Finite Group Databasebeta

← #26111146

Subgroups — #26111146

Order 12032.

500 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#10625329C211yesmaximal, characteristic, complemented
2#20244308C211yesmaximal, characteristic
3C4×#13093948C211yesmaximal, characteristic, complemented
4#20241839C211yesmaximal, characteristic
5#20241290C211yesmaximal, characteristic, complemented
6#20241224C211yesmaximal, characteristic, complemented
7C47×#3103C211yesmaximal, characteristic
8#15689254—22no—
9#15689254—21no—
10Q8×C47⋊C8—21no—
11#15689147—21no—
12#15689147C2211yescharacteristic
13C2×#13093951—21no—
14C2×#13093951—21no—
15C2×#13093948—21no—
16C4×#10625329—21no—
17C4×#10625329C2211yescharacteristic
18C2×C4×C47⋊C8C2211yescharacteristic
19C4×C4⋊Dic47C2211yescharacteristic
20#15688885—21no—
21D4⋊(C47⋊C8)—22no—
22C47×#532C2211yescharacteristic
23C2×C4×Q8×C47C2211yescharacteristic
24C47×#481—21no—
25C47×#475C411yescharacteristic, complemented
26C47×#475C411yescharacteristic, complemented
27C47×#460C411yescharacteristic, complemented
28C47×#460C411yescharacteristic, complemented
29C47×#448—21no—
30C4×SD16×C47—21no—
31C4×SD16×C47—21no—
32C47×#431C2211yescharacteristic
33C2×#10625329—21no—
34C2×#10625329D411yescharacteristic
35C2×#10625329D411yescharacteristic
36C22×C47⋊C8—21no—
37C2×C4⋊Dic47—21no—
38C2×C4×Dic47—21no—
39C4⋊(C47⋊C8)—22no—
40C4⋊(C47⋊C8)—22no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)D412yes—
43#13093951—42no—
44#13093951—42no—
45#13093948—24no—
46#13093935—22no—
47#13093935—21no—
48C4×C47⋊C8—22no—
49C4×C47⋊C8—21no—
50C4×C47⋊C8—21no—
51C4×C47⋊C8—21no—
52C47⋊#95—21no—
53C2×#41×C47—21no—
54C22×Q8×C47D411yescharacteristic
55C2×SD16×C47—21no—
56C2×SD16×C47—21no—
57C4⋊Q8×C47C2×C411yescharacteristic
58#127×C47C2×C411yescharacteristic
59#124×C47—21no—
60#121×C47—21no—
61#121×C47—21no—
62C4×Q8×C47—21no—
63C4×Q8×C47—21no—
64C4×Q8×C47C2×C411yescharacteristic
65C4×Q8×C47C2×C411yescharacteristic
66C4×Q8×C47D411yescharacteristic
67C4×Q8×C47D411yescharacteristic
68C4×D4×C47—21no—
69C4×D4×C47—21no—
70C2×C4⋊C4×C47—21no—
71C2×C4⋊C4×C47D411yescharacteristic
72C2×C42×C47—21no—
73C2×C42×C47C2311yescharacteristic
74#106×C47—21no—
75#106×C47—21no—
76C4⋊C8×C47C2×C411yescharacteristic
77C4⋊C8×C47C2×C411yescharacteristic
78#103×C47—21no—
79#103×C47—21no—
80#102×C47—21no—
81#102×C47—21no—
82C22⋊C8×C47—21no—
83C4×C8×C47—21no—
84#41×C47—41no—
85#41×C47—41no—
86#41×C47—41no—
87#41×C47—41no—
88C2×Q8×C47—21no—
89C2×Q8×C47—22no—
90C2×Q8×C47—21no—
91C2×Q8×C47—41no—
92C2×Q8×C47C22⋊C411yescharacteristic
93C2×Q8×C47C22⋊C411yescharacteristic
94C2×D4×C47—21no—
95C2×D4×C47—21no—
96C2×D4×C47—21no—
97C2×D4×C47—41no—
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—41no—
101C22×C4×C47C2×D411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47#4111yescharacteristic
104C22×Dic47—41no—
105SD16×C47—42no—
106SD16×C47—42no—
107C2×C8×C47—21no—
108C2×C8×C47—21no—
109C4⋊C4×C47—21no—
110C4⋊C4×C47—21no—
111C4⋊C4×C47—21no—
112C4⋊C4×C47—21no—
113C4⋊C4×C47—21no—
114C4⋊C4×C47—21no—
115C4⋊C4×C47—21no—
116C4⋊C4×C47C22⋊C411yescharacteristic
117C4⋊C4×C47C22⋊C411yescharacteristic
118C22⋊C4×C47—41no—
119C22⋊C4×C47—41no—
120C42×C47—21no—
121C42×C47—21no—
122C42×C47—21no—
123C42×C47—21no—
124C42×C47C22×C411yescharacteristic
125C42×C47C2×D411yescharacteristic
126C42×C47C2×D411yescharacteristic
127C42×C47#4111yescharacteristic
128C4⋊Dic47—22no—
129C4⋊Dic47—41no—
130C4×Dic47—41no—
131C4×Dic47—41no—
132#10625329—21no—
133#10625329—21no—
134#10625329—21no—
135#10625329—41no—
136#10625329—41no—
137#10625329SD1612yescomplemented
138C2×C47⋊C8—21no—
139C2×C47⋊C8—22no—
140C2×C47⋊C8—21no—
141C2×C47⋊C8—22no—
142C2×C47⋊C8—41no—
143C2×C47⋊C8—41no—
144C23×C47—41no—
145C23×C47#12111yescharacteristic
146Q8×C47—21no—
147Q8×C47—21no—
148Q8×C47—21no—
149Q8×C47—42no—
150Q8×C47—41no—
151Q8×C47—41no—
152Q8×C47#10412yescomplemented
153D4×C47—42no—
154D4×C47—41no—
155D4×C47—41no—
156D4×C47—41no—
157D4×C47—41no—
158D4×C47—81no—
159D4×C47—81no—
160C2×C4×C47—21no—
161C2×C4×C47—21no—
162C2×C4×C47—21no—
163C2×C4×C47—21no—
164C2×C4×C47—21no—
165C2×C4×C47—21no—
166C2×C4×C47—41no—
167C2×C4×C47—42no—
168C2×C4×C47—41no—
169C2×C4×C47—81no—
170C2×C4×C47—81no—
171C2×C4×C47—81no—
172C2×C4×C47C2×C22⋊C411yescharacteristic
173C2×C4×C47C4×D411yescharacteristic
174C2×C4×C47C4×D411yescharacteristic
175C2×C4×C47#12011yescharacteristic
176C2×C4×C47#12111yescharacteristic
177C2×C4×C47#12311yescharacteristic
178C2×C4×C47C2×SD1611yescharacteristic
179C2×C4×C47#13511yescharacteristic
180C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C47#13711yescharacteristic
182C2×Dic47—42no—
183C2×Dic47—81no—
184C2×Dic47—81no—
185C8×C47—41no—
186C8×C47—41no—
187C47⋊C8—22no—
188C47⋊C8—42no—
189C47⋊C8—41no—
190C47⋊C8—42no—
191#8992—471nomaximal, Sylow
192C22×C47—21no—
193C22×C47—21no—
194C22×C47—81no—
195C22×C47—81no—
196C22×C47—81no—
197C22×C47#38611yescharacteristic
198C22×C47#46011yescharacteristic
199C22×C47#46411yescharacteristic
200C4×C47—21no—
201C4×C47—22no—
202C4×C47—21no—
203C4×C47—21no—
204C4×C47—41no—
205C4×C47—41no—
206C4×C47—41no—
207C4×C47—81no—
208C4×C47C2×#10411yescharacteristic
209C4×C47#42111yescharacteristic
210C4×C47C4×SD1611yescharacteristic
211C4×C47#44011yescharacteristic
212C4×C47#44811yescharacteristic
213C4×C47#44911yescharacteristic
214Dic47—82no—
215#3103D4711yescharacteristic
216Q8×M16—471no—
217#1980—471no—
218C4×#103—471no—
219#1492—471no—
220#1303—471no—
221#1278—471no—
222C2×C47—21no—
223C2×C47—21no—
224C2×C47—161no—
225C2×C47#187111yescharacteristic
226C2×C47#188011yescharacteristic
227C2×C47#188111yescharacteristic
228#532C2×D4711yescharacteristic
229C2×C4×Q8C2×D4711yescharacteristic
230#481—21no—
231#475Dic4711yescharacteristic, complemented
232#475Dic4711yescharacteristic, complemented
233#460Dic4711yescharacteristic, complemented
234#460Dic4711yescharacteristic, complemented
235#448—21no—
236#446—942no—
237#446—941no—
238C8×Q8—941no—
239C4×SD16—21no—
240C4×SD16—21no—
241#431C2×D4711yescharacteristic
242#423—471no—
243#423—941no—
244C2×#104—941no—
245C2×#104—941no—
246C2×#103—941no—
247C4×M16—471no—
248C4×M16—941no—
249C2×C4×C8—471no—
250C4×C4⋊C4—471no—
251#340—941no—
252D4⋊C8—942no—
253C47#899211yescharacteristic, complemented, Sylow
254C2×#41—21no—
255C22×Q8C22⋊D4711yescharacteristic
256C2×SD16—21no—
257C2×SD16—21no—
258C2×M16—471no—
259C2×M16—471no—
260C2×M16—941no—
261C22×C8—941no—
262C4⋊Q8C2×Dic4711yescharacteristic
263#127C2×Dic4711yescharacteristic
264#124—21no—
265#121—21no—
266#121—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C2×Dic4711yescharacteristic
270C4×Q8C2×Dic4711yescharacteristic
271C4×Q8C22⋊D4711yescharacteristic
272C4×Q8C22⋊D4711yescharacteristic
273C4×D4—21no—
274C4×D4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—941no—
277C2×C4⋊C4C22⋊D4711yescharacteristic
278C2×C42—21no—
279C2×C42—941no—
280C2×C42C22×D4711yescharacteristic
281#106—21no—
282#106—21no—
283C4⋊C8—472no—
284C4⋊C8—942no—
285C4⋊C8—942no—
286C4⋊C8—941no—
287C4⋊C8C2×Dic4711yescharacteristic
288C4⋊C8C2×Dic4711yescharacteristic
289#104—1882no—
290#104—1882no—
291#103—21no—
292#103—21no—
293#103—944no—
294#102—21no—
295#102—21no—
296C22⋊C8—21no—
297#97—942no—
298#97—941no—
299C4×C8—21no—
300C4×C8—942no—
301C4×C8—941no—
302C4×C8—941no—
303C4×C8—941no—
304#95—941no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×Q8—21no—
312C2×Q8—41no—
313C2×Q8C22⋊Dic4711yescharacteristic
314C2×Q8C22⋊Dic4711yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—1881no—
323C22×C4D4×D4711yescharacteristic
324C22×C4#1062535211yescharacteristic
325C22×C4C2×C22⋊D4711yescharacteristic
326SD16—42no—
327SD16—42no—
328M16—472no—
329M16—941no—
330M16—941no—
331M16—941no—
332M16—1881no—
333M16—1881no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—941no—
337C2×C8—942no—
338C2×C8—942no—
339C2×C8—941no—
340C2×C8—1881no—
341C2×C8—1881no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—942no—
350C4⋊C4—1881no—
351C4⋊C4C22⋊Dic4711yescharacteristic
352C4⋊C4C22⋊Dic4711yescharacteristic
353C22⋊C4—41no—
354C22⋊C4—41no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—1881no—
360C42—1881no—
361C42D4×D4711yescharacteristic
362C42#1062535211yescharacteristic
363C42C22×Dic4711yescharacteristic
364C42C2×C22⋊D4711yescharacteristic
365C23—41no—
366C23#1309399611yescharacteristic
367Q8—21no—
368Q8—21no—
369Q8—21no—
370Q8—42no—
371Q8—41no—
372Q8—41no—
373Q8#1309395112yescomplemented
374D4—42no—
375D4—41no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—81no—
380D4—81no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—3761no—
394C2×C4—3761no—
395C2×C4—1882no—
396C2×C4C2×C22⋊Dic4711yescharacteristic
397C2×C4D4×Dic4711yescharacteristic
398C2×C4D4×Dic4711yescharacteristic
399C2×C4#1309399211yescharacteristic
400C2×C4#1309399611yescharacteristic
401C2×C4#1309400511yescharacteristic
402C2×C4SD16×D4711yescharacteristic
403C2×C4#1309404511yescharacteristic
404C2×C4#1309405111yescharacteristic
405C2×C4#1309405311yescharacteristic
406C8—41no—
407C8—41no—
408C8—942no—
409C8—1882no—
410C8—1881no—
411C8—1882no—
412C22—21no—
413C22—21no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22#1568900311yescharacteristic
418C22#1568932111yescharacteristic
419C22#1568934311yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—81no—
428C4—3762no—
429C4C2×#1309395111yescharacteristic
430C4#1568914011yescharacteristic
431C4SD16×Dic4711yescharacteristic
432C4#1568922411yescharacteristic
433C4#1568926011yescharacteristic
434C4#1568926711yescharacteristic
435C2—21no—
436C2—21no—
437C2—161no—
438C2#2024369111yescharacteristic
439C2#2024374511yescharacteristic
440C2#2024375211yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3103C2
socleC22×C47#386
prime radicalC47×#532C22
cosocleC2×C42×C47C23
hypercenter#3103D47
hypocenterC47#8992
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
1. Q8×#10625329C2semidirect
3. C4×#13093948C2semidirect
5. #20241290C2semidirect
6. #20241224C2semidirect
25. C47×#475C4semidirect
26. C47×#475C4semidirect
27. C47×#460C4semidirect
28. C47×#460C4semidirect
137. #10625329SD16semidirect
152. Q8×C47#104semidirect
231. #475Dic47semidirect
232. #475Dic47semidirect
233. #460Dic47semidirect
234. #460Dic47semidirect
253. C47#8992semidirect
373. Q8#13093951semidirect

Maximal subgroups

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

subgroupindex
Q8×#106253292
#202443082
C4×#130939482
#202418392
#202412902
#202412242
C47×#31032
#899247