Finite Group Databasebeta

← #26112171

Subgroups — #26112171

Order 12032.

562 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252868C211yesmaximal, characteristic
2#20242057C211yesmaximal, characteristic, complemented
3C2×#15688866C211yesmaximal, characteristic, complemented
4#20241775C211yesmaximal, characteristic, complemented
5#20241285C211yesmaximal, characteristic, complemented
6#20241262C211yesmaximal, characteristic, complemented
7C47×#2906C211yesmaximal, characteristic, complemented
8C2×#13094049—21no—
9C2×#13094021C2211yescharacteristic
10#15689323—21no—
11#15689323C412yes—
12#15689318—21no—
13#15689318C412yes—
14#15689176C2211yescharacteristic
15#15689147C2211yescharacteristic
16C2×C4⋊(C47⋊C8)C2211yescharacteristic
17#15689127—21no—
18#15688866—21no—
19#15688866—21no—
20#15688866—22no—
21Q8⋊(C47⋊C8)—21no—
22D4⋊(C47⋊C8)—21no—
23C2×#131×C47—21no—
24C4×#41×C47C2211yescharacteristic
25C47×#424—21no—
26C47×#424—22no—
27C47×#423—21no—
28C47×#423C2211yescharacteristic
29C2×C4⋊C8×C47—21no—
30C2×C4⋊C8×C47C2211yescharacteristic
31C22×D188—21no—
32C22×D188—21no—
33C22×D188—21no—
34#13094049—42no—
35#13094049—42no—
36C2×#10625336—22no—
37C2×#10625334—22no—
38C2×#10625329—21no—
39C22×C47⋊C8—21no—
40#13094021—21no—
41#13094021—21no—
42#13094021—21no—
43#13094021C2×C412yes—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)—21no—
47C4⋊(C47⋊C8)—21no—
48C4⋊(C47⋊C8)C2×C412yes—
49#13093944—22no—
50#13093944—22no—
51C22⋊(C47⋊C8)—21no—
52C4×C47⋊C8—21no—
53C2×#41×C47D411yescharacteristic
54#131×C47—42no—
55#131×C47—42no—
56C2×M16×C47—22no—
57C2×M16×C47—21no—
58C22×C8×C47—22no—
59C22×C8×C47—21no—
60C4×Q8×C47C2×C411yescharacteristic
61C4×Q8×C47D411yescharacteristic
62C4×D4×C47—21no—
63C4×D4×C47—21no—
64C4×D4×C47C2×C411yescharacteristic
65C4×D4×C47D411yescharacteristic
66#117×C47—21no—
67#117×C47D411yescharacteristic
68C2×C42×C47—21no—
69C2×C42×C47C2311yescharacteristic
70C4⋊C8×C47—22no—
71C4⋊C8×C47—22no—
72C4⋊C8×C47—21no—
73C4⋊C8×C47—21no—
74C4⋊C8×C47D411yescharacteristic
75C4⋊C8×C47D411yescharacteristic
76C4⋊C8×C47D412yes—
77C23×D47—41no—
78#41×C47—21no—
79#41×C47—21no—
80#41×C47D812yes—
81#41×C47SD1612yes—
82C2×Q8×C47C22⋊C411yescharacteristic
83C2×D4×C47—21no—
84C2×D4×C47C22⋊C411yescharacteristic
85C22×C4×C47—21no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47#4111yescharacteristic
90C22×C4×C47#4111yescharacteristic
91C2×D188—22no—
92C2×D188—22no—
93C2×D188—22no—
94C2×D188—41no—
95C2×D188—41no—
96C2×D188—41no—
97C2×D188—41no—
98C2×D188—41no—
99C2×D188—41no—
100C2×D188—41no—
101C2×D188—82no—
102C2×D188—81no—
103C2×D188—81no—
104M16×C47—44no—
105M16×C47—41no—
106M16×C47—41no—
107C2×C8×C47—21no—
108C2×C8×C47—22no—
109C2×C8×C47—21no—
110C2×C8×C47—44no—
111C2×C8×C47—41no—
112C2×C8×C47—41no—
113C4⋊C4×C47—21no—
114C4⋊C4×C47—21no—
115C4⋊C4×C47C22⋊C411yescharacteristic
116C4⋊C4×C47C22⋊C411yescharacteristic
117C22⋊C4×C47—21no—
118C22⋊C4×C47—21no—
119C22⋊C4×C47—21no—
120C42×C47—21no—
121C42×C47—21no—
122C42×C47—21no—
123C42×C47C22×C411yescharacteristic
124C42×C47C2×D411yescharacteristic
125C42×C47C2×D411yescharacteristic
126C42×C47C2×D411yescharacteristic
127#10625336—44no—
128#10625334—44no—
129#10625329—41no—
130#10625329—41no—
131C2×C47⋊C8—22no—
132C2×C47⋊C8—41no—
133C2×C47⋊C8—41no—
134C2×C47⋊C8—81no—
135C23×C47—21no—
136C23×C47#12311yescharacteristic
137C22×D47—42no—
138C22×D47—82no—
139C22×D47—82no—
140C22×D47—81no—
141C22×D47—81no—
142Q8×C47—21no—
143Q8×C47#10212yes—
144D4×C47—22no—
145D4×C47—22no—
146D4×C47—21no—
147D4×C47#10212yes—
148C2×C4×C47—21no—
149C2×C4×C47—22no—
150C2×C4×C47—22no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—21no—
154C2×C4×C47—21no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—81no—
159C2×C4×C47C2×C22⋊C411yescharacteristic
160C2×C4×C47C4×D411yescharacteristic
161C2×C4×C47C4×D411yescharacteristic
162C2×C4×C47#12011yescharacteristic
163C2×C4×C47#12111yescharacteristic
164C2×C4×C47#12111yescharacteristic
165C2×C4×C47C2×D811yescharacteristic
166C2×C4×C47C2×SD1611yescharacteristic
167C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C47#13711yescharacteristic
169D188—42no—
170D188—42no—
171D188—82no—
172D188—82no—
173D188—82no—
174D188—81no—
175D188—81no—
176D188—81no—
177D188—81no—
178C8×C47—41no—
179C8×C47—42no—
180C8×C47—41no—
181C47⋊C8—42no—
182C47⋊C8—81no—
183C47⋊C8—81no—
184#9173—471nomaximal, Sylow
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—22no—
188C22×C47—41no—
189C22×C47—41no—
190C22×C47#38611yescharacteristic
191C22×C47#48011yescharacteristic
192C22×C47#48111yescharacteristic
193C2×D47—82no—
194C2×D47—82no—
195C2×D47—82no—
196C2×D47—162no—
197C2×D47—162no—
198C4×C47—22no—
199C4×C47—21no—
200C4×C47—21no—
201C4×C47—21no—
202C4×C47—41no—
203C4×C47—41no—
204C4×C47C2×#10211yescharacteristic
205C4×C47#41711yescharacteristic
206C4×C47#44711yescharacteristic
207C4×C47#44811yescharacteristic
208C4×C47#47111yescharacteristic
209C4×C47#47211yescharacteristic
210#3038—471no—
211#2906D4711yescharacteristic, complemented
212#1545—471no—
213C2×#331—471no—
214#1477—471no—
215#1301—471no—
216#1292—471no—
217C2×C47—21no—
218C2×C47—21no—
219C2×C47—42no—
220C2×C47#187411yescharacteristic
221C2×C47#190311yescharacteristic
222C2×C47#190711yescharacteristic
223D47—162no—
224C2×AGL(1,ℤ/8)—941no—
225C2×#131—21no—
226C2×#127—471no—
227C4×#41C2×D4711yescharacteristic
228#460—472no—
229#460—941no—
230#459—472no—
231#459—941no—
232#431—471no—
233#424—21no—
234#424—22no—
235#423—21no—
236#423—471no—
237#423C2×D4711yescharacteristic
238C2×C4⋊C8—21no—
239C2×C4⋊C8—471no—
240C2×C4⋊C8C2×D4711yescharacteristic
241#418—941no—
242#331—941no—
243#331—941no—
244#331—942no—
245Q8⋊C8—941no—
246D4⋊C8—941no—
247C47#917311yescharacteristic, complemented, Sylow
248C2×#41C22⋊D4711yescharacteristic
249C22×D4—941no—
250C22×D4—941no—
251C22×D4—941no—
252AGL(1,ℤ/8)—1882no—
253AGL(1,ℤ/8)—1882no—
254C2×SD16—942no—
255C2×D8—942no—
256#131—42no—
257#131—42no—
258C2×M16—22no—
259C2×M16—21no—
260C2×M16—941no—
261C22×C8—22no—
262C22×C8—21no—
263C22×C8—941no—
264#127—472no—
265#127—941no—
266#127—941no—
267#127—941no—
268C4×Q8C4×D4711yescharacteristic
269C4×Q8D18811yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4C4×D4711yescharacteristic
273C4×D4D18811yescharacteristic
274#117—21no—
275#117C22⋊D4711yescharacteristic
276C2×C42—21no—
277C2×C42C22×D4711yescharacteristic
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—472no—
283C4⋊C8—941no—
284C4⋊C8—941no—
285C4⋊C8—941no—
286C4⋊C8—941no—
287C4⋊C8C22⋊D4711yescharacteristic
288C4⋊C8C22⋊D4711yescharacteristic
289C4⋊C8C22⋊D4712yes—
290#102—942no—
291#102—942no—
292C22⋊C8—941no—
293C4×C8—941no—
294C24—1881no—
295#41—21no—
296#41—21no—
297#41#1062533412yes—
298#41#1062533612yes—
299C2×Q8(C2×D47)⋊C411yescharacteristic
300C2×D4—21no—
301C2×D4—942no—
302C2×D4—942no—
303C2×D4—942no—
304C2×D4—3761no—
305C2×D4—3761no—
306C2×D4—3762no—
307C2×D4—1881no—
308C2×D4—1881no—
309C2×D4—1881no—
310C2×D4—1881no—
311C2×D4—1881no—
312C2×D4—1881no—
313C2×D4—1881no—
314C2×D4(C2×D47)⋊C411yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4#1062535011yescharacteristic
319C22×C4#1062535011yescharacteristic
320C22×C4C2×C22⋊D4711yescharacteristic
321SD16—1884no—
322D8—1884no—
323M16—44no—
324M16—41no—
325M16—41no—
326M16—1881no—
327M16—1881no—
328C2×C8—21no—
329C2×C8—22no—
330C2×C8—21no—
331C2×C8—44no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—942no—
335C2×C8—3761no—
336C2×C8—1881no—
337C2×C8—1881no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4(C2×D47)⋊C411yescharacteristic
341C4⋊C4(C2×D47)⋊C411yescharacteristic
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42C2×C4×D4711yescharacteristic
349C42C2×D18811yescharacteristic
350C42C2×C22⋊D4711yescharacteristic
351C42C2×C22⋊D4711yescharacteristic
352C23—21no—
353C23—3762no—
354C23—3762no—
355C23—3761no—
356C23—3761no—
357C23—1882no—
358C23#1309400711yescharacteristic
359Q8—21no—
360Q8#1309394412yes—
361D4—22no—
362D4—22no—
363D4—21no—
364D4—3762no—
365D4—3762no—
366D4—3762no—
367D4—3761no—
368D4—3761no—
369D4—3761no—
370D4—3761no—
371D4—1882no—
372D4—1882no—
373D4#1309394412yes—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—81no—
385C2×C4C2×(C2×D47)⋊C411yescharacteristic
386C2×C4C4×C22⋊D4711yescharacteristic
387C2×C4C4×C22⋊D4711yescharacteristic
388C2×C4#1309399311yescharacteristic
389C2×C4#1309399811yescharacteristic
390C2×C4#1309399811yescharacteristic
391C2×C4C2×#1062533411yescharacteristic
392C2×C4C2×#1062533611yescharacteristic
393C2×C4#1309405011yescharacteristic
394C2×C4#1309405511yescharacteristic
395C8—41no—
396C8—42no—
397C8—41no—
398C8—3761no—
399C8—3761no—
400C8—1882no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—3762no—
407C22—3762no—
408C22—3762no—
409C22—7522no—
410C22—7522no—
411C22#1568900211yescharacteristic
412C22#1568943611yescharacteristic
413C22#1568944111yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—22no—
418C4—41no—
419C4—41no—
420C4C2×#1309394411yescharacteristic
421C4#1568912211yescharacteristic
422C4#1568925711yescharacteristic
423C4#1568926211yescharacteristic
424C4#1568938611yescharacteristic
425C4#1568939311yescharacteristic
426C2—21no—
427C2—21no—
428C2—42no—
429C2—7522no—
430C2#2024370511yescharacteristic
431C2#2024386811yescharacteristic
432C2#2024389611yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2906C2
socleC22×C47#386
prime radical#20252868C2
cosocleC2×C42×C47C23
hypercenter#2906D47
hypocenterC47#9173
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2906

Complements

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

NH ≅ G/Ntype
2. #20242057C2semidirect
3. C2×#15688866C2semidirect
4. #20241775C2semidirect
5. #20241285C2semidirect
6. #20241262C2semidirect
7. C47×#2906C2semidirect
211. #2906D47semidirect
247. C47#9173semidirect

Maximal subgroups

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

subgroupindex
#202528682
#202420572
C2×#156888662
#202417752
#202412852
#202412622
C47×#29062
#917347