Finite Group Databasebeta

← #35297743

Subgroups — #35297743

Order 21248.

830 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C2×#19127561C211yesmaximal, characteristic
2#24907373C211yesmaximal, characteristic
3#24905305C214yesmaximal, complemented
4C83×#1779C211yesmaximal, characteristic
5C22×C4⋊Dic83C411yescharacteristic
6C22×C4⋊Dic83C411yescharacteristic
7C22×C22⋊Dic83C412yes—
8C2×C22⋊(C83⋊C8)—21no—
9C2×C22⋊(C83⋊C8)—21no—
10C2×C22⋊(C83⋊C8)C2212yes—
11#19127561—21no—
12#19127561—21no—
13#19127561—21no—
14#19127561—21no—
15C2×C83⋊#95C2212yes—
16C24×C4×C83C2211yescharacteristic
17C2×C22⋊C8×C83—21no—
18C2×C22⋊C8×C83—21no—
19C2×C22⋊C8×C83C2212yes—
20C23×Dic83C2×C412yes—
21C23×Dic83C2×C412yes—
22C22×C83⋊C8—22no—
23C2×C4⋊Dic83—21no—
24C2×C4⋊Dic83—21no—
25C2×C4⋊Dic83—21no—
26C2×C4⋊Dic83—21no—
27C2×C4⋊Dic83—42no—
28C2×C4⋊Dic83—42no—
29C2×C22⋊Dic83—22no—
30C2×C22⋊Dic83—22no—
31C2×C22⋊Dic83—44no—
32C22⋊(C83⋊C8)—24no—
33C22⋊(C83⋊C8)—42no—
34C22⋊(C83⋊C8)—42no—
35C83⋊#95—22no—
36C83⋊#95—22no—
37C25×C83C2×C411yescharacteristic
38C23×C4×C83—24no—
39C23×C4×C83—42no—
40C23×C4×C83—42no—
41C23×C4×C83C2×C411yescharacteristic
42C23×C4×C83D411yescharacteristic
43C23×C4×C83D411yescharacteristic
44C23×C4×C83D411yescharacteristic
45C23×C4×C83D411yescharacteristic
46C23×C4×C83C2311yescharacteristic
47C22×C8×C83—22no—
48C22⋊C8×C83—24no—
49C22⋊C8×C83—42no—
50C22⋊C8×C83—42no—
51C22×Dic83—22no—
52C22×Dic83—22no—
53C22×Dic83—24no—
54C22×Dic83—24no—
55C22×Dic83—42no—
56C22×Dic83—42no—
57C22×Dic83—42no—
58C22×Dic83—42no—
59C2×C83⋊C8—42no—
60C2×C83⋊C8—42no—
61C2×C83⋊C8—42no—
62C4⋊Dic83—42no—
63C4⋊Dic83—42no—
64C4⋊Dic83—42no—
65C4⋊Dic83—42no—
66C22⋊Dic83—44no—
67C22⋊Dic83—44no—
68C24×C83—22no—
69C24×C83—22no—
70C24×C83—22no—
71C24×C83—22no—
72C24×C83—22no—
73C24×C83—41no—
74C24×C83—41no—
75C24×C83C22⋊C411yescharacteristic
76C24×C83C22⋊C411yescharacteristic
77C24×C83C22×C411yescharacteristic
78C22×C4×C83—22no—
79C22×C4×C83—22no—
80C22×C4×C83—22no—
81C22×C4×C83—22no—
82C22×C4×C83—22no—
83C22×C4×C83—22no—
84C22×C4×C83—22no—
85C22×C4×C83—22no—
86C22×C4×C83—22no—
87C22×C4×C83—22no—
88C22×C4×C83—22no—
89C22×C4×C83—41no—
90C22×C4×C83—41no—
91C22×C4×C83—41no—
92C22×C4×C83—41no—
93C22×C4×C83—41no—
94C22×C4×C83—41no—
95C22×C4×C83—416no—
96C22×C4×C83C22⋊C411yescharacteristic
97C22×C4×C83C22⋊C411yescharacteristic
98C22×C4×C83C2×D411yescharacteristic
99C22×C4×C83C2×D411yescharacteristic
100C22×C4×C83#4114yes—
101C2×C8×C83—42no—
102C2×C8×C83—42no—
103C2×C8×C83—42no—
104C23×C83—21no—
105C23×C83—21no—
106C23×C83—21no—
107C23×C83—21no—
108C23×C83—28no—
109C23×C83—21no—
110C23×C83—21no—
111C23×C83—21no—
112C23×C83—21no—
113C23×C83—21no—
114C23×C83—21no—
115C23×C83—21no—
116C23×C83—21no—
117C23×C83—21no—
118C23×C83—21no—
119C23×C83—21no—
120C23×C83—21no—
121C23×C83—44no—
122C23×C83—44no—
123C23×C83—44no—
124C23×C83—44no—
125C23×C83—44no—
126C23×C83—44no—
127C23×C83#10012yes—
128C23×C83#10112yes—
129C23×C83C2×C22⋊C411yescharacteristic
130C23×C83#11712yes—
131C23×C83#13112yes—
132C23×C83#13112yes—
133C2×C4×C83—28no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—21no—
137C2×C4×C83—21no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83—44no—
143C2×C4×C83—44no—
144C2×C4×C83—44no—
145C2×C4×C83—44no—
146C2×C4×C83—44no—
147C2×C4×C83#9912yes—
148C2×C4×C83#9912yes—
149C2×C4×C83#12312yes—
150C2×C4×C83#12312yes—
151C2×Dic83—42no—
152C2×Dic83—42no—
153C2×Dic83—42no—
154C2×Dic83—42no—
155C2×Dic83—42no—
156C2×Dic83—42no—
157C2×Dic83—84no—
158C2×Dic83—84no—
159C8×C83—82no—
160C83⋊C8—82no—
161C22×C83—22no—
162C22×C83—22no—
163C22×C83—22no—
164C22×C83—22no—
165C22×C83—22no—
166C22×C83—22no—
167C22×C83—22no—
168C22×C83—22no—
169C22×C83—22no—
170C22×C83—22no—
171C22×C83—22no—
172C22×C83—22no—
173C22×C83—416no—
174C22×C83—41no—
175C22×C83—41no—
176C22×C83—41no—
177C22×C83—41no—
178C22×C83—41no—
179C22×C83—41no—
180C22×C83—41no—
181C22×C83—41no—
182C22×C83#38111yescharacteristic
183C22×C83#40811yescharacteristic
184C22×C83#40811yescharacteristic
185C22×C83C2×#9911yescharacteristic
186C22×C83C2×#9911yescharacteristic
187C22×C83C2×#10011yescharacteristic
188C22×C83C2×#10111yescharacteristic
189C22×C83#43314yes—
190C4×C83—22no—
191C4×C83—22no—
192C4×C83—22no—
193C4×C83—22no—
194Dic83—82no—
195Dic83—82no—
196#8704—831nomaximal, Sylow
197C2×C83—24no—
198C2×C83—42no—
199C2×C83—42no—
200C2×C83#147111yescharacteristic
201C2×C83#147111yescharacteristic
202C2×C83#147211yescharacteristic
203C2×C83#147211yescharacteristic
204C2×C83#181811yescharacteristic
205C2×C83#181911yescharacteristic
206C2×C83#182111yescharacteristic
207C2×#380—831no—
208#1779—831no—
209#1779D8311yescharacteristic
210#1280—834no—
211C83#870411yescharacteristic, complemented, Sylow
212C24×C4C2×D8311yescharacteristic
213C22×C4⋊C4—831no—
214C22×C4⋊C4—831no—
215C22×C22⋊C4—832no—
216C2×C22⋊C8—21no—
217C2×C22⋊C8—21no—
218C2×C22⋊C8—832no—
219C2×C22⋊C8—1661no—
220C2×C22⋊C8—1661no—
221C2×C22⋊C8C2×D8312yes—
222#380—1661no—
223#380—1661no—
224#380—1661no—
225#380—1661no—
226C2×#95—832no—
227C25C4×D8311yescharacteristic
228C23×C4—24no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—832no—
232C23×C4—832no—
233C23×C4C4×D8311yescharacteristic
234C23×C4D33211yescharacteristic
235C23×C4D33211yescharacteristic
236C23×C4C22⋊D8311yescharacteristic
237C23×C4C22⋊D8311yescharacteristic
238C23×C4C22×D8311yescharacteristic
239C22×C8—22no—
240C22×C8—1662no—
241C2×C4⋊C4—3322no—
242C2×C4⋊C4—3322no—
243C2×C4⋊C4—1661no—
244C2×C4⋊C4—1661no—
245C2×C4⋊C4—1661no—
246C2×C4⋊C4—1661no—
247C2×C22⋊C4—3324no—
248C2×C22⋊C4—1662no—
249C2×C22⋊C4—1662no—
250C22⋊C8—24no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253C22⋊C8—3322no—
254C22⋊C8—3322no—
255C22⋊C8—1664no—
256#95—1662no—
257#95—1662no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—41no—
264C24—41no—
265C24(C2×D83)⋊C411yescharacteristic
266C24(C2×D83)⋊C411yescharacteristic
267C24C2×C4×D8311yescharacteristic
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—416no—
286C22×C4—3322no—
287C22×C4—3322no—
288C22×C4—3322no—
289C22×C4—3322no—
290C22×C4—1662no—
291C22×C4—1662no—
292C22×C4—1664no—
293C22×C4—1664no—
294C22×C4(C2×D83)⋊C411yescharacteristic
295C22×C4(C2×D83)⋊C411yescharacteristic
296C22×C4C2×C22⋊D8311yescharacteristic
297C22×C4C2×D33211yescharacteristic
298C22×C4#1305058914yes—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—3322no—
303C2×C8—3322no—
304C2×C8—3322no—
305C4⋊C4—3322no—
306C4⋊C4—3322no—
307C4⋊C4—3322no—
308C4⋊C4—3322no—
309C22⋊C4—3324no—
310C22⋊C4—3324no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—28no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—44no—
329C23—44no—
330C23—44no—
331C23—44no—
332C23—44no—
333C23—44no—
334C23#1458170812yes—
335C23#1458171012yes—
336C23C2×(C2×D83)⋊C411yescharacteristic
337C23#1458174812yes—
338C23#1458179812yes—
339C23#1458180012yes—
340C2×C4—28no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—3322no—
355C2×C4—3322no—
356C2×C4—3322no—
357C2×C4—3322no—
358C2×C4—3322no—
359C2×C4—3322no—
360C2×C4—6644no—
361C2×C4—6644no—
362C2×C4#1458170612yes—
363C2×C4#1458170612yes—
364C2×C4#1458177312yes—
365C2×C4#1458177612yes—
366C8—82no—
367C8—6642no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—416no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22#1912756211yescharacteristic
390C22#1912766711yescharacteristic
391C22#1912766911yescharacteristic
392C22C2×#1458170611yescharacteristic
393C22C2×#1458170611yescharacteristic
394C22C2×#1458170811yescharacteristic
395C22C2×#1458171011yescharacteristic
396C22#1912775914yes—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—22no—
401C4—6642no—
402C4—6642no—
403C2—24no—
404C2—42no—
405C2—42no—
406C2#2490582211yescharacteristic
407C2#2490582411yescharacteristic
408C2#2490582611yescharacteristic
409C2#2490582811yescharacteristic
410C2#2490753211yescharacteristic
411C2#2490753511yescharacteristic
412C2#2490754111yescharacteristic
subgroupG/H
centerC23C2×(C2×D83)⋊C4
commutatorC24×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#1779C2
socleC23×C83C2×C22⋊C4
prime radicalC25×C83C2×C4
cosocleC23×C4×C83C23
hypercenter#1779D83
hypocenterC83#8704
LCSG ▷ C24×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
3. #24905305C2semidirect
211. C83#8704semidirect

Maximal subgroups

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

subgroupindex
C2×#191275612
#249073732
#249053052
C83×#17792
#870483