×C47#12012yes—
166C2×C4×C47C2×D814yes—
167C2×C4×C47C22×D412yes—
168C22⋊D47—44no—
169C22⋊D47—44no—
170C22⋊D47—48no—
171C2×Dic47—24no—
172C2×Dic47—44no—
173D188—44no—
174D188—48no—
175D188—82no—
176D188—84no—
177C4×D47—416no—
178Dic94—48no—
179C47⋊C8—44no—
180C2×#1610—471nomaximal, Sylow
181C22×C47—24no—
182C22×C47—24no—
183C22×C47—21no—
184C22×C47—21no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47#44712yes—
188C22×C47#44712yes—
189C22×C47#45612yes—
190C22×C47#45812yes—
191C22×C47C2×#12011yescharacteristic
192C22×C47C22×D811yescharacteristic
193C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
194C2×D47—44no—
195C2×D47—44no—
196C2×D47—44no—
197C2×D47—88no—
198C4×C47—24no—
199C4×C47—22no—
200C4×C47—22no—
201C4×C47—44no—
202C4×C47—42no—
203C4×C47—82no—
204Dic47—44no—
205C2×#537—471no—
206C2×#480—472no—
207C2×#398D4711yescharacteristic, complemented
208#1610—478no—
209C2×#327—472no—
210C2×#324—471no—
211C2×C47—21no—
212C2×C47—21no—
213C2×C47—22no—
214C2×C47#161014yescomplemented
215C2×C47C2×#44711yescharacteristic
216C2×C47C2×#45611yescharacteristic
217C2×C47C2×#45811yescharacteristic
218D47—84no—
219C22×#41—941no—
220#537—474no—
221#537—944no—
222#537—942no—
223C2×C22⋊Q8—942no—
224C2×#121—472no—
225C2×#121—942no—
226C2×#121—941no—
227C22×C4⋊C4—21no—
228C22×C4⋊C4C2×D4711yescharacteristic
229C22×C4⋊C4C2×D4711yescharacteristic
230#480—478no—
231#480—944no—
232C2×#107—942no—
233C2×#102—942no—
234C2×C22⋊C8—472no—
235#398—22no—
236#398C2×D4714yescomplemented
237C2×#95—21no—
238C2×#95C2×D4711yescharacteristic
239#327—478no—
240#324—474no—
241C47C2×#161011yescharacteristic, complemented, Sylow
242C2×#41—944no—
243C2×#41—944no—
244C2×#41—1884no—
245C2×#41—1884no—
246C22×Q8—941no—
247C22×D4—472no—
248C22×D4—941no—
249C22×D4—941no—
250C22×D4—942no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—942no—
254C23×C4C22⋊D4711yescharacteristic
255C23×C4C22⋊D4711yescharacteristic
256C23×C4C22×D4711yescharacteristic
257C22×C8—942no—
258C22⋊Q8—948no—
259C22⋊Q8—948no—
260#121—474no—
261#121—474no—
262#121—948no—
263#121—948no—
264#121—944no—
265#121—942no—
266#121—942no—
267#121—944no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—472no—
276C2×C4⋊C4—941no—
277C2×C4⋊C4C22⋊D4712yes—
278C2×C4⋊C4C22×D4712yes—
279C2×C4⋊C4C22×D4712yes—
280C2×C22⋊C4—942no—
281C2×C22⋊C4—942no—
282#107—948no—
283#102—948no—
284C22⋊C8—474no—
285C22⋊C8—942no—
286#95—24no—
287#95—21no—
288#95C22×D4712yes—
289C24—942no—
290C24C2×C22⋊D4711yescharacteristic
291#41—18816no—
292#41—1888no—
293#41—1888no—
294C2×Q8—944no—
295C2×Q8—1884no—
296C2×D4—474no—
297C2×D4—944no—
298C2×D4—944no—
299C2×D4—944no—
300C2×D4—944no—
301C2×D4—942no—
302C2×D4—3762no—
303C2×D4—1884no—
304C2×D4—1884no—
305C2×D4—1884no—
306C2×D4—1884no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—948no—
324C22×C4—942no—
325C22×C4—1888no—
326C22×C4#1062533414yes—
327C22×C4D4×D4712yes—
328C22×C4D4×D4712yes—
329C22×C4C2×C22⋊D4712yes—
330C22×C4C2×C22⋊D4712yes—
331C22×C4C2×C22⋊D4711yescharacteristic
332C22×C4C23×D4711yescharacteristic
333C2×C8—944no—
334C2×C8—1884no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—42no—
339C4⋊C4—48no—
340C4⋊C4—474no—
341C4⋊C4—944no—
342C4⋊C4—942no—
343C22⋊C4—948no—
344C22⋊C4—944no—
345C22⋊C4—944no—
346C23—22no—
347C23—22no—
348C23—944no—
349C23—1884no—
350C23—1884no—
351C23—1884no—
352C23#1309399211yescharacteristic
353C23#1309399211yescharacteristic
354C23C2×#1062533412yes—
355C23#1309404912yes—
356C23C22×C22⋊D4711yescharacteristic
357Q8—1888no—
358D4—3762no—
359D4—3764no—
360D4—1884no—
361D4—1888no—
362D4—1888no—
363D4—1884no—
364D4—1884no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—48no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—944no—
387C2×C4—18816no—
388C2×C4—1884no—
389C2×C4#1309399212yes—
390C2×C4C2×#1062533414yes—
391C2×C4C2×D4×D4712yes—
392C8—1884no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—24no—
398C22—24no—
399C22—3768no—
400C22—1884no—
401C22—1884no—
402C22—1884no—
403C22#1568925612yes—
404C22#1568925612yes—
405C22#1568930512yes—
406C22#1568931212yes—
407C22C2×#1309399211yescharacteristic
408C22C22×#1062533411yescharacteristic
409C22C2×#1309404911yescharacteristic
410C4—22no—
411C4—22no—
412C4—24no—
413C4—44no—
414C4—42no—
415C4—82no—
416C4—1884no—
417C2—21no—
418C2—22no—
419C2—21no—
420C2—3764no—
421C2#2024239014yesdirect product factor
422C2C2×#1568925611yescharacteristic
423C2C2×#1568930511yescharacteristic
424C2C2×#1568931211yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#398C2
socleC23×C47#120
prime radicalC2×#15689704C2
hypercenterC2×#398D47
hypocenterC47C2×#1610
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
2. C2×#15689436C2semidirect
3. #20242390C2direct (N × H)
4. C2×#15688856C2semidirect
5. C2×#15688849C2semidirect
6. C2×C47×#398C2semidirect
16. #15689436C22semidirect
20. #15688856C22semidirect
21. #15688849C22semidirect
26. C47×#398C22semidirect
207. C2×#398D47semidirect
214. C2×C47#1610semidirect
236. #398C2×D47semidirect
241. C47C2×#1610semidirect
421. C2#20242390direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156897042
C2×#156894362
#202423902
C2×#156888562
C2×#156888492
C2×C47×#3982
C2×#161047