Finite Group Databasebeta

← #35313581

Subgroups — #35313581

Order 21248.

562 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 115 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#24910075C211yesmaximal, characteristic
2C2×#19127425C211yesmaximal, characteristic
3#24905412C212yesmaximal, complemented
4#24905410C212yesmaximal, complemented
5C2×#323×C83C211yesmaximal, characteristic
6C22×C22⋊Dic83C411yescharacteristic
7C22×C22⋊Dic83C411yescharacteristic
8C2×C22⋊(C83⋊C8)—21no—
9#19127574—21no—
10#19127574—21no—
11#19127573C411yescharacteristic
12#19127573C411yescharacteristic
13#19127573C2211yescharacteristic
14#19127573C2211yescharacteristic
15#19127559—21no—
16#19127559—21no—
17#19127425—21no—
18#19127425C2212yes—
19C22×C22⋊C4×C83C2211yescharacteristic
20C2×C22⋊C8×C83—21no—
21#323×C83—21no—
22#323×C83C2212yes—
23C23×Dic83—21no—
24C23×Dic83C2×C411yescharacteristic
25C23×Dic83C2×C411yescharacteristic
26C22×C83⋊C8—41no—
27C2×Dic83⋊C4—21no—
28C2×Dic83⋊C4—21no—
29C2×Dic83⋊C4—21no—
30C2×Dic83⋊C4—21no—
31C2×C22⋊Dic83—21no—
32C2×C22⋊Dic83—21no—
33C2×C22⋊Dic83—21no—
34C2×C22⋊Dic83—22no—
35C2×C22⋊Dic83—41no—
36C2×C22⋊Dic83—41no—
37C2×C22⋊Dic83C814yescomplemented
38C2×C22⋊Dic83C2×C411yescharacteristic
39C2×C22⋊Dic83C2×C411yescharacteristic
40C22⋊(C83⋊C8)—22no—
41C22⋊(C83⋊C8)—41no—
42C83⋊#95—21no—
43C83⋊#95—21no—
44C83⋊#95—21no—
45C83⋊#95—21no—
46C25×C83C2×C411yescharacteristic
47C23×C4×C83D411yescharacteristic
48C23×C4×C83D411yescharacteristic
49C22×C8×C83—41no—
50C2×C22⋊C4×C83—41no—
51C2×C22⋊C4×C83—41no—
52C2×C22⋊C4×C83C2×C411yescharacteristic
53C2×C22⋊C4×C83D411yescharacteristic
54C2×C22⋊C4×C83D411yescharacteristic
55C2×C22⋊C4×C83C2311yescharacteristic
56C22⋊C8×C83—22no—
57C22⋊C8×C83—41no—
58C22×Dic83—21no—
59C22×Dic83—21no—
60C22×Dic83—24no—
61C22×Dic83—21no—
62C22×Dic83—21no—
63C22×Dic83—41no—
64C22×Dic83—41no—
65C22×Dic83—81no—
66C22×Dic83—81no—
67C22×Dic83—81no—
68C22×Dic83—81no—
69C2×C83⋊C8—42no—
70C2×C83⋊C8—82no—
71Dic83⋊C4—81no—
72Dic83⋊C4—81no—
73Dic83⋊C4—81no—
74Dic83⋊C4—81no—
75C22⋊Dic83—24no—
76C22⋊Dic83—21no—
77C22⋊Dic83—21no—
78C22⋊Dic83—42no—
79C22⋊Dic83—41no—
80C22⋊Dic83—41no—
81C22⋊Dic83C2×C812yescomplemented
82C22⋊Dic83C2×C812yescomplemented
83C24×C83—21no—
84C24×C83—21no—
85C24×C83—22no—
86C24×C83—81no—
87C24×C83—81no—
88C24×C83C22⋊C411yescharacteristic
89C24×C83C22⋊C411yescharacteristic
90C24×C83C2×C812yes—
91C24×C83M1612yes—
92C24×C83C22×C411yescharacteristic
93C22×C4×C83—21no—
94C22×C4×C83—21no—
95C22×C4×C83—21no—
96C22×C4×C83—21no—
97C22×C4×C83—81no—
98C22×C4×C83—81no—
99C22×C4×C83C22⋊C411yescharacteristic
100C22×C4×C83C22⋊C411yescharacteristic
101C22×C4×C83C2×D411yescharacteristic
102C22×C4×C83C2×D411yescharacteristic
103C2×C8×C83—42no—
104C2×C8×C83—82no—
105C22⋊C4×C83—41no—
106C22⋊C4×C83—41no—
107C22⋊C4×C83—41no—
108C22⋊C4×C83—41no—
109C23×C83—21no—
110C23×C83—21no—
111C23×C83—21no—
112C23×C83—21no—
113C23×C83—22no—
114C23×C83—22no—
115C23×C83—22no—
116C23×C83—22no—
117C23×C83—41no—
118C23×C83—41no—
119C23×C83—41no—
120C23×C83—41no—
121C23×C83—41no—
122C23×C83—41no—
123C23×C83—81no—
124C23×C83—81no—
125C23×C83—81no—
126C23×C83—81no—
127C23×C83—82no—
128C23×C83—82no—
129C23×C83—84no—
130C23×C83C22⋊C812yes—
131C23×C83C22⋊C812yes—
132C23×C83#9911yescharacteristic
133C23×C83#9911yescharacteristic
134C23×C83#10112yes—
135C23×C83C2×C22⋊C411yescharacteristic
136C23×C83C22×C811yescharacteristic
137C23×C83C2×M1611yescharacteristic
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83—42no—
143C2×C4×C83—42no—
144C2×C4×C83—41no—
145C2×C4×C83—41no—
146C2×C4×C83—41no—
147C2×C4×C83—41no—
148C2×C4×C83—81no—
149C2×C4×C83—81no—
150C2×Dic83—22no—
151C2×Dic83—22no—
152C2×Dic83—41no—
153C2×Dic83—41no—
154C2×Dic83—41no—
155C2×Dic83—41no—
156C2×Dic83—41no—
157C2×Dic83—41no—
158C2×Dic83—41no—
159C2×Dic83—41no—
160C2×Dic83—41no—
161C2×Dic83—41no—
162C2×Dic83—84no—
163C2×Dic83—82no—
164C2×Dic83—82no—
165C8×C83—82no—
166C83⋊C8—82no—
167C22×C83—21no—
168C22×C83—21no—
169C22×C83—21no—
170C22×C83—21no—
171C22×C83—21no—
172C22×C83—21no—
173C22×C83—22no—
174C22×C83—21no—
175C22×C83—21no—
176C22×C83—42no—
177C22×C83—42no—
178C22×C83—42no—
179C22×C83—41no—
180C22×C83—41no—
181C22×C83—42no—
182C22×C83—42no—
183C22×C83—41no—
184C22×C83—41no—
185C22×C83—41no—
186C22×C83—41no—
187C22×C83—82no—
188C22×C83—82no—
189C22×C83—82no—
190C22×C83—82no—
191C22×C83#32412yes—
192C22×C83#32412yes—
193C22×C83C2×C22⋊C811yescharacteristic
194C22×C83C2×#9911yescharacteristic
195C22×C83C2×#10111yescharacteristic
196C4×C83—41no—
197C4×C83—41no—
198C4×C83—41no—
199C4×C83—41no—
200Dic83—42no—
201Dic83—42no—
202Dic83—81no—
203Dic83—81no—
204#11453—831nomaximal, Sylow
205C2×C83—21no—
206C2×C83—21no—
207C2×C83—22no—
208C2×C83—22no—
209C2×C83—42no—
210C2×C83—41no—
211C2×C83—41no—
212C2×C83C2×#32411yescharacteristic
213C2×C83#212111yescharacteristic
214C2×C83#212111yescharacteristic
215#2315—831no—
216C2×#323—831no—
217C2×#323D8311yescharacteristic
218#1326—832no—
219#1326—832no—
220C83#1145311yescharacteristic, complemented, Sylow
221C22×C22⋊C4—831no—
222C22×C22⋊C4—831no—
223C22×C22⋊C4C2×D8311yescharacteristic
224C2×C22⋊C8—21no—
225C2×C22⋊C8—1661no—
226#385—831no—
227#385—831no—
228#385—831no—
229#385—831no—
230#385—1661no—
231#385—1661no—
232#380—1661no—
233#380—1661no—
234#323—21no—
235#323—832no—
236#323—1661no—
237#323C2×D8312yes—
238C25C4×D8311yescharacteristic
239C23×C4—831no—
240C23×C4—831no—
241C23×C4—1661no—
242C23×C4D33211yescharacteristic
243C23×C4C22⋊D8311yescharacteristic
244C22×C8—41no—
245C22×C8—3321no—
246C2×C4⋊C4—1661no—
247C2×C4⋊C4—1661no—
248C2×C4⋊C4—1661no—
249C2×C4⋊C4—1661no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—3321no—
253C2×C22⋊C4—3321no—
254C2×C22⋊C4—834no—
255C2×C22⋊C4—831no—
256C2×C22⋊C4—831no—
257C2×C22⋊C4—1661no—
258C2×C22⋊C4—1661no—
259C2×C22⋊C4—1661no—
260C2×C22⋊C4—1662no—
261C2×C22⋊C4C4×D8311yescharacteristic
262C2×C22⋊C4D33211yescharacteristic
263C2×C22⋊C4C22⋊D8311yescharacteristic
264C2×C22⋊C4C22×D8311yescharacteristic
265C22⋊C8—22no—
266C22⋊C8—41no—
267C22⋊C8—3321no—
268C22⋊C8—1662no—
269#95—1661no—
270#95—1661no—
271#95—1661no—
272#95—1661no—
273C24—21no—
274C24—21no—
275C24—22no—
276C24—81no—
277C24—81no—
278C24(C2×D83)⋊C411yescharacteristic
279C24(C2×D83)⋊C411yescharacteristic
280C24C8×D8312yes—
281C24#1305057312yes—
282C24C2×C4×D8311yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4—3321no—
290C22×C4—3321no—
291C22×C4—6641no—
292C22×C4—6641no—
293C22×C4—6641no—
294C22×C4—6641no—
295C22×C4—1661no—
296C22×C4—1661no—
297C22×C4—1661no—
298C22×C4—1661no—
299C22×C4—1664no—
300C22×C4(C2×D83)⋊C411yescharacteristic
301C22×C4(C2×D83)⋊C411yescharacteristic
302C22×C4C2×C22⋊D8311yescharacteristic
303C22×C4C2×D33211yescharacteristic
304C2×C8—42no—
305C2×C8—82no—
306C2×C8—3322no—
307C2×C8—6642no—
308C4⋊C4—6641no—
309C4⋊C4—6641no—
310C4⋊C4—6641no—
311C4⋊C4—6641no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—3322no—
317C22⋊C4—3321no—
318C22⋊C4—3321no—
319C22⋊C4—832no—
320C22⋊C4—832no—
321C22⋊C4—1661no—
322C22⋊C4—1661no—
323C22⋊C4—1664no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—82no—
343C23—82no—
344C23—84no—
345C23(C2×D83)⋊C812yes—
346C23(C2×D83)⋊C812yes—
347C23#1458170611yescharacteristic
348C23#1458170611yescharacteristic
349C23#1458171012yes—
350C23C2×(C2×D83)⋊C411yescharacteristic
351C23C2×C8×D8311yescharacteristic
352C23C2×#1305057311yescharacteristic
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—3321no—
366C2×C4—3321no—
367C2×C4—3321no—
368C2×C4—3321no—
369C2×C4—3321no—
370C2×C4—3321no—
371C2×C4—3321no—
372C2×C4—3321no—
373C2×C4—3321no—
374C2×C4—3321no—
375C2×C4—6644no—
376C2×C4—6642no—
377C2×C4—6642no—
378C2×C4—1662no—
379C2×C4—1662no—
380C8—82no—
381C8—6642no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—22no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—41no—
395C22—41no—
396C22—42no—
397C22—42no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—82no—
403C22—82no—
404C22—82no—
405C22—82no—
406C22#1912742612yes—
407C22#1912742612yes—
408C22C2×(C2×D83)⋊C811yescharacteristic
409C22C2×#1458170611yescharacteristic
410C22C2×#1458171011yescharacteristic
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—3322no—
416C4—3322no—
417C4—6641no—
418C4—6641no—
419C2—21no—
420C2—21no—
421C2—22no—
422C2—22no—
423C2—41no—
424C2—41no—
425C2—42no—
426C2C2×#1912742611yescharacteristic
427C2#2490910711yescharacteristic
428C2#2490911011yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC23×C83C22×C8
FrattiniC2×C22⋊C4C22×D83
FittingC2×#323×C83C2
socleC22×C83C2×#99
prime radicalC25×C83C2×C4
cosocleC2×C22⋊C4×C83C23
hypercenterC2×#323D83
hypocenterC83#11453
LCSG ▷ C23×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #24905412C2semidirect
4. #24905410C2semidirect
37. C2×C22⋊Dic83C8semidirect
81. C22⋊Dic83C2×C8semidirect
82. C22⋊Dic83C2×C8semidirect
220. C83#11453semidirect

Maximal subgroups

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

subgroupindex
#249100752
C2×#191274252
#249054122
#249054102
C2×#323×C832
#1145383