Finite Group Databasebeta

← #35300676

Subgroups — #35300676

Order 21248.

604 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917494C211yesmaximal, characteristic
2#24908291C211yesmaximal, characteristic
3#24905957C211yesmaximal, characteristic
4#24905892C211yesmaximal, characteristic, complemented
5#24905366C211yesmaximal, characteristic, complemented
6#24905296C211yesmaximal, characteristic, complemented
7C42×Q8×C83C211yesmaximal, characteristic, complemented
8#19128251C2211yescharacteristic
9#19128019—21no—
10#19127986C412yescomplemented
11#19127901C412yescomplemented
12#19127840—21no—
13C4×#13050577—22no—
14#19127754C2211yescharacteristic
15#19127754C2211yescharacteristic
16#19127725C2211yescharacteristic
17C2×#14581720—21no—
18C2×#14581720—21no—
19#19127434—22no—
20Q8⋊(C83⋊C8)—22no—
21C2×C4×Q8×C83—21no—
22C2×C4×Q8×C83C2211yescharacteristic
23C4×C4⋊C4×C83—21no—
24C4×C4⋊C4×C83—21no—
25C4×C4⋊C4×C83—22no—
26C4×C4⋊C4×C83C2211yescharacteristic
27C43×C83—21no—
28C43×C83C2211yescharacteristic
29C2×#13050590—21no—
30C2×#13050577—22no—
31C2×#13050574—21no—
32#14581792C2×C411yescharacteristic
33#14581789C2×C411yescharacteristic
34#14581780—21no—
35#14581766—21no—
36#14581766—21no—
37C4×D332—21no—
38C4×D332—21no—
39#14581723—22no—
40C4⋊(C83⋊C8)—21no—
41C4⋊(C83⋊C8)—21no—
42C4⋊(C83⋊C8)—21no—
43C4⋊(C83⋊C8)C2×C412yes—
44#14581720—42no—
45#14581720—42no—
46#14581716—22no—
47#14581712—22no—
48C22⋊(C83⋊C8)—21no—
49C22⋊(C83⋊C8)—21no—
50C4×C83⋊C8—21no—
51C4×C83⋊C8—21no—
52C22×Q8×C83D411yescharacteristic
53C4×Q8×C83—28no—
54C4×Q8×C83—21no—
55C4×Q8×C83—21no—
56C4×Q8×C83C2×C412yes—
57C4×Q8×C83D412yes—
58C2×C4⋊C4×C83—22no—
59C2×C4⋊C4×C83—21no—
60C2×C4⋊C4×C83—21no—
61C2×C4⋊C4×C83D411yescharacteristic
62C2×C42×C83—22no—
63C2×C42×C83—21no—
64C2×C42×C83—21no—
65C2×C42×C83—22no—
66C2×C42×C83—21no—
67C2×C42×C83—21no—
68C2×C42×C83D411yescharacteristic
69C2×C42×C83D411yescharacteristic
70C2×C42×C83C2311yescharacteristic
71#95×C83—22no—
72#95×C83D412yes—
73#13050590—41no—
74#13050590—41no—
75#13050590—41no—
76#13050590—41no—
77C2×Dic166—21no—
78C2×D332—21no—
79C2×D332—21no—
80C2×D332—21no—
81C2×C22⋊D83—41no—
82C2×C4×D83—41no—
83#13050577—44no—
84#13050574—41no—
85#13050574—41no—
86C2×C83⋊C8—22no—
87C2×C83⋊C8—41no—
88C2×C83⋊C8—41no—
89C4⋊Dic83—21no—
90C4⋊Dic83—21no—
91(C2×D83)⋊C4—41no—
92(C2×D83)⋊C4—41no—
93C2×Q8×C83—24no—
94C2×Q8×C83—21no—
95C2×Q8×C83C22⋊C412yes—
96C22×C4×C83—21no—
97C22×C4×C83—21no—
98C22×C4×C83—21no—
99C22×C4×C83—21no—
100C22×C4×C83—22no—
101C22×C4×C83C2×D411yescharacteristic
102C22×C4×C83C2×D411yescharacteristic
103C22×C4×C83C2×D411yescharacteristic
104C4⋊C4×C83—28no—
105C4⋊C4×C83—24no—
106C4⋊C4×C83—22no—
107C4⋊C4×C83—22no—
108C4⋊C4×C83—21no—
109C4⋊C4×C83C22⋊C412yes—
110C42×C83—28no—
111C42×C83—24no—
112C42×C83—22no—
113C42×C83—22no—
114C42×C83—21no—
115C42×C83—21no—
116C42×C83—21no—
117C42×C83—42no—
118C42×C83—44no—
119C42×C83—41no—
120C42×C83—41no—
121C42×C83SD1612yescomplemented
122C42×C83C22×C411yescharacteristic
123C42×C83C2×D411yescharacteristic
124C42×C83#4111yescharacteristic
125C42×C83#4111yescharacteristic
126C23×C83#12011yescharacteristic
127C22×D83—41no—
128Q8×C83—24no—
129Q8×C83—22no—
130Q8×C83#10414yescomplemented
131C2×C4×C83—22no—
132C2×C4×C83—24no—
133C2×C4×C83—22no—
134C2×C4×C83—22no—
135C2×C4×C83—22no—
136C2×C4×C83—22no—
137C2×C4×C83—22no—
138C2×C4×C83—24no—
139C2×C4×C83—22no—
140C2×C4×C83—41no—
141C2×C4×C83—41no—
142C2×C4×C83—42no—
143C2×C4×C83—42no—
144C2×C4×C83—41no—
145C2×C4×C83—42no—
146C2×C4×C83C2×C22⋊C411yescharacteristic
147C2×C4×C83C4×D411yescharacteristic
148C2×C4×C83C4×D411yescharacteristic
149C2×C4×C83#12111yescharacteristic
150C2×C4×C83#12111yescharacteristic
151C2×C4×C83#12311yescharacteristic
152C2×C4×C83C2×SD1611yescharacteristic
153C2×C4×C83#13511yescharacteristic
154C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C83#13711yescharacteristic
156C22⋊D83—81no—
157C22⋊D83—81no—
158C2×Dic83—41no—
159D332—42no—
160D332—41no—
161D332—41no—
162D332—81no—
163C4×D83—81no—
164C4×D83—81no—
165Dic166—41no—
166Dic166—41no—
167C83⋊C8—42no—
168C83⋊C8—81no—
169C83⋊C8—81no—
170C22×C83—21no—
171C22×C83—21no—
172C22×C83#38611yescharacteristic
173C22×C83#44811yescharacteristic
174C22×C83#44911yescharacteristic
175C2×D83—81no—
176C2×D83—81no—
177C2×D83—81no—
178C4×C83—24no—
179C4×C83—24no—
180C4×C83—21no—
181C4×C83—21no—
182C4×C83—21no—
183C4×C83—42no—
184C4×C83—44no—
185C4×C83—42no—
186C4×C83C2×#10412yes—
187C4×C83C4×SD1611yescharacteristic
188C4×C83#44011yescharacteristic
189C4×C83#46011yescharacteristic
190C4×C83#46411yescharacteristic
191Dic83—81no—
192#9234—831nomaximal, Sylow
193C2×C83—21no—
194C2×C83—21no—
195C2×C83#188011yescharacteristic
196C2×C83#189611yescharacteristic
197C2×C83#190611yescharacteristic
198D83—161no—
199#3103—831no—
200C42×Q8D8311yescharacteristic, complemented
201#1962—831no—
202#1502—831no—
203#1487—831no—
204#1303—831no—
205#1276—831no—
206C83#923411yescharacteristic, complemented, Sylow
207#532—831no—
208C2×C4×Q8—21no—
209C2×C4×Q8C2×D8311yescharacteristic
210#481—1661no—
211#475—832no—
212#460—832no—
213#448—1661no—
214C4×SD16—1662no—
215#431—831no—
216#431—831no—
217#423—831no—
218C2×#104—1661no—
219C2×#104—1661no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4C2×D8311yescharacteristic
224C43—21no—
225C43C2×D8311yescharacteristic
226#327—1662no—
227Q8⋊C8—1662no—
228C2×#41—1661no—
229C22×Q8C22⋊D8311yescharacteristic
230C2×SD16—1662no—
231C2×M16—1661no—
232C4⋊Q8—831no—
233#127—831no—
234#124—1661no—
235#121—1661no—
236#121—1661no—
237C4×Q8—28no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8C4×D8312yes—
241C4×Q8D33212yes—
242C4×D4—1661no—
243C4×D4—1661no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C22⋊D8311yescharacteristic
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42C22⋊D8311yescharacteristic
255C2×C42C22⋊D8311yescharacteristic
256C2×C42C22×D8311yescharacteristic
257#106—1662no—
258C4⋊C8—832no—
259C4⋊C8—1661no—
260C4⋊C8—1661no—
261C4⋊C8—1661no—
262#104—3322no—
263#104—3322no—
264#103—1662no—
265#102—1662no—
266C22⋊C8—1661no—
267C22⋊C8—1661no—
268C4×C8—1661no—
269C4×C8—1661no—
270#95—22no—
271#95C22⋊D8312yes—
272#41—3321no—
273#41—3321no—
274#41—3321no—
275#41—3321no—
276C2×Q8—24no—
277C2×Q8—21no—
278C2×Q8—1661no—
279C2×Q8(C2×D83)⋊C412yes—
280C2×D4—3321no—
281C2×D4—1661no—
282C2×D4—1661no—
283C2×D4—1661no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—3321no—
290C22×C4C2×C22⋊D8311yescharacteristic
291C22×C4C2×C22⋊D8311yescharacteristic
292C22×C4C2×C22⋊D8311yescharacteristic
293SD16—3324no—
294M16—3321no—
295M16—3321no—
296C2×C8—3321no—
297C2×C8—3321no—
298C2×C8—1662no—
299C4⋊C4—28no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—1661no—
305C4⋊C4—1661no—
306C4⋊C4(C2×D83)⋊C412yes—
307C22⋊C4—3321no—
308C22⋊C4—3321no—
309C42—28no—
310C42—24no—
311C42—22no—
312C42—21no—
313C42—22no—
314C42—21no—
315C42—21no—
316C42—42no—
317C42—44no—
318C42—41no—
319C42—41no—
320C42#1305057712yescomplemented
321C42C2×C4×D8311yescharacteristic
322C42C2×D33211yescharacteristic
323C42#1305059011yescharacteristic
324C42#1305059011yescharacteristic
325C23—3321no—
326C23#1458176111yescharacteristic
327Q8—24no—
328Q8—22no—
329Q8—3321no—
330Q8—3321no—
331Q8#1458172014yescomplemented
332D4—3322no—
333D4—3321no—
334D4—3321no—
335D4—6641no—
336D4—6641no—
337D4—6641no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—3321no—
354C2×C4—6641no—
355C2×C4—6641no—
356C2×C4C2×(C2×D83)⋊C411yescharacteristic
357C2×C4C4×C22⋊D8311yescharacteristic
358C2×C4C4×C22⋊D8311yescharacteristic
359C2×C4#1458176611yescharacteristic
360C2×C4#1458176611yescharacteristic
361C2×C4#1458177511yescharacteristic
362C2×C4C2×#1305057711yescharacteristic
363C2×C4#1458181211yescharacteristic
364C2×C4#1458181711yescharacteristic
365C2×C4#1458182311yescharacteristic
366C8—3322no—
367C8—6641no—
368C8—6641no—
369C22—21no—
370C22—21no—
371C22—6641no—
372C22—6641no—
373C22—6641no—
374C22#1912758011yescharacteristic
375C22#1912784011yescharacteristic
376C22#1912784611yescharacteristic
377C4—24no—
378C4—24no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—42no—
383C4—44no—
384C4—42no—
385C4—6641no—
386C4C2×#1458172012yes—
387C4C4×#1305057711yescharacteristic
388C4#1912779911yescharacteristic
389C4#1912790111yescharacteristic
390C4#1912792311yescharacteristic
391C2—21no—
392C2—21no—
393C2—13281no—
394C2#2490781811yescharacteristic
395C2#2490790611yescharacteristic
396C2#2490796411yescharacteristic
subgroupG/H
centerC2×C4#14581766
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC42×Q8×C83C2
socleC22×C83#386
prime radical#19128251C22
cosocleC2×C42×C83C23
hypercenterC42×Q8D83
hypocenterC83#9234
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
4. #24905892C2semidirect
5. #24905366C2semidirect
6. #24905296C2semidirect
7. C42×Q8×C83C2semidirect
10. #19127986C4semidirect
11. #19127901C4semidirect
121. C42×C83SD16semidirect
130. Q8×C83#104semidirect
200. C42×Q8D83semidirect
206. C83#9234semidirect
320. C42#13050577semidirect
331. Q8#14581720semidirect

Maximal subgroups

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

subgroupindex
#249174942
#249082912
#249059572
#249058922
#249053662
#249052962
C42×Q8×C832
#923483