Finite Group Databasebeta

← #35307938

Subgroups — #35307938

Order 21248.

546 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24912381C211yesmaximal, characteristic
2#24907620C211yesmaximal, characteristic, complemented
3#24907468C211yesmaximal, characteristic, complemented
4#24907464C211yesmaximal, characteristic
5C2×#19127430C211yesmaximal, characteristic
6#24905347C211yesmaximal, characteristic, complemented
7C2×Q8⋊C8×C83C211yesmaximal, characteristic, complemented
8C2×#14581768—21no—
9C2×C4×D332C2211yescharacteristic
10C2×#14581724—22no—
11C2×C4⋊(C83⋊C8)C2211yescharacteristic
12C2×#14581716—21no—
13C2×#14581714—21no—
14#19127623—22no—
15#19127603—21no—
16#19127599—21no—
17#19127571C2211yescharacteristic
18#19127571C2211yescharacteristic
19#19127558—21no—
20#19127464—21no—
21#19127463—21no—
22#19127430—22no—
23C2×C4×Q8×C83C2211yescharacteristic
24C2×C4⋊C8×C83C2211yescharacteristic
25C2×C4×C8×C83C2211yescharacteristic
26Q8⋊C8×C83—22no—
27C22×D332D411yescharacteristic
28C22×C4×D83—21no—
29C22×C83⋊C8—21no—
30#14581768—42no—
31#14581768—42no—
32C4×D332—21no—
33C4×D332—21no—
34C4×D332—21no—
35C4×D332—21no—
36C2×C4⋊Dic83—21no—
37C2×C4⋊Dic83—21no—
38C2×C4⋊Dic83—21no—
39C2×C4⋊Dic83—21no—
40C2×C4⋊Dic83—21no—
41C2×C4⋊Dic83D411yescharacteristic
42C2×C4⋊Dic83D412yes—
43C2×Dic83⋊C4—22no—
44C2×(C2×D83)⋊C4—22no—
45C2×(C2×D83)⋊C4—22no—
46C2×(C2×D83)⋊C4—21no—
47#14581724—44no—
48C4⋊(C83⋊C8)—21no—
49C4⋊(C83⋊C8)—21no—
50#14581716—42no—
51#14581714—42no—
52C83⋊#95—22no—
53C22×Q8×C83D411yescharacteristic
54C22×C8×C83—21no—
55C22×C8×C83—21no—
56C4×Q8×C83—21no—
57C4×Q8×C83—21no—
58C4×Q8×C83—21no—
59C4×Q8×C83—21no—
60C2×C4⋊C4×C83—21no—
61C2×C4⋊C4×C83D411yescharacteristic
62C2×C42×C83—21no—
63C2×C42×C83C2311yescharacteristic
64C4⋊C8×C83—21no—
65C4⋊C8×C83—21no—
66C4×C8×C83—21no—
67C4×C8×C83D412yes—
68C23×D83—21no—
69C2×D332—21no—
70C2×D332—21no—
71C2×D332—21no—
72C2×D332—21no—
73C2×D332—21no—
74C2×D332SD1612yes—
75C22×Dic83—21no—
76C22×Dic83—22no—
77C22×Dic83—41no—
78C2×C4×D83—41no—
79C2×C4×D83—41no—
80C2×C4×D83—41no—
81C2×C4×D83—41no—
82C2×C4×D83—41no—
83C2×C4×D83—41no—
84C2×C83⋊C8—41no—
85C2×C83⋊C8—41no—
86C2×C83⋊C8—41no—
87C4⋊Dic83—21no—
88C4⋊Dic83—21no—
89C4⋊Dic83—41no—
90C4⋊Dic83—41no—
91C4⋊Dic83—41no—
92C4⋊Dic83—41no—
93C4⋊Dic83—42no—
94C4⋊Dic83—42no—
95C4⋊Dic83—41no—
96C4⋊Dic83—41no—
97C4⋊Dic83—41no—
98C4⋊Dic83—41no—
99Dic83⋊C4—44no—
100(C2×D83)⋊C4—44no—
101(C2×D83)⋊C4—41no—
102(C2×D83)⋊C4—41no—
103(C2×D83)⋊C4—82no—
104C2×Q8×C83—21no—
105C2×Q8×C83—21no—
106C2×Q8×C83—21no—
107C2×Q8×C83—21no—
108C2×Q8×C83—21no—
109C2×Q8×C83SD1612yes—
110C22×C4×C83—21no—
111C22×C4×C83—21no—
112C22×C4×C83C2×D411yescharacteristic
113C22×C4×C83C2×D411yescharacteristic
114C22×C4×C83C2×D411yescharacteristic
115C2×C8×C83—22no—
116C2×C8×C83—22no—
117C2×C8×C83—22no—
118C2×C8×C83—41no—
119C2×C8×C83—41no—
120C2×C8×C83—41no—
121C4⋊C4×C83—21no—
122C4⋊C4×C83—21no—
123C4⋊C4×C83—41no—
124C4⋊C4×C83—41no—
125C42×C83—41no—
126C42×C83—41no—
127C42×C83C2×D411yescharacteristic
128C42×C83#4111yescharacteristic
129C42×C83#4111yescharacteristic
130C42×C83#4111yescharacteristic
131C23×C83#12011yescharacteristic
132C22×D83—22no—
133C22×D83—41no—
134C22×D83—41no—
135C22×D83—41no—
136C22×D83—41no—
137C22×D83—41no—
138C22×D83—41no—
139Q8×C83—22no—
140Q8×C83—22no—
141Q8×C83—41no—
142Q8×C83—41no—
143C2×C4×C83—21no—
144C2×C4×C83—21no—
145C2×C4×C83—22no—
146C2×C4×C83—41no—
147C2×C4×C83—41no—
148C2×C4×C83—41no—
149C2×C4×C83—41no—
150C2×C4×C83—41no—
151C2×C4×C83#12111yescharacteristic
152C2×C4×C83#12111yescharacteristic
153C2×C4×C83#12111yescharacteristic
154C2×C4×C83#12311yescharacteristic
155C2×C4×C83#12311yescharacteristic
156C2×C4×C83#12411yescharacteristic
157C2×C4×C83C2×SD1611yescharacteristic
158C2×C4×C83C2×SD1611yescharacteristic
159C2×C4×C83#13511yescharacteristic
160C2×C4×C83#13511yescharacteristic
161C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C83#13711yescharacteristic
164C2×C4×C83#13711yescharacteristic
165C2×Dic83—41no—
166C2×Dic83—41no—
167C2×Dic83—42no—
168C2×Dic83—41no—
169C2×Dic83—42no—
170C2×Dic83—42no—
171C2×Dic83—81no—
172C2×Dic83—81no—
173C2×Dic83—81no—
174D332—22no—
175D332—22no—
176D332—41no—
177D332—41no—
178C4×D83—81no—
179C4×D83—81no—
180C4×D83—81no—
181C4×D83—81no—
182C8×C83—24no—
183C8×C83—81no—
184C83⋊C8—81no—
185C22×C83#39411yescharacteristic
186C22×C83#44811yescharacteristic
187C22×C83#44911yescharacteristic
188C22×C83#45011yescharacteristic
189C22×C83#45211yescharacteristic
190C22×C83#45311yescharacteristic
191C22×C83#45611yescharacteristic
192C2×D83—42no—
193C2×D83—42no—
194C2×D83—42no—
195C2×D83—81no—
196C2×D83—81no—
197C2×D83—81no—
198C2×D83—81no—
199C4×C83—21no—
200C4×C83—21no—
201C4×C83—42no—
202C4×C83—81no—
203C4×C83#46511yescharacteristic
204C4×C83#46511yescharacteristic
205C4×C83#46911yescharacteristic
206C4×C83#47011yescharacteristic
207C4×C83#48111yescharacteristic
208C4×C83#48211yescharacteristic
209C4×C83#48311yescharacteristic
210C4×C83#48511yescharacteristic
211Dic83—82no—
212Dic83—81no—
213Dic83—161no—
214#10435—831nomaximal, Sylow
215C2×C83#162211yescharacteristic
216C2×C83#163111yescharacteristic
217C2×C83#165111yescharacteristic
218C2×C83#165211yescharacteristic
219C2×C83#203411yescharacteristic
220C2×C83#203511yescharacteristic
221C2×C83#203911yescharacteristic
222D83—82no—
223#2538—831no—
224#1844—831no—
225#1803—831no—
226#1802—831no—
227C2×Q8⋊C8D8311yescharacteristic, complemented
228C2×D4⋊C8—831no—
229#1296—831no—
230C83#1043511yescharacteristic, complemented, Sylow
231C2×C22⋊Q8—1661no—
232C2×C4×Q8C2×D8311yescharacteristic
233C2×C4×D4—831no—
234C2×#106—1662no—
235C2×C4⋊C8—831no—
236C2×C4⋊C8C2×D8311yescharacteristic
237C2×#103—1661no—
238C2×#102—1661no—
239C2×C4×C8C2×D8311yescharacteristic
240#396—1662no—
241#390—1661no—
242#389—1661no—
243#384—831no—
244#384—831no—
245#380—1661no—
246#340—1661no—
247#340—1661no—
248Q8⋊C8—22no—
249D4⋊C8—1662no—
250C22×Q8D33211yescharacteristic
251C22×D4—831no—
252C23×C4—1661no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—1661no—
256C22⋊Q8—3322no—
257C22⋊Q8—3322no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×D4—1661no—
263C4×D4—1661no—
264C4×D4—1661no—
265C4×D4—1661no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—832no—
268C2×C4⋊C4—831no—
269C2×C4⋊C4—1662no—
270C2×C4⋊C4—1661no—
271C2×C4⋊C4—1661no—
272C2×C4⋊C4—1661no—
273C2×C4⋊C4—1661no—
274C2×C4⋊C4—1661no—
275C2×C4⋊C4D33211yescharacteristic
276C2×C22⋊C4—1662no—
277C2×C22⋊C4—1661no—
278C2×C22⋊C4—1662no—
279C2×C42—21no—
280C2×C42C22×D8311yescharacteristic
281#106—3324no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—1661no—
285C4⋊C8—1661no—
286#103—3322no—
287#102—3322no—
288C4×C8—21no—
289C4×C8C22⋊D8312yes—
290#95—1662no—
291C24—1661no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8#1305057912yes—
298C2×D4—832no—
299C2×D4—1661no—
300C2×D4—1661no—
301C2×D4—1661no—
302C2×D4—1661no—
303C2×D4—1661no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—3321no—
307C22×C4—3321no—
308C22×C4—3321no—
309C22×C4—3321no—
310C22×C4—3321no—
311C22×C4—3321no—
312C22×C4—3321no—
313C22×C4—1661no—
314C22×C4—1662no—
315C22×C4D4×D8311yescharacteristic
316C22×C4D4×D8311yescharacteristic
317C22×C4C2×D33211yescharacteristic
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—3321no—
325C2×C8—3321no—
326C2×C8—3321no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—3322no—
332C4⋊C4—3324no—
333C4⋊C4—3322no—
334C4⋊C4—3321no—
335C4⋊C4—3321no—
336C4⋊C4—3321no—
337C4⋊C4—3321no—
338C4⋊C4—3321no—
339C4⋊C4—3321no—
340C4⋊C4—3321no—
341C4⋊C4—3321no—
342C4⋊C4—1661no—
343C4⋊C4—1661no—
344C22⋊C4—3324no—
345C22⋊C4—3321no—
346C22⋊C4—3321no—
347C22⋊C4—6642no—
348C42—41no—
349C42—41no—
350C42C2×C22⋊D8311yescharacteristic
351C42#1305058911yescharacteristic
352C42#1305058911yescharacteristic
353C42#1305059011yescharacteristic
354C23—3321no—
355C23—3321no—
356C23—3321no—
357C23—3321no—
358C23—3321no—
359C23—3321no—
360C23—1662no—
361C23#1458176211yescharacteristic
362Q8—22no—
363Q8—22no—
364Q8—41no—
365Q8—41no—
366D4—3321no—
367D4—3321no—
368D4—1662no—
369D4—1662no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—3321no—
379C2×C4—3322no—
380C2×C4—3321no—
381C2×C4—3322no—
382C2×C4—3322no—
383C2×C4—3321no—
384C2×C4—6641no—
385C2×C4—6641no—
386C2×C4—6641no—
387C2×C4—6641no—
388C2×C4—6641no—
389C2×C4—6641no—
390C2×C4—6641no—
391C2×C4#1458176311yescharacteristic
392C2×C4#1458176411yescharacteristic
393C2×C4#1458176611yescharacteristic
394C2×C4#1458177411yescharacteristic
395C2×C4#1458177611yescharacteristic
396C2×C4#1458177811yescharacteristic
397C2×C4SD16×D8311yescharacteristic
398C2×C4C2×#1305057911yescharacteristic
399C2×C4#1458181311yescharacteristic
400C2×C4#1458181411yescharacteristic
401C2×C4#1458182011yescharacteristic
402C2×C4#1458182011yescharacteristic
403C2×C4#1458182411yescharacteristic
404C2×C4#1458182411yescharacteristic
405C8—24no—
406C8—81no—
407C8—6641no—
408C22—3322no—
409C22—3322no—
410C22—3322no—
411C22—6641no—
412C22—6641no—
413C22—6641no—
414C22—6641no—
415C22#1912761711yescharacteristic
416C22#1912784111yescharacteristic
417C22#1912784811yescharacteristic
418C22#1912785411yescharacteristic
419C22#1912786611yescharacteristic
420C22#1912787011yescharacteristic
421C22#1912788511yescharacteristic
422C4—21no—
423C4—21no—
424C4—42no—
425C4—81no—
426C4—13281no—
427C4—6642no—
428C4—6641no—
429C4#1912792711yescharacteristic
430C4#1912793011yescharacteristic
431C4#1912795411yescharacteristic
432C4#1912795611yescharacteristic
433C4#1912802011yescharacteristic
434C4#1912802611yescharacteristic
435C4#1912803011yescharacteristic
436C4#1912804011yescharacteristic
437C2—6642no—
438C2#2490653811yescharacteristic
439C2#2490659211yescharacteristic
440C2#2490672911yescharacteristic
441C2#2490673511yescharacteristic
442C2#2490865311yescharacteristic
443C2#2490865811yescharacteristic
444C2#2490868611yescharacteristic
subgroupG/H
centerC23#14581762
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC2×Q8⋊C8×C83C2
socleC23×C83#120
prime radicalC22×D332D4
hypercenterC2×Q8⋊C8D83
hypocenterC83#10435
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #24907620C2semidirect
3. #24907468C2semidirect
6. #24905347C2semidirect
7. C2×Q8⋊C8×C83C2semidirect
227. C2×Q8⋊C8D83semidirect
230. C83#10435semidirect

Maximal subgroups

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

subgroupindex
#249123812
#249076202
#249074682
#249074642
C2×#191274302
#249053472
C2×Q8⋊C8×C832
#1043583