Finite Group Databasebeta

← #26134610

Subgroups — #26134610

Order 12032.

490 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 107 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×#15688922C212yesmaximal, complemented
2C2×#15688920C212yesmaximal
3#20243384C211yesmaximal, characteristic, complemented
4C4×#13093943C211yesmaximal, characteristic, complemented
5C47×#2545C211yesmaximal, characteristic
6#15689147—21no—
7C2×#13093943C2212yes—
8C2×#13093943C2212yes—
9C4×#10625329—21no—
10#15688922—24no—
11#15688920—24no—
12#15688892—21no—
13#15688892—21no—
14C2×C4⋊Q8×C47C2211yescharacteristic
15C2×#124×C47C411yescharacteristic
16C2×#124×C47C2211yescharacteristic
17C2×C4×Q8×C47C2211yescharacteristic
18C47×#396—21no—
19C47×#396—21no—
20C47×#391C411yescharacteristic
21C47×#390C411yescharacteristic
22C47×#388—21no—
23C47×#388—21no—
24C4×C4⋊C4×C47C411yescharacteristic
25C2×#10625329—22no—
26C2×#10625329—22no—
27C4⋊(C47⋊C8)—41no—
28C4⋊(C47⋊C8)—41no—
29#13093943—22no—
30#13093943—22no—
31#13093943—22no—
32#13093943D414yes—
33#13093935—41no—
34C4×C47⋊C8—41no—
35C22×Q8×C47C2×C411yescharacteristic
36C22×Q8×C47C2311yescharacteristic
37C22×D4×C47C2×C411yescharacteristic
38C4⋊Q8×C47—22no—
39C4⋊Q8×C47—22no—
40#124×C47—22no—
41#124×C47—22no—
42#124×C47—41no—
43#124×C47—41no—
44C4×Q8×C47—41no—
45C4×Q8×C47—41no—
46C2×C4⋊C4×C47—21no—
47C2×C4⋊C4×C47—21no—
48C2×C4⋊C4×C47—21no—
49C2×C4⋊C4×C47C2×C411yescharacteristic
50C2×C4⋊C4×C47D411yescharacteristic
51C2×C4⋊C4×C47D411yescharacteristic
52C2×C22⋊C4×C47—21no—
53C2×C22⋊C4×C47—21no—
54C2×C22⋊C4×C47—21no—
55C2×C22⋊C4×C47—21no—
56C2×C22⋊C4×C47—21no—
57C2×C22⋊C4×C47—21no—
58C2×C42×C47C2×C411yescharacteristic
59C2×C42×C47C2×C411yescharacteristic
60C2×C42×C47C2×C411yescharacteristic
61C2×C42×C47D411yescharacteristic
62C2×C42×C47D411yescharacteristic
63#95×C47—21no—
64#95×C47—21no—
65#95×C47—21no—
66C24×C47—21no—
67C2×Q8×C47—22no—
68C2×Q8×C47—41no—
69C2×Q8×C47—41no—
70C2×Q8×C47—41no—
71C2×Q8×C47—41no—
72C2×Q8×C47C2×D412yes—
73C2×Q8×C47#4112yes—
74C2×D4×C47—22no—
75C2×D4×C47—41no—
76C2×D4×C47—41no—
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47—21no—
80C22×C4×C47C22⋊C411yescharacteristic
81C22×C4×C47C22⋊C411yescharacteristic
82C22×C4×C47C22⋊C411yescharacteristic
83C22×C4×C47C22⋊C411yescharacteristic
84C22×C4×C47C22×C411yescharacteristic
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C4⋊C4×C47—41no—
88C4⋊C4×C47—41no—
89C4⋊C4×C47—41no—
90C4⋊C4×C47—42no—
91C4⋊C4×C47—42no—
92C4⋊C4×C47—41no—
93C4⋊C4×C47—41no—
94C22⋊C4×C47—42no—
95C22⋊C4×C47—42no—
96C22⋊C4×C47—42no—
97C22⋊C4×C47—41no—
98C22⋊C4×C47—41no—
99C22⋊C4×C47—42no—
100C22⋊C4×C47—41no—
101C22⋊C4×C47—41no—
102C42×C47—21no—
103C42×C47—21no—
104C42×C47—21no—
105C42×C47—21no—
106C42×C47—21no—
107C42×C47—21no—
108C42×C47—22no—
109C42×C47—22no—
110#10625329—24no—
111#10625329—42no—
112#10625329—42no—
113#10625329—42no—
114C2×C47⋊C8—42no—
115C2×C47⋊C8—42no—
116C23×C47—41no—
117C23×C47—41no—
118C23×C47—41no—
119C23×C47—41no—
120C23×C47—41no—
121C23×C47—41no—
122C23×C47—41no—
123C23×C47C2×C22⋊C411yescharacteristic
124Q8×C47—42no—
125Q8×C47—42no—
126Q8×C47—82no—
127D4×C47—82no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—22no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—21no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47#9911yescharacteristic
151C2×C4×C47#9911yescharacteristic
152C2×C4×C47C4×D412yes—
153C2×C4×C47#12112yes—
154C2×C4×C47#12411yescharacteristic
155C2×C4×C47#12711yescharacteristic
156C47⋊C8—44no—
157C47⋊C8—82no—
158#13228—471nomaximal, Sylow
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—81no—
162C22×C47—81no—
163C22×C47—81no—
164C22×C47—81no—
165C22×C47—81no—
166C22×C47—81no—
167C22×C47—81no—
168C22×C47#39011yescharacteristic
169C22×C47C2×#9911yescharacteristic
170C22×C47#41011yescharacteristic
171C4×C47—21no—
172C4×C47—21no—
173C4×C47—22no—
174C4×C47—21no—
175C4×C47—22no—
176C4×C47—21no—
177C4×C47—42no—
178C4×C47—41no—
179C4×C47—41no—
180C4×C47—41no—
181C4×C47—81no—
182C4×C47—81no—
183#2545D4711yescharacteristic
184C2×#356—472no—
185C2×#355—472no—
186#1801—471no—
187C4×#101—471no—
188C2×C47—21no—
189C2×C47—21no—
190C2×C47—161no—
191C2×C47#196411yescharacteristic
192C2×C47#213211yescharacteristic
193C2×C47#213311yescharacteristic
194C2×C4⋊Q8C2×D4711yescharacteristic
195C2×#124Dic4711yescharacteristic
196C2×#124C2×D4711yescharacteristic
197C2×C4×Q8C2×D4711yescharacteristic
198#423—941no—
199C2×#101—472no—
200C2×#101—472no—
201C4×M16—941no—
202#396—21no—
203#396—21no—
204#391Dic4711yescharacteristic
205#390Dic4711yescharacteristic
206#388—21no—
207#388—21no—
208C4×C4⋊C4Dic4711yescharacteristic
209#356—944no—
210#355—944no—
211#343—941no—
212#343—941no—
213C47#1322811yescharacteristic, complemented, Sylow
214C22×Q8C2×Dic4711yescharacteristic
215C22×Q8C22×D4711yescharacteristic
216C22×D4C2×Dic4711yescharacteristic
217C2×M16—942no—
218C2×M16—942no—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221#124—22no—
222#124—22no—
223#124—41no—
224#124—41no—
225C4×Q8—41no—
226C4×Q8—41no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C2×Dic4711yescharacteristic
231C2×C4⋊C4C22⋊D4711yescharacteristic
232C2×C4⋊C4C22⋊D4711yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C42C2×Dic4711yescharacteristic
240C2×C42C2×Dic4711yescharacteristic
241C2×C42C2×Dic4711yescharacteristic
242C2×C42C22⋊D4711yescharacteristic
243C2×C42C22⋊D4711yescharacteristic
244C4⋊C8—1881no—
245C4⋊C8—1881no—
246#101—474no—
247#101—942no—
248#101—942no—
249#101—942no—
250#97—1881no—
251C4×C8—1881no—
252#95—21no—
253#95—21no—
254#95—21no—
255C24—21no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8D4×D4712yes—
262C2×Q8#1062535212yes—
263C2×D4—22no—
264C2×D4—41no—
265C2×D4—41no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4C22⋊Dic4711yescharacteristic
270C22×C4C22⋊Dic4711yescharacteristic
271C22×C4C22⋊Dic4711yescharacteristic
272C22×C4C22⋊Dic4711yescharacteristic
273C22×C4C22×Dic4711yescharacteristic
274C22×C4C2×C22⋊D4711yescharacteristic
275C22×C4C2×C22⋊D4711yescharacteristic
276M16—944no—
277M16—1882no—
278M16—1882no—
279M16—1882no—
280C2×C8—1882no—
281C2×C8—1882no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—22no—
304C42—22no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23C2×C22⋊Dic4711yescharacteristic
313Q8—42no—
314Q8—42no—
315Q8—82no—
316D4—82no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4#1309393911yescharacteristic
340C2×C4#1309393911yescharacteristic
341C2×C4D4×Dic4712yes—
342C2×C4#1309399512yes—
343C2×C4#1309401111yescharacteristic
344C2×C4#1309402011yescharacteristic
345C8—3762no—
346C8—1884no—
347C22—21no—
348C22—21no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#1568902611yescharacteristic
357C22C2×#1309393911yescharacteristic
358C22#1568909711yescharacteristic
359C4—21no—
360C4—22no—
361C4—22no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—81no—
370C4—81no—
371C2—21no—
372C2—21no—
373C2—161no—
374C2#2024422511yescharacteristic
375C2#2024507311yescharacteristic
376C2#2024507711yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC47×#2545C2
socleC22×C47C2×#99
prime radicalC22×D4×C47C2×C4
cosocleC22×Q8×C47C23
hypercenter#2545D47
hypocenterC47#13228
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2545

Complements

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

NH ≅ G/Ntype
1. C2×#15688922C2semidirect
3. #20243384C2semidirect
4. C4×#13093943C2semidirect
213. C47#13228semidirect

Maximal subgroups

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

subgroupindex
C2×#156889222
C2×#156889202
#202433842
C4×#130939432
C47×#25452
#1322847