Finite Group Databasebeta

← #35299379

Subgroups — #35299379

Order 21248.

782 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916504C211yesmaximal, characteristic, complemented
2#24907407C212yesmaximal, complemented
3C4×#14581707C211yesmaximal, characteristic, complemented
4#24905349C212yesmaximal, complemented
5C83×#2388C211yesmaximal, characteristic
6#19127779—24no—
7#19127754—21no—
8C2×#14581719—24no—
9C2×#14581707—21no—
10#19127665—21no—
11#19127665C2212yes—
12C4×C22⋊Dic83C2212yes—
13#19127468—21no—
14C22×#41×C83C2211yescharacteristic
15C2×C4×D4×C83—21no—
16C2×C4×D4×C83C2211yescharacteristic
17#395×C83C411yescharacteristic, complemented
18#394×C83—21no—
19#394×C83C411yescharacteristic, complemented
20#393×C83C411yescharacteristic, complemented
21#392×C83—21no—
22#392×C83C411yescharacteristic, complemented
23C4×C22⋊C4×C83—21no—
24C4×C22⋊C4×C83C2211yescharacteristic
25C2×#13050574—22no—
26C2×#13050574D414yescomplemented
27C2×C22⋊Dic83—22no—
28C2×C4×Dic83—22no—
29C4⋊(C83⋊C8)—22no—
30#14581719—48no—
31#14581707—24no—
32C22⋊(C83⋊C8)—22no—
33C22⋊(C83⋊C8)—22no—
34C22⋊(C83⋊C8)—22no—
35C4×C83⋊C8—22no—
36C83⋊#95—22no—
37C2×#41×C83—22no—
38C2×#41×C83—22no—
39C2×#41×C83—44no—
40C22×Q8×C83C2×C411yescharacteristic
41C22×D4×C83—21no—
42C22×D4×C83—21no—
43C22×D4×C83C2×C411yescharacteristic
44C22×D4×C83D411yescharacteristic
45C23×C4×C83—21no—
46C23×C4×C83D411yescharacteristic
47C23×C4×C83C2311yescharacteristic
48C4×D4×C83—22no—
49C4×D4×C83—22no—
50C4×D4×C83—44no—
51C2×C4⋊C4×C83—21no—
52C2×C4⋊C4×C83C2×C411yescharacteristic
53C2×C22⋊C4×C83—21no—
54C2×C22⋊C4×C83—21no—
55C2×C22⋊C4×C83—21no—
56C2×C22⋊C4×C83—21no—
57C2×C22⋊C4×C83—21no—
58C2×C22⋊C4×C83C2×C411yescharacteristic
59C2×C22⋊C4×C83D411yescharacteristic
60C2×C42×C83—21no—
61C2×C42×C83D411yescharacteristic
62#95×C83—21no—
63#95×C83—21no—
64#95×C83C2×C411yescharacteristic
65#95×C83C2×C411yescharacteristic
66C22×Dic83—42no—
67#13050574—44no—
68#13050574—82no—
69C2×C83⋊C8—24no—
70C22⋊Dic83—24no—
71C22⋊Dic83—42no—
72C4×Dic83—44no—
73C24×C83—21no—
74C24×C83C22⋊C411yescharacteristic
75C24×C83C2×D411yescharacteristic
76#41×C83—28no—
77#41×C83—44no—
78#41×C83—44no—
79#41×C83—44no—
80C2×Q8×C83—22no—
81C2×Q8×C83—42no—
82C2×D4×C83—22no—
83C2×D4×C83—22no—
84C2×D4×C83—22no—
85C2×D4×C83—41no—
86C2×D4×C83—42no—
87C2×D4×C83—42no—
88C2×D4×C83—42no—
89C2×D4×C83—42no—
90C2×D4×C83—42no—
91C2×D4×C83—42no—
92C2×D4×C83—42no—
93C22×C4×C83—24no—
94C22×C4×C83—21no—
95C22×C4×C83—21no—
96C22×C4×C83—21no—
97C22×C4×C83—21no—
98C22×C4×C83—21no—
99C22×C4×C83—42no—
100C22×C4×C83—42no—
101C22×C4×C83—42no—
102C22×C4×C83—41no—
103C22×C4×C83—41no—
104C22×C4×C83C22⋊C411yescharacteristic
105C22×C4×C83C22⋊C411yescharacteristic
106C22×C4×C83C22⋊C411yescharacteristic
107C22×C4×C83C22×C411yescharacteristic
108C22×C4×C83C2×D412yes—
109C22×C4×C83C2×D411yescharacteristic
110C22×C4×C83#4112yes—
111C4⋊C4×C83—22no—
112C4⋊C4×C83—42no—
113C22⋊C4×C83—22no—
114C22⋊C4×C83—22no—
115C22⋊C4×C83—42no—
116C22⋊C4×C83—42no—
117C22⋊C4×C83—42no—
118C22⋊C4×C83—42no—
119C22⋊C4×C83—42no—
120C42×C83—21no—
121C42×C83—21no—
122C42×C83—42no—
123C23×C83—21no—
124C23×C83—21no—
125C23×C83—22no—
126C23×C83—21no—
127C23×C83—21no—
128C23×C83—21no—
129C23×C83—41no—
130C23×C83—41no—
131C23×C83—41no—
132C23×C83—41no—
133C23×C83—41no—
134C23×C83—41no—
135C23×C83—41no—
136C23×C83—41no—
137C23×C83—41no—
138C23×C83—41no—
139C23×C83C2×C22⋊C411yescharacteristic
140C23×C83#12112yes—
141Q8×C83—24no—
142Q8×C83—42no—
143D4×C83—24no—
144D4×C83—44no—
145D4×C83—42no—
146D4×C83—48no—
147D4×C83—48no—
148C2×C4×C83—21no—
149C2×C4×C83—21no—
150C2×C4×C83—22no—
151C2×C4×C83—24no—
152C2×C4×C83—22no—
153C2×C4×C83—22no—
154C2×C4×C83—21no—
155C2×C4×C83—21no—
156C2×C4×C83—41no—
157C2×C4×C83—42no—
158C2×C4×C83—44no—
159C2×C4×C83—41no—
160C2×C4×C83—41no—
161C2×C4×C83—41no—
162C2×C4×C83—41no—
163C2×C4×C83—44no—
164C2×C4×C83—42no—
165C2×C4×C83—41no—
166C2×C4×C83—41no—
167C2×C4×C83—41no—
168C2×C4×C83—41no—
169C2×C4×C83—41no—
170C2×C4×C83—81no—
171C2×C4×C83#10412yescomplemented
172C2×C4×C83C4×D412yes—
173C2×C4×C83#12011yescharacteristic
174C2×C4×C83#12311yescharacteristic
175C2×Dic83—44no—
176C2×Dic83—82no—
177C2×Dic83—82no—
178C83⋊C8—44no—
179C22×C83—21no—
180C22×C83—21no—
181C22×C83—22no—
182C22×C83—22no—
183C22×C83—22no—
184C22×C83—42no—
185C22×C83—42no—
186C22×C83—42no—
187C22×C83—41no—
188C22×C83—41no—
189C22×C83—42no—
190C22×C83—41no—
191C22×C83—81no—
192C22×C83—81no—
193C22×C83—81no—
194C22×C83—81no—
195C22×C83—81no—
196C22×C83—81no—
197C22×C83—81no—
198C22×C83#38611yescharacteristic
199C22×C83C2×#10411yescharacteristic
200C22×C83#42111yescharacteristic
201C4×C83—22no—
202C4×C83—21no—
203C4×C83—22no—
204C4×C83—42no—
205C4×C83—41no—
206C4×C83—42no—
207C4×C83—41no—
208C4×C83—81no—
209C4×C83B(4,𝔽2)12yes—
210Dic83—84no—
211#9010—831nomaximal, Sylow
212C2×C83—21no—
213C2×C83—21no—
214C2×C83—42no—
215C2×C83—81no—
216C2×C83—82no—
217C2×C83#187111yescharacteristic
218C2×C83#188411yescharacteristic
219C2×C83#188911yescharacteristic
220#2975—831no—
221#2388D8311yescharacteristic
222#1789—832no—
223C4×#99—831no—
224#1297—832no—
225C83#901011yescharacteristic, complemented, Sylow
226C22×#41C2×D8311yescharacteristic
227C2×C4×D4—21no—
228C2×C4×D4C2×D8311yescharacteristic
229#436—1664no—
230#431—1661no—
231C2×#104—1664no—
232C2×#99—1661no—
233#407—832no—
234#407—1661no—
235#395Dic8311yescharacteristic, complemented
236#394—21no—
237#394Dic8311yescharacteristic, complemented
238#393Dic8311yescharacteristic, complemented
239#392—21no—
240#392Dic8311yescharacteristic, complemented
241C4×C22⋊C4—21no—
242C4×C22⋊C4—832no—
243C4×C22⋊C4C2×D8311yescharacteristic
244#342—1661no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—44no—
248C22×Q8C2×Dic8311yescharacteristic
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2×Dic8311yescharacteristic
252C22×D4C22⋊D8311yescharacteristic
253C23×C4—21no—
254C23×C4C22⋊D8311yescharacteristic
255C23×C4C22×D8311yescharacteristic
256C2×M16—834no—
257C2×M16—1662no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—44no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4C2×Dic8311yescharacteristic
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—1662no—
269C2×C22⋊C4C2×Dic8311yescharacteristic
270C2×C22⋊C4C22⋊D8311yescharacteristic
271C2×C42—21no—
272C2×C42—1662no—
273C2×C42C22⋊D8311yescharacteristic
274C4⋊C8—1662no—
275#104—3328no—
276#99—1664no—
277C22⋊C8—1662no—
278C22⋊C8—1662no—
279C22⋊C8—1662no—
280C4×C8—1662no—
281#95—21no—
282#95—21no—
283#95—1662no—
284#95C2×Dic8311yescharacteristic
285#95C2×Dic8311yescharacteristic
286C24—21no—
287C24C22⋊Dic8311yescharacteristic
288C24C2×C22⋊D8311yescharacteristic
289#41—28no—
290#41—44no—
291#41—44no—
292#41—44no—
293C2×Q8—22no—
294C2×Q8—42no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—3322no—
318C22×C4C22⋊Dic8311yescharacteristic
319C22×C4C22⋊Dic8311yescharacteristic
320C22×C4C22⋊Dic8311yescharacteristic
321C22×C4C22×Dic8311yescharacteristic
322C22×C4D4×D8312yes—
323C22×C4C2×C22⋊D8311yescharacteristic
324C22×C4#1305058912yes—
325M16—3324no—
326M16—6642no—
327C2×C8—1664no—
328C4⋊C4—22no—
329C4⋊C4—42no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—3322no—
338C22⋊C4—1664no—
339C42—21no—
340C42—21no—
341C42—42no—
342C42—3324no—
343C23—21no—
344C23—21no—
345C23—22no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23C2×C22⋊Dic8311yescharacteristic
360C23#1458176412yes—
361Q8—24no—
362Q8—42no—
363D4—24no—
364D4—44no—
365D4—42no—
366D4—48no—
367D4—48no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—3324no—
392C2×C4—6642no—
393C2×C4—6642no—
394C2×C4#1458171912yescomplemented
395C2×C4D4×Dic8312yes—
396C2×C4#1458176011yescharacteristic
397C2×C4#1458177311yescharacteristic
398C8—3324no—
399C22—21no—
400C22—21no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—41no—
408C22—42no—
409C22—41no—
410C22—41no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22#1912758111yescharacteristic
419C22C2×#1458171911yescharacteristic
420C22#1912771811yescharacteristic
421C4—21no—
422C4—22no—
423C4—22no—
424C4—42no—
425C4—41no—
426C4—42no—
427C4—41no—
428C4—81no—
429C4—6644no—
430C4#1912788712yes—
431C2—21no—
432C2—21no—
433C2—42no—
434C2—81no—
435C2—82no—
436C2#2490776611yescharacteristic
437C2#2490784711yescharacteristic
438C2#2490787211yescharacteristic
subgroupG/H
centerC2×C4#14581760
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
Fitting, prime radicalC83×#2388C2
socleC22×C83#386
cosocleC23×C4×C83C23
hypercenter#2388D83
hypocenterC83#9010
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2388

Complements

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

NH ≅ G/Ntype
1. #24916504C2semidirect
2. #24907407C2semidirect
3. C4×#14581707C2semidirect
4. #24905349C2semidirect
17. #395×C83C4semidirect
19. #394×C83C4semidirect
20. #393×C83C4semidirect
22. #392×C83C4semidirect
26. C2×#13050574D4semidirect
171. C2×C4×C83#104semidirect
225. C83#9010semidirect
235. #395Dic83semidirect
237. #394Dic83semidirect
238. #393Dic83semidirect
240. #392Dic83semidirect
394. C2×C4#14581719semidirect

Maximal subgroups

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

subgroupindex
#249165042
#249074072
C4×#145817072
#249053492
C83×#23882
#901083