Finite Group Databasebeta

← #35308231

Subgroups — #35308231

Order 21248.

506 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127907C211yesmaximal, characteristic
2#24908424C211yesmaximal, characteristic, complemented
3#24908257C211yesmaximal, characteristic
4#24907464C211yesmaximal, characteristic
5C2×#19127448C211yesmaximal, characteristic, complemented
6#24905347C211yesmaximal, characteristic, complemented
7C2×#334×C83C211yesmaximal, characteristic, complemented
8C22×#13050579—21no—
9C2×#14581792C2211yescharacteristic
10C2×C4×D332C2211yescharacteristic
11#19127907—22no—
12#19127907—22no—
13C2×C4⋊(C83⋊C8)C2211yescharacteristic
14C2×#14581717—21no—
15C2×#14581714—21no—
16C2×(C2×D83)⋊C8—21no—
17C2×(C2×D83)⋊C8—21no—
18#19127571C2211yescharacteristic
19#19127464—21no—
20#19127464—21no—
21#19127464—21no—
22#19127448—22no—
23C2×C4⋊C8×C83C2211yescharacteristic
24C2×C4⋊C8×C83C2211yescharacteristic
25C2×C4×C8×C83C2211yescharacteristic
26#334×C83—22no—
27C22×Dic166—21no—
28C22×D332D411yescharacteristic
29C22×C4×D83—21no—
30C2×#13050579—24no—
31C2×#13050579—42no—
32C2×#13050579—42no—
33C22×C83⋊C8—21no—
34#14581792—21no—
35#14581792—21no—
36#14581792—21no—
37#14581792D412yes—
38C4×D332—21no—
39C4×D332—21no—
40C4×D332—21no—
41C4×D332—21no—
42C2×C4⋊Dic83—21no—
43C2×C4⋊Dic83—21no—
44C2×C4⋊Dic83—21no—
45C2×C4⋊Dic83—21no—
46C2×C4⋊Dic83—21no—
47C2×C4⋊Dic83D411yescharacteristic
48C2×(C2×D83)⋊C4—21no—
49C4⋊(C83⋊C8)—21no—
50C4⋊(C83⋊C8)—21no—
51#14581717—24no—
52#14581714—42no—
53(C2×D83)⋊C8—42no—
54(C2×D83)⋊C8—42no—
55C22×C8×C83—21no—
56C22×C8×C83—21no—
57C22×C8×C83D411yescharacteristic
58C22×C8×C83D411yescharacteristic
59C2×C42×C83C2311yescharacteristic
60C4⋊C8×C83—21no—
61C4⋊C8×C83—21no—
62C4⋊C8×C83—21no—
63C4⋊C8×C83D412yes—
64C4×C8×C83—21no—
65C4×C8×C83—21no—
66C23×D83—21no—
67C2×Dic166—22no—
68C2×Dic166—41no—
69C2×Dic166—41no—
70C2×Dic166—41no—
71C2×Dic166—41no—
72C2×Dic166—41no—
73C2×D332—21no—
74C2×D332—21no—
75C2×D332—21no—
76C2×D332—21no—
77C2×D332—21no—
78C2×D332SD1612yes—
79C22×Dic83—21no—
80C22×Dic83—41no—
81C22×Dic83—41no—
82C2×C4×D83—41no—
83C2×C4×D83—41no—
84C2×C4×D83—41no—
85C2×C4×D83—41no—
86C2×C4×D83—41no—
87C2×C4×D83—41no—
88#13050579—48no—
89C2×C83⋊C8—41no—
90C2×C83⋊C8—41no—
91C2×C83⋊C8—41no—
92C4⋊Dic83—21no—
93C4⋊Dic83—22no—
94C4⋊Dic83—22no—
95C4⋊Dic83—21no—
96C4⋊Dic83—41no—
97C4⋊Dic83—41no—
98C4⋊Dic83—41no—
99C4⋊Dic83—41no—
100C4⋊Dic83—41no—
101C4⋊Dic83—41no—
102C4⋊Dic83—81no—
103(C2×D83)⋊C4—41no—
104(C2×D83)⋊C4—41no—
105C22×C4×C83C2×D411yescharacteristic
106C22×C4×C83C2×D411yescharacteristic
107C22×C4×C83#4111yescharacteristic
108C2×C8×C83—21no—
109C2×C8×C83—21no—
110C2×C8×C83—22no—
111C2×C8×C83—21no—
112C2×C8×C83—21no—
113C2×C8×C83—21no—
114C2×C8×C83—41no—
115C2×C8×C83—41no—
116C2×C8×C83—41no—
117C2×C8×C83—41no—
118C2×C8×C83—41no—
119C2×C8×C83SD1612yescomplemented
120C42×C83C2×D411yescharacteristic
121C42×C83C2×D411yescharacteristic
122C42×C83#4111yescharacteristic
123C42×C83#4111yescharacteristic
124C23×C83#12111yescharacteristic
125C22×D83—22no—
126C22×D83—41no—
127C22×D83—41no—
128C22×D83—41no—
129C22×D83—41no—
130C22×D83—41no—
131C22×D83—41no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83#12011yescharacteristic
135C2×C4×C83#12111yescharacteristic
136C2×C4×C83#12111yescharacteristic
137C2×C4×C83#12311yescharacteristic
138C2×C4×C83#12311yescharacteristic
139C2×C4×C83#12411yescharacteristic
140C2×C4×C83C2×SD1611yescharacteristic
141C2×C4×C83C2×SD1611yescharacteristic
142C2×C4×C83#13511yescharacteristic
143C2×C4×C83#13511yescharacteristic
144C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C83#13711yescharacteristic
148C2×Dic83—41no—
149C2×Dic83—42no—
150C2×Dic83—41no—
151C2×Dic83—41no—
152C2×Dic83—81no—
153C2×Dic83—81no—
154C2×Dic83—81no—
155C2×Dic83—81no—
156C2×Dic83—81no—
157D332—22no—
158D332—22no—
159D332—41no—
160D332—41no—
161C4×D83—81no—
162C4×D83—81no—
163C4×D83—81no—
164C4×D83—81no—
165Dic166—42no—
166Dic166—42no—
167Dic166—81no—
168Dic166—81no—
169C8×C83—22no—
170C8×C83—42no—
171C8×C83—41no—
172C8×C83—81no—
173C83⋊C8—81no—
174C22×C83#39411yescharacteristic
175C22×C83#46111yescharacteristic
176C22×C83#46311yescharacteristic
177C22×C83#46511yescharacteristic
178C22×C83#46911yescharacteristic
179C22×C83#47111yescharacteristic
180C22×C83#47311yescharacteristic
181C2×D83—42no—
182C2×D83—42no—
183C2×D83—42no—
184C2×D83—81no—
185C2×D83—81no—
186C2×D83—81no—
187C2×D83—81no—
188C4×C83—21no—
189C4×C83—21no—
190C4×C83#44911yescharacteristic
191C4×C83#45011yescharacteristic
192C4×C83#46011yescharacteristic
193C4×C83#46411yescharacteristic
194C4×C83#46511yescharacteristic
195C4×C83#46811yescharacteristic
196C4×C83#48111yescharacteristic
197C4×C83#48211yescharacteristic
198Dic83—81no—
199Dic83—82no—
200Dic83—161no—
201#10477—831nomaximal, Sylow
202C2×C83#167111yescharacteristic
203C2×C83#167911yescharacteristic
204C2×C83#169311yescharacteristic
205C2×C83#170111yescharacteristic
206C2×C83#203411yescharacteristic
207C2×C83#205211yescharacteristic
208C2×C83#205511yescharacteristic
209D83—82no—
210C2×#461—831no—
211#1990—831no—
212#1954—831no—
213#1802—831no—
214C2×#334D8311yescharacteristic, complemented
215C2×#332—831no—
216#1296—831no—
217C83#1047711yescharacteristic, complemented, Sylow
218C22×SD16—1661no—
219C2×C4⋊Q8—831no—
220C2×C4×D4—831no—
221#461—1662no—
222#461—1662no—
223C2×C4⋊C8—831no—
224C2×C4⋊C8C2×D8311yescharacteristic
225C2×C4⋊C8C2×D8311yescharacteristic
226C2×#103—1661no—
227C2×#102—1661no—
228C2×C22⋊C8—1661no—
229C2×C22⋊C8—1661no—
230C2×C4×C8C2×D8311yescharacteristic
231#384—831no—
232#340—1661no—
233#340—1661no—
234#340—1661no—
235#334—22no—
236#332—1662no—
237C22×Q8—1661no—
238C22×D4—831no—
239C23×C4—1661no—
240C2×SD16—3322no—
241C2×SD16—3322no—
242C2×SD16—1664no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—1661no—
246C22×C8D33211yescharacteristic
247C22×C8D33211yescharacteristic
248C4⋊Q8—832no—
249C4⋊Q8—1661no—
250C4⋊Q8—1661no—
251C4⋊Q8—1661no—
252C4×D4—1661no—
253C4×D4—1661no—
254C4×D4—1661no—
255C4×D4—1661no—
256C2×C4⋊C4—831no—
257C2×C4⋊C4—1661no—
258C2×C4⋊C4—1661no—
259C2×C4⋊C4—1661no—
260C2×C4⋊C4—1661no—
261C2×C4⋊C4—1661no—
262C2×C22⋊C4—1661no—
263C2×C42C22×D8311yescharacteristic
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—1661no—
268C4⋊C8—1661no—
269C4⋊C8C22⋊D8312yes—
270#103—1664no—
271#102—3322no—
272C22⋊C8—3322no—
273C22⋊C8—3322no—
274C4×C8—21no—
275C4×C8—21no—
276C24—1661no—
277C2×Q8—3321no—
278C2×Q8—3321no—
279C2×Q8—3321no—
280C2×Q8—3321no—
281C2×Q8—3321no—
282C2×Q8—1662no—
283C2×D4—832no—
284C2×D4—1661no—
285C2×D4—1661no—
286C2×D4—1661no—
287C2×D4—1661no—
288C2×D4—1661no—
289C22×C4—3321no—
290C22×C4—3321no—
291C22×C4—3321no—
292C22×C4—3321no—
293C22×C4—3321no—
294C22×C4—3321no—
295C22×C4—3321no—
296C22×C4—3321no—
297C22×C4—1661no—
298C22×C4D4×D8311yescharacteristic
299C22×C4C2×D33211yescharacteristic
300C22×C4#1305059111yescharacteristic
301SD16—3328no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—22no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—3321no—
314C2×C8—3321no—
315C2×C8—3321no—
316C2×C8#1305057912yescomplemented
317C4⋊C4—3321no—
318C4⋊C4—3321no—
319C4⋊C4—3321no—
320C4⋊C4—3321no—
321C4⋊C4—3321no—
322C4⋊C4—3321no—
323C4⋊C4—6641no—
324C4⋊C4—1661no—
325C4⋊C4—1662no—
326C4⋊C4—1661no—
327C4⋊C4—1662no—
328C22⋊C4—3321no—
329C22⋊C4—3321no—
330C42D4×D8311yescharacteristic
331C42C2×C22⋊D8311yescharacteristic
332C42#1305059011yescharacteristic
333C42#1305059111yescharacteristic
334C23—3321no—
335C23—3321no—
336C23—3321no—
337C23—3321no—
338C23—3321no—
339C23—3321no—
340C23—1662no—
341C23#1458176711yescharacteristic
342Q8—3322no—
343Q8—3322no—
344Q8—6641no—
345Q8—6641no—
346D4—3321no—
347D4—3321no—
348D4—1662no—
349D4—1662no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—3321no—
353C2×C4—3322no—
354C2×C4—3321no—
355C2×C4—3321no—
356C2×C4—6641no—
357C2×C4—6641no—
358C2×C4—6641no—
359C2×C4—6641no—
360C2×C4—6641no—
361C2×C4—6641no—
362C2×C4—6641no—
363C2×C4—6641no—
364C2×C4—6641no—
365C2×C4#1458176011yescharacteristic
366C2×C4#1458176611yescharacteristic
367C2×C4#1458176711yescharacteristic
368C2×C4#1458177711yescharacteristic
369C2×C4#1458177711yescharacteristic
370C2×C4#1458178111yescharacteristic
371C2×C4SD16×D8311yescharacteristic
372C2×C4C2×#1305057911yescharacteristic
373C2×C4#1458181311yescharacteristic
374C2×C4#1458181411yescharacteristic
375C2×C4#1458181911yescharacteristic
376C2×C4#1458181911yescharacteristic
377C2×C4#1458182011yescharacteristic
378C2×C4#1458182511yescharacteristic
379C8—22no—
380C8—42no—
381C8—41no—
382C8—81no—
383C8—6641no—
384C22—3322no—
385C22—3322no—
386C22—3322no—
387C22—6641no—
388C22—6641no—
389C22—6641no—
390C22—6641no—
391C22#1912761811yescharacteristic
392C22#1912790711yescharacteristic
393C22#1912791711yescharacteristic
394C22#1912793111yescharacteristic
395C22#1912795511yescharacteristic
396C22#1912796511yescharacteristic
397C22#1912797711yescharacteristic
398C4—21no—
399C4—21no—
400C4—13281no—
401C4—6642no—
402C4—6641no—
403C4#1912784511yescharacteristic
404C4#1912785011yescharacteristic
405C4#1912790211yescharacteristic
406C4#1912792411yescharacteristic
407C4#1912793011yescharacteristic
408C4#1912794711yescharacteristic
409C4#1912802111yescharacteristic
410C4#1912802711yescharacteristic
411C2—6642no—
412C2#2490683911yescharacteristic
413C2#2490688711yescharacteristic
414C2#2490697711yescharacteristic
415C2#2490702511yescharacteristic
416C2#2490865411yescharacteristic
417C2#2490876411yescharacteristic
418C2#2490878111yescharacteristic
subgroupG/H
centerC23#14581767
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC2×#334×C83C2
socleC23×C83#121
prime radicalC22×D332D4
hypercenterC2×#334D83
hypocenterC83#10477
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#334

Complements

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

NH ≅ G/Ntype
2. #24908424C2semidirect
5. C2×#19127448C2semidirect
6. #24905347C2semidirect
7. C2×#334×C83C2semidirect
119. C2×C8×C83SD16semidirect
214. C2×#334D83semidirect
217. C83#10477semidirect
316. C2×C8#13050579semidirect

Maximal subgroups

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

subgroupindex
C2×#191279072
#249084242
#249082572
#249074642
C2×#191274482
#249053472
C2×#334×C832
#1047783