Finite Group Databasebeta

← #35307784

Subgroups — #35307784

Order 21248.

572 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24912462C211yesmaximal, characteristic
2#24908269C211yesmaximal, characteristic
3#24908222C211yesmaximal, characteristic, complemented
4#24907468C211yesmaximal, characteristic, complemented
5C2×#19127444C211yesmaximal, characteristic
6#24905303C211yesmaximal, characteristic, complemented
7C2×Q8⋊C8×C83C211yesmaximal, characteristic, complemented
8C2×#14581789C2211yescharacteristic
9C2×#14581781—21no—
10C2×#14581726—21no—
11C2×C4⋊(C83⋊C8)C2211yescharacteristic
12C2×#14581716—21no—
13C2×#14581714—22no—
14#19127618—22no—
15#19127603—21no—
16#19127603—21no—
17#19127602—21no—
18#19127571C2211yescharacteristic
19C4×C4⋊Dic83C2211yescharacteristic
20#19127464—21no—
21#19127463—21no—
22#19127444—22no—
23C2×C4×Q8×C83C2211yescharacteristic
24C2×C4⋊C8×C83C2211yescharacteristic
25C2×C4×C8×C83C2211yescharacteristic
26Q8⋊C8×C83—22no—
27C22×D332—21no—
28C22×D332—21no—
29C22×D332D412yes—
30C22×C83⋊C8—21no—
31#14581789—21no—
32#14581789—21no—
33#14581789—21no—
34#14581789—21no—
35#14581781—42no—
36#14581781—42no—
37C2×C4⋊Dic83—21no—
38C2×C4⋊Dic83—21no—
39C2×C4⋊Dic83—21no—
40C2×C4⋊Dic83D411yescharacteristic
41C2×C4⋊Dic83D411yescharacteristic
42C2×(C2×D83)⋊C4—22no—
43C2×(C2×D83)⋊C4—22no—
44C2×(C2×D83)⋊C4—22no—
45C2×(C2×D83)⋊C4—22no—
46C2×C4×Dic83—21no—
47#14581726—42no—
48C4⋊(C83⋊C8)—21no—
49C4⋊(C83⋊C8)—21no—
50#14581716—42no—
51#14581714—44no—
52C83⋊#95—21no—
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×C83D412yes—
66C4×C8×C83—21no—
67C4×C8×C83—21no—
68C23×D83—22no—
69C2×D332—22no—
70C2×D332—22no—
71C2×D332—42no—
72C2×D332—42no—
73C2×D332—41no—
74C2×D332—41no—
75C2×D332—41no—
76C2×D332—41no—
77C2×D332—41no—
78C2×D332—41no—
79C2×D332—41no—
80C2×D332—41no—
81C2×D332—41no—
82C2×D332—41no—
83C2×D332—41no—
84C2×D332—41no—
85C22×Dic83—21no—
86C22×Dic83—21no—
87C22×Dic83—41no—
88C2×C83⋊C8—41no—
89C2×C83⋊C8—41no—
90C2×C83⋊C8—41no—
91C4⋊Dic83—21no—
92C4⋊Dic83—21no—
93C4⋊Dic83—21no—
94C4⋊Dic83—21no—
95C4⋊Dic83—42no—
96C4⋊Dic83—41no—
97C4⋊Dic83—41no—
98C4⋊Dic83—41no—
99C4⋊Dic83—41no—
100(C2×D83)⋊C4—44no—
101(C2×D83)⋊C4—44no—
102(C2×D83)⋊C4—82no—
103(C2×D83)⋊C4—82no—
104C4×Dic83—41no—
105C4×Dic83—41no—
106C2×Q8×C83—21no—
107C2×Q8×C83—21no—
108C2×Q8×C83—21no—
109C2×Q8×C83—21no—
110C2×Q8×C83—21no—
111C2×Q8×C83D812yes—
112C22×C4×C83—21no—
113C22×C4×C83—21no—
114C22×C4×C83C2×D411yescharacteristic
115C22×C4×C83C2×D411yescharacteristic
116C22×C4×C83C2×D411yescharacteristic
117C2×C8×C83—22no—
118C2×C8×C83—22no—
119C2×C8×C83—41no—
120C2×C8×C83—41no—
121C2×C8×C83—41no—
122C2×C8×C83—41no—
123C4⋊C4×C83—21no—
124C4⋊C4×C83—21no—
125C4⋊C4×C83—41no—
126C4⋊C4×C83—41no—
127C42×C83—41no—
128C42×C83—41no—
129C42×C83C2×D411yescharacteristic
130C42×C83#4111yescharacteristic
131C42×C83#4111yescharacteristic
132C42×C83#4111yescharacteristic
133C23×C83#12011yescharacteristic
134C22×D83—42no—
135C22×D83—42no—
136C22×D83—42no—
137C22×D83—42no—
138C22×D83—42no—
139C22×D83—42no—
140C22×D83—42no—
141Q8×C83—22no—
142Q8×C83—22no—
143Q8×C83—41no—
144Q8×C83—41no—
145C2×C4×C83—21no—
146C2×C4×C83—21no—
147C2×C4×C83—22no—
148C2×C4×C83—41no—
149C2×C4×C83—41no—
150C2×C4×C83—41no—
151C2×C4×C83—41no—
152C2×C4×C83—41no—
153C2×C4×C83#12111yescharacteristic
154C2×C4×C83#12111yescharacteristic
155C2×C4×C83#12111yescharacteristic
156C2×C4×C83#12311yescharacteristic
157C2×C4×C83#12311yescharacteristic
158C2×C4×C83#12411yescharacteristic
159C2×C4×C83C2×D811yescharacteristic
160C2×C4×C83#13511yescharacteristic
161C2×C4×C83#13511yescharacteristic
162C2×C4×C83#13511yescharacteristic
163C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C83#13711yescharacteristic
166C2×C4×C83#13711yescharacteristic
167C2×Dic83—41no—
168C2×Dic83—41no—
169C2×Dic83—41no—
170C2×Dic83—41no—
171C2×Dic83—41no—
172C2×Dic83—41no—
173C2×Dic83—81no—
174C2×Dic83—81no—
175C2×Dic83—81no—
176D332—44no—
177D332—82no—
178D332—81no—
179D332—81no—
180D332—81no—
181D332—81no—
182D332—81no—
183D332—81no—
184D332—81no—
185D332—81no—
186C8×C83—42no—
187C8×C83—42no—
188C83⋊C8—81no—
189C22×C83#39411yescharacteristic
190C22×C83#44711yescharacteristic
191C22×C83#44911yescharacteristic
192C22×C83#45211yescharacteristic
193C22×C83#45211yescharacteristic
194C22×C83#45311yescharacteristic
195C22×C83#45511yescharacteristic
196C2×D83—82no—
197C2×D83—82no—
198C2×D83—82no—
199C2×D83—82no—
200C2×D83—82no—
201C2×D83—82no—
202C2×D83—82no—
203C4×C83—21no—
204C4×C83—21no—
205C4×C83—42no—
206C4×C83—81no—
207C4×C83#46311yescharacteristic
208C4×C83#46411yescharacteristic
209C4×C83#46611yescharacteristic
210C4×C83#46811yescharacteristic
211C4×C83#48011yescharacteristic
212C4×C83#48211yescharacteristic
213C4×C83#48811yescharacteristic
214C4×C83#48911yescharacteristic
215Dic83—81no—
216Dic83—81no—
217Dic83—161no—
218#10411—831nomaximal, Sylow
219C2×C83#162111yescharacteristic
220C2×C83#163011yescharacteristic
221C2×C83#164011yescharacteristic
222C2×C83#164111yescharacteristic
223C2×C83#203311yescharacteristic
224C2×C83#203711yescharacteristic
225C2×C83#204211yescharacteristic
226D83—162no—
227#2548—831no—
228#1957—831no—
229#1947—831no—
230#1803—831no—
231C2×#331—831no—
232C2×Q8⋊C8D8311yescharacteristic, complemented
233#1279—831no—
234C83#1041111yescharacteristic, complemented, Sylow
235C2×#127—831no—
236C2×#124—1661no—
237C2×C4×Q8C2×D8311yescharacteristic
238C2×#107—1661no—
239C2×C4⋊C8—831no—
240C2×C4⋊C8C2×D8311yescharacteristic
241C2×#103—1661no—
242C2×#102—1662no—
243C2×C4×C8C2×D8311yescharacteristic
244#394—1662no—
245#390—1661no—
246#390—1661no—
247#390—1661no—
248#384—831no—
249C4×C4⋊C4—831no—
250#340—1661no—
251#340—1661no—
252#331—1662no—
253Q8⋊C8—22no—
254C22×Q8D33211yescharacteristic
255C22×D4—832no—
256C22×D4—1661no—
257C22×D4—1661no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—1661no—
261#127—1661no—
262#127—1661no—
263#127—1661no—
264#127—1661no—
265#124—3322no—
266#124—3322no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—831no—
273C2×C4⋊C4—831no—
274C2×C4⋊C4—1661no—
275C2×C4⋊C4—1661no—
276C2×C4⋊C4—1661no—
277C2×C4⋊C4D33211yescharacteristic
278C2×C22⋊C4—1662no—
279C2×C22⋊C4—1662no—
280C2×C22⋊C4—1662no—
281C2×C22⋊C4—1662no—
282C2×C42—21no—
283C2×C42—1661no—
284C2×C42C22×D8311yescharacteristic
285#107—3322no—
286C4⋊C8—21no—
287C4⋊C8—1661no—
288C4⋊C8—1661no—
289C4⋊C8C22⋊D8312yes—
290#103—3322no—
291#102—3324no—
292C4×C8—21no—
293C4×C8—21no—
294#95—1661no—
295C24—1662no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8D66412yes—
302C2×D4—3322no—
303C2×D4—3322no—
304C2×D4—3321no—
305C2×D4—3321no—
306C2×D4—3321no—
307C2×D4—3321no—
308C2×D4—3321no—
309C2×D4—3321no—
310C2×D4—3321no—
311C2×D4—3321no—
312C2×D4—3321no—
313C2×D4—3321no—
314C2×D4—3321no—
315C2×D4—3321no—
316C2×D4—1662no—
317C2×D4—1662no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—3321no—
321C22×C4—1661no—
322C22×C4—1661no—
323C22×C4D4×D8311yescharacteristic
324C22×C4D4×D8311yescharacteristic
325C22×C4C2×D33211yescharacteristic
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—3321no—
333C2×C8—3321no—
334C2×C8—3321no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—3322no—
340C4⋊C4—3321no—
341C4⋊C4—3321no—
342C4⋊C4—3321no—
343C4⋊C4—3321no—
344C4⋊C4—1661no—
345C4⋊C4—1661no—
346C4⋊C4—1661no—
347C4⋊C4—1661no—
348C22⋊C4—3324no—
349C22⋊C4—3324no—
350C22⋊C4—6642no—
351C22⋊C4—6642no—
352C42—41no—
353C42—41no—
354C42—3321no—
355C42—3321no—
356C42C2×C22⋊D8311yescharacteristic
357C42#1305058911yescharacteristic
358C42#1305058911yescharacteristic
359C42#1305059011yescharacteristic
360C23—3322no—
361C23—3322no—
362C23—3322no—
363C23—3322no—
364C23—3322no—
365C23—3322no—
366C23—3322no—
367C23#1458176211yescharacteristic
368Q8—22no—
369Q8—22no—
370Q8—41no—
371Q8—41no—
372D4—3324no—
373D4—6642no—
374D4—6641no—
375D4—6641no—
376D4—6641no—
377D4—6641no—
378D4—6641no—
379D4—6641no—
380D4—6641no—
381D4—6641no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—3321no—
391C2×C4—3321no—
392C2×C4—3321no—
393C2×C4—3321no—
394C2×C4—3321no—
395C2×C4—3321no—
396C2×C4—6641no—
397C2×C4—6641no—
398C2×C4—6641no—
399C2×C4#1458176311yescharacteristic
400C2×C4#1458176411yescharacteristic
401C2×C4#1458176611yescharacteristic
402C2×C4#1458177411yescharacteristic
403C2×C4#1458177611yescharacteristic
404C2×C4#1458177811yescharacteristic
405C2×C4C2×D66411yescharacteristic
406C2×C4#1458181311yescharacteristic
407C2×C4#1458181411yescharacteristic
408C2×C4#1458181511yescharacteristic
409C2×C4#1458181911yescharacteristic
410C2×C4#1458182011yescharacteristic
411C2×C4#1458182411yescharacteristic
412C2×C4#1458182511yescharacteristic
413C8—42no—
414C8—42no—
415C8—6641no—
416C22—6642no—
417C22—6642no—
418C22—6642no—
419C22—6642no—
420C22—6642no—
421C22—6642no—
422C22—6642no—
423C22#1912761711yescharacteristic
424C22#1912783711yescharacteristic
425C22#1912784911yescharacteristic
426C22#1912786511yescharacteristic
427C22#1912786611yescharacteristic
428C22#1912787011yescharacteristic
429C22#1912788011yescharacteristic
430C4—21no—
431C4—21no—
432C4—42no—
433C4—81no—
434C4—13281no—
435C4—6641no—
436C4—6641no—
437C4#1912791811yescharacteristic
438C4#1912792511yescharacteristic
439C4#1912793711yescharacteristic
440C4#1912794711yescharacteristic
441C4#1912801511yescharacteristic
442C4#1912802611yescharacteristic
443C4#1912805711yescharacteristic
444C4#1912806211yescharacteristic
445C2—13282no—
446C2#2490653011yescharacteristic
447C2#2490658611yescharacteristic
448C2#2490665211yescharacteristic
449C2#2490665811yescharacteristic
450C2#2490864611yescharacteristic
451C2#2490867211yescharacteristic
452C2#2490870111yescharacteristic
subgroupG/H
centerC23#14581762
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC2×Q8⋊C8×C83C2
socleC23×C83#120
prime radicalC2×#14581789C22
hypercenterC2×Q8⋊C8D83
hypocenterC83#10411
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
3. #24908222C2semidirect
4. #24907468C2semidirect
6. #24905303C2semidirect
7. C2×Q8⋊C8×C83C2semidirect
232. C2×Q8⋊C8D83semidirect
234. C83#10411semidirect

Maximal subgroups

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

subgroupindex
#249124622
#249082692
#249082222
#249074682
C2×#191274442
#249053032
C2×Q8⋊C8×C832
#1041183