Finite Group Databasebeta

← #26123605

Subgroups — #26123605

Order 12032.

874 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689172C211yesmaximal, characteristic, complemented
2C2×#15689140C212yesmaximal
3C2×#15688880C211yesmaximal, characteristic, complemented
4#20243084C212yesmaximal, complemented
5C2×C47×#408C211yesmaximal, characteristic
6C22×#10625329C2212yes—
7C2×#13093983C2212yes—
8#15689172—24no—
9C2×#13093960—22no—
10#15689140—24no—
11#15689140—24no—
12C2×#13093951—24no—
13#15688880—22no—
14#15688877—24no—
15C22×#41×C47C2211yescharacteristic
16C22×M16×C47C2211yescharacteristic
17C23×C8×C47C2211yescharacteristic
18C47×#408—22no—
19C47×#408C414yescomplemented
20C2×C22⋊C8×C47—22no—
21C2×#10625329—22no—
22C2×#10625329—22no—
23C2×#10625329—22no—
24C2×#10625329—22no—
25C2×#10625329—22no—
26C2×#10625329D414yes—
27C22×C47⋊C8—22no—
28#13093983—22no—
29#13093983—22no—
30#13093983—22no—
31#13093983—22no—
32C2×C4⋊Dic47—22no—
33C2×C22⋊Dic47—22no—
34C2×C4×Dic47D414yes—
35#13093960—28no—
36#13093951—216no—
37C2×#41×C47—21no—
38C2×#41×C47—21no—
39C2×#41×C47—22no—
40C2×#41×C47—22no—
41C2×#41×C47—22no—
42C2×#41×C47—22no—
43C2×#41×C47C2×C412yes—
44C2×#41×C47D412yes—
45C22×Q8×C47Q811yescharacteristic
46C22×D4×C47—21no—
47C22×D4×C47Q811yescharacteristic
48C23×C4×C47—21no—
49C23×C4×C47C2311yescharacteristic
50C2×M16×C47—22no—
51C2×M16×C47—22no—
52C2×M16×C47C2×C412yes—
53C2×M16×C47D412yes—
54C22×C8×C47—22no—
55C22×C8×C47—21no—
56C22×C8×C47—21no—
57C22×C8×C47—22no—
58C22×C8×C47C2×C412yes—
59C22×C8×C47D412yes—
60C22⋊C8×C47—28no—
61C24×C47—21no—
62C24×C47#4111yescharacteristic
63#41×C47—24no—
64#41×C47—24no—
65#41×C47—24no—
66#41×C47—24no—
67#41×C47—42no—
68#41×C47—42no—
69#41×C47—42no—
70#41×C47—42no—
71C2×Q8×C47—21no—
72C2×Q8×C47—22no—
73C2×Q8×C47—22no—
74C2×Q8×C47C4⋊C412yes—
75C2×D4×C47—22no—
76C2×D4×C47—22no—
77C2×D4×C47—22no—
78C2×D4×C47—22no—
79C2×D4×C47—21no—
80C2×D4×C47—42no—
81C2×D4×C47—42no—
82C2×D4×C47C4⋊C412yes—
83C22×C4×C47—21no—
84C22×C4×C47—22no—
85C22×C4×C47—22no—
86C22×C4×C47—41no—
87C22×C4×C47—41no—
88C22×C4×C47—41no—
89C22×C4×C47—41no—
90C22×C4×C47C22×C411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47C2×D411yescharacteristic
94C22×C4×C47C2×D412yes—
95C22×C4×C47C2×D412yes—
96C22×C4×C47C2×Q811yescharacteristic
97C22×C4×C47#4112yes—
98C22×C4×C47#4112yes—
99C22×C4×C47#4111yescharacteristic
100C22×Dic47—24no—
101M16×C47—24no—
102M16×C47—24no—
103C2×C8×C47—22no—
104C2×C8×C47—22no—
105C2×C8×C47—22no—
106C2×C8×C47—24no—
107C2×C8×C47—42no—
108C2×C8×C47—41no—
109C2×C8×C47—41no—
110C2×C8×C47C22⋊C414yes—
111C22⋊Dic47—42no—
112C22⋊Dic47—42no—
113C4⋊Dic47—42no—
114C4⋊Dic47—42no—
115C4×Dic47—24no—
116C4×Dic47—24no—
117#10625329—24no—
118#10625329—24no—
119#10625329—42no—
120#10625329—42no—
121C2×C47⋊C8—24no—
122C2×C47⋊C8—42no—
123C2×C47⋊C8—42no—
124C23×C47—22no—
125C23×C47—42no—
126C23×C47—44no—
127C23×C47—41no—
128C23×C47—41no—
129C23×C47#11711yescharacteristic
130C23×C47#12111yescharacteristic
131C23×C47C22⋊Q811yescharacteristic
132C23×C47#12312yes—
133C23×C47#12412yes—
134Q8×C47—24no—
135Q8×C47—42no—
136D4×C47—24no—
137D4×C47—44no—
138D4×C47—44no—
139D4×C47—42no—
140C2×C4×C47—22no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—42no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
146C2×C4×C47—42no—
147C2×C4×C47C2×C22⋊C411yescharacteristic
148C2×C4×C47C2×C4⋊C411yescharacteristic
149C2×C4×C47C4×D412yes—
150C2×C4×C47C4×D412yes—
151C2×C4×C47#12011yescharacteristic
152C2×C4×C47#12012yes—
153C2×C4×C47#12112yes—
154C2×C4×C47#12112yes—
155C2×C4×C47#12112yes—
156C2×C4×C47#12111yescharacteristic
157C2×C4×C47C22⋊Q812yes—
158C2×C4×C47C22⋊Q812yes—
159C2×C4×C47C22⋊Q811yescharacteristic
160C2×C4×C47#12311yescharacteristic
161C2×C4×C47#12312yes—
162C2×C4×C47#12411yescharacteristic
163C2×C4×C47#12612yes—
164C2×C4×C47#12711yescharacteristic
165C2×Dic47—48no—
166C2×Dic47—44no—
167C8×C47—24no—
168C8×C47—42no—
169C47⋊C8—44no—
170#11112—471nomaximal, Sylow
171C22×C47—24no—
172C22×C47—22no—
173C22×C47—41no—
174C22×C47—41no—
175C22×C47—42no—
176C22×C47—44no—
177C22×C47—42no—
178C22×C47—82no—
179C22×C47—82no—
180C22×C47#38011yescharacteristic
181C22×C47#38812yes—
182C22×C47#39311yescharacteristic
183C22×C47#39412yes—
184C22×C47#39911yescharacteristic
185C4×C47—42no—
186C4×C47#38512yes—
187C4×C47#38611yescharacteristic
188C4×C47#39011yescharacteristic
189C4×C47#39211yescharacteristic
190C4×C47#39612yes—
191C4×C47#39711yescharacteristic
192Dic47—84no—
193C2×#430—471no—
194C2×#421—472no—
195C2×#408D4711yescharacteristic
196C2×H(3,ℤ/4)—471no—
197#1729—472no—
198C2×C47—22no—
199C2×C47—21no—
200C2×C47—21no—
201C2×C47—22no—
202C2×C47—82no—
203C2×C47#143911yescharacteristic
204C2×C47#209812yes—
205C22×#41C2×D4711yescharacteristic
206C22×M16—472no—
207C22×M16C2×D4711yescharacteristic
208C23×C8C2×D4711yescharacteristic
209C2×#117—472no—
210#430—944no—
211C2×#108—942no—
212#421—944no—
213#421—944no—
214C2×#104—944no—
215#408—22no—
216#408Dic4714yescomplemented
217C2×C22⋊C8—22no—
218H(3,ℤ/4)—942no—
219#336—944no—
220C47#1111211yescharacteristic, complemented, Sylow
221C2×#41—21no—
222C2×#41—21no—
223C2×#41—22no—
224C2×#41—22no—
225C2×#41—22no—
226C2×#41—22no—
227C2×#41D18812yes—
228C2×#41C2×Dic4712yes—
229C22×Q8Dic9411yescharacteristic
230C22×D4—21no—
231C22×D4Dic9411yescharacteristic
232C23×C4—21no—
233C23×C4C22×D4711yescharacteristic
234C2×M16—22no—
235C2×M16—22no—
236C2×M16—474no—
237C2×M16—942no—
238C2×M16—942no—
239C2×M16—942no—
240C2×M16—942no—
241C2×M16—942no—
242C2×M16C2×Dic4712yes—
243C2×M16C22⋊D4712yes—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—942no—
249C22×C8C2×Dic4712yes—
250C22×C8C22⋊D4712yes—
251#117—942no—
252#117—942no—
253#117—942no—
254#117—942no—
255C2×C4⋊C4—942no—
256C2×C22⋊C4—942no—
257C2×C42—474no—
258#108—948no—
259#104—9416no—
260C22⋊C8—28no—
261C24—21no—
262C24#1062535011yescharacteristic
263#41—24no—
264#41—24no—
265#41—24no—
266#41—24no—
267#41—42no—
268#41—42no—
269#41—42no—
270#41—42no—
271C2×Q8—21no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8C4⋊Dic4712yes—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—21no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4C4⋊Dic4712yes—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—944no—
291C22×C4C2×Dic9411yescharacteristic
292C22×C4C2×D18811yescharacteristic
293C22×C4#1062535011yescharacteristic
294C22×C4D4×D4712yes—
295C22×C4D4×D4712yes—
296C22×C4#1062535212yes—
297C22×C4#1062535212yes—
298C22×C4C22×Dic4711yescharacteristic
299C22×C4C2×C22⋊D4711yescharacteristic
300C22×C4C2×C22⋊D4711yescharacteristic
301M16—24no—
302M16—24no—
303M16—944no—
304M16—944no—
305M16—1882no—
306M16—1882no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—24no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—944no—
315C2×C8—1882no—
316C2×C8—1882no—
317C2×C8C22⋊Dic4714yes—
318C4⋊C4—1882no—
319C4⋊C4—1882no—
320C22⋊C4—1882no—
321C22⋊C4—1882no—
322C42—944no—
323C42—944no—
324C23—22no—
325C23—42no—
326C23—44no—
327C23—41no—
328C23—41no—
329C23#1309398311yescharacteristic
330C23#1309399811yescharacteristic
331C23#1309400311yescharacteristic
332C23#1309400612yes—
333C23#1309401012yes—
334Q8—24no—
335Q8—42no—
336D4—24no—
337D4—44no—
338D4—44no—
339D4—42no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—1888no—
348C2×C4—1884no—
349C2×C4C2×C22⋊Dic4711yescharacteristic
350C2×C4C2×C4⋊Dic4711yescharacteristic
351C2×C4D4×Dic4712yes—
352C2×C4D4×Dic4712yes—
353C2×C4#1309399211yescharacteristic
354C2×C4#1309399412yes—
355C2×C4#1309399512yes—
356C2×C4#1309399612yes—
357C2×C4#1309399712yes—
358C2×C4#1309399811yescharacteristic
359C2×C4#1309400112yes—
360C2×C4#1309400112yes—
361C2×C4#1309400311yescharacteristic
362C2×C4#1309400511yescharacteristic
363C2×C4#1309400812yes—
364C2×C4#1309401111yescharacteristic
365C2×C4#1309401712yes—
366C2×C4#1309402011yescharacteristic
367C8—24no—
368C8—42no—
369C8—1884no—
370C22—24no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—44no—
376C22—42no—
377C22—82no—
378C22—82no—
379C22#1568898311yescharacteristic
380C22#1568901612yes—
381C22#1568903511yescharacteristic
382C22#1568903912yes—
383C22#1568905911yescharacteristic
384C4—42no—
385C4—3764no—
386C4#1568899812yes—
387C4#1568900311yescharacteristic
388C4#1568902611yescharacteristic
389C4#1568903211yescharacteristic
390C4#1568904612yes—
391C4#1568904811yescharacteristic
392C2—22no—
393C2—21no—
394C2—21no—
395C2—22no—
396C2—82no—
397C2#2024163311yescharacteristic
398C2#2024494112yes—
subgroupG/H
centerC2×C4#13094020
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#408C2
socleC22×C47#399
prime radicalC22×#41×C47C22
cosocleC23×C4×C47C23
hypercenterC2×#408D47
hypocenterC47#11112
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
1. C2×#15689172C2semidirect
3. C2×#15688880C2semidirect
4. #20243084C2semidirect
19. C47×#408C4semidirect
216. #408Dic47semidirect
220. C47#11112semidirect

Maximal subgroups

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

subgroupindex
C2×#156891722
C2×#156891402
C2×#156888802
#202430842
C2×C47×#4082
#1111247