Finite Group Databasebeta

← #35299975

Subgroups — #35299975

Order 21248.

694 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127754C211yesmaximal, characteristic, complemented
2C2×#19127435C212yesmaximal
3#24905690C211yesmaximal, characteristic, complemented
4#24905341C212yesmaximal, complemented
5C83×#2854C211yesmaximal, characteristic
6#19127754—24no—
7C2×C4⋊(C83⋊C8)—21no—
8C2×C22⋊(C83⋊C8)C2212yes—
9C2×C4×C83⋊C8—21no—
10C2×C83⋊#95—21no—
11C2×C83⋊#95—21no—
12C2×C83⋊#95C2212yes—
13#19127435—24no—
14C2×#123×C83—21no—
15C2×#121×C83C2211yescharacteristic
16C22×C42×C83C2211yescharacteristic
17#390×C83C411yescharacteristic, complemented
18#390×C83C411yescharacteristic, complemented
19#389×C83C411yescharacteristic, complemented
20#389×C83C411yescharacteristic, complemented
21#388×C83—21no—
22#388×C83—21no—
23#386×C83—21no—
24#380×C83C2211yescharacteristic
25C23×Dic83—22no—
26C22×C83⋊C8—22no—
27C4⋊(C83⋊C8)—24no—
28C22⋊(C83⋊C8)—22no—
29C22⋊(C83⋊C8)D414yes—
30C4×C83⋊C8—24no—
31C83⋊#95—24no—
32C83⋊#95—22no—
33C83⋊#95—22no—
34C83⋊#95—42no—
35C22×D4×C83—21no—
36C22×D4×C83C2×C411yescharacteristic
37C23×C4×C83D411yescharacteristic
38C23×C4×C83D411yescharacteristic
39C23×C4×C83C2311yescharacteristic
40#123×C83—42no—
41#123×C83—42no—
42#121×C83—22no—
43#121×C83—22no—
44C2×C4⋊C4×C83—21no—
45C2×C4⋊C4×C83—21no—
46C2×C4⋊C4×C83C2×C411yescharacteristic
47C2×C4⋊C4×C83C2×C411yescharacteristic
48C2×C4⋊C4×C83C2×C411yescharacteristic
49C2×C22⋊C4×C83—21no—
50C2×C22⋊C4×C83—21no—
51C2×C22⋊C4×C83—21no—
52C2×C22⋊C4×C83—21no—
53C2×C22⋊C4×C83—21no—
54C2×C22⋊C4×C83—21no—
55C2×C42×C83—22no—
56C2×C42×C83—22no—
57C2×C42×C83C2×C411yescharacteristic
58C2×C42×C83C2×C411yescharacteristic
59C2×C42×C83D411yescharacteristic
60C2×C42×C83D411yescharacteristic
61#95×C83—21no—
62#95×C83—21no—
63#95×C83—21no—
64C22×Dic83—24no—
65C22×Dic83—42no—
66C22×Dic83—42no—
67C22×Dic83—42no—
68C22×Dic83—42no—
69C22×Dic83—42no—
70C22×Dic83—42no—
71C2×C83⋊C8—24no—
72C2×C83⋊C8—42no—
73C2×C83⋊C8—42no—
74C24×C83—21no—
75C24×C83C2×D411yescharacteristic
76C2×D4×C83—22no—
77C2×D4×C83—41no—
78C2×D4×C83—41no—
79C2×D4×C83—42no—
80C2×D4×C83—41no—
81C2×D4×C83—41no—
82C2×D4×C83—42no—
83C22×C4×C83—22no—
84C22×C4×C83—22no—
85C22×C4×C83—22no—
86C22×C4×C83—22no—
87C22×C4×C83—21no—
88C22×C4×C83—21no—
89C22×C4×C83—21no—
90C22×C4×C83—41no—
91C22×C4×C83—41no—
92C22×C4×C83—41no—
93C22×C4×C83C22⋊C411yescharacteristic
94C22×C4×C83C22⋊C411yescharacteristic
95C22×C4×C83C22⋊C411yescharacteristic
96C22×C4×C83C22⋊C411yescharacteristic
97C22×C4×C83D812yes—
98C22×C4×C83SD1612yes—
99C22×C4×C83C22×C411yescharacteristic
100C22×C4×C83C2×D412yes—
101C22×C4×C83C2×D411yescharacteristic
102C22×C4×C83#4112yes—
103C4⋊C4×C83—22no—
104C4⋊C4×C83—41no—
105C4⋊C4×C83—41no—
106C4⋊C4×C83—42no—
107C4⋊C4×C83—42no—
108C22⋊C4×C83—42no—
109C22⋊C4×C83—41no—
110C22⋊C4×C83—41no—
111C22⋊C4×C83—42no—
112C22⋊C4×C83—42no—
113C22⋊C4×C83—42no—
114C22⋊C4×C83—41no—
115C22⋊C4×C83—41no—
116C42×C83—22no—
117C42×C83—22no—
118C42×C83—21no—
119C42×C83—21no—
120C42×C83—21no—
121C42×C83—21no—
122C23×C83—21no—
123C23×C83—21no—
124C23×C83—21no—
125C23×C83—21no—
126C23×C83—41no—
127C23×C83—41no—
128C23×C83—41no—
129C23×C83—41no—
130C23×C83—41no—
131C23×C83—41no—
132C23×C83—41no—
133C23×C83C2×C22⋊C411yescharacteristic
134C23×C83#12112yes—
135C23×C83C2×D811yescharacteristic
136C23×C83C2×SD1611yescharacteristic
137C23×C83#13512yes—
138D4×C83—82no—
139D4×C83—82no—
140D4×C83—82no—
141C2×C4×C83—21no—
142C2×C4×C83—21no—
143C2×C4×C83—21no—
144C2×C4×C83—21no—
145C2×C4×C83—21no—
146C2×C4×C83—21no—
147C2×C4×C83—21no—
148C2×C4×C83—21no—
149C2×C4×C83—21no—
150C2×C4×C83—21no—
151C2×C4×C83—24no—
152C2×C4×C83—24no—
153C2×C4×C83—21no—
154C2×C4×C83—21no—
155C2×C4×C83—21no—
156C2×C4×C83—42no—
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—41no—
164C2×C4×C83—41no—
165C2×C4×C83—41no—
166C2×C4×C83—41no—
167C2×C4×C83—41no—
168C2×C4×C83#10212yes—
169C2×C4×C83#10212yes—
170C2×C4×C83#10412yes—
171C2×C4×C83C4×D412yes—
172C2×C4×C83#12011yescharacteristic
173C2×C4×C83#12311yescharacteristic
174C2×Dic83—44no—
175C2×Dic83—44no—
176C2×Dic83—44no—
177C2×Dic83—82no—
178C2×Dic83—82no—
179C2×Dic83—82no—
180C2×Dic83—82no—
181C83⋊C8—44no—
182C22×C83—22no—
183C22×C83—22no—
184C22×C83—22no—
185C22×C83—21no—
186C22×C83—21no—
187C22×C83—21no—
188C22×C83—21no—
189C22×C83—22no—
190C22×C83—81no—
191C22×C83—81no—
192C22×C83—81no—
193C22×C83—81no—
194C22×C83—81no—
195C22×C83—81no—
196C22×C83—81no—
197C22×C83#38611yescharacteristic
198C22×C83#41011yescharacteristic
199C22×C83#41011yescharacteristic
200C22×C83C2×#10211yescharacteristic
201C22×C83#41611yescharacteristic
202C22×C83C2×#10411yescharacteristic
203C22×C83#42111yescharacteristic
204C22×C83#46512yes—
205C22×C83#46612yes—
206C4×C83—22no—
207C4×C83—22no—
208C4×C83—22no—
209C4×C83—21no—
210C4×C83—22no—
211C4×C83—21no—
212C4×C83—41no—
213C4×C83—81no—
214C4×C83—81no—
215C4×C83—81no—
216Dic83—84no—
217#9114—831nomaximal, Sylow
218C2×C83—22no—
219C2×C83—22no—
220C2×C83—161no—
221C2×C83#150011yescharacteristic
222C2×C83#150211yescharacteristic
223C2×C83#151711yescharacteristic
224C2×C83#152111yescharacteristic
225C2×C83#187111yescharacteristic
226C2×C83#189311yescharacteristic
227C2×C83#189911yescharacteristic
228C2×#431—831no—
229#2854D8311yescharacteristic
230C2×#327—832no—
231#1434—831no—
232#1294—832no—
233C83#911411yescharacteristic, complemented, Sylow
234C2×#123—21no—
235C2×#121C2×D8311yescharacteristic
236C22×C42C2×D8311yescharacteristic
237#431—1664no—
238C2×C4⋊C8—1661no—
239C2×C22⋊C8—832no—
240C2×C4×C8—1661no—
241#390Dic8311yescharacteristic, complemented
242#390Dic8311yescharacteristic, complemented
243#389Dic8311yescharacteristic, complemented
244#389Dic8311yescharacteristic, complemented
245#388—21no—
246#388—21no—
247#386—21no—
248#380C2×D8311yescharacteristic
249C2×#95—832no—
250C2×#95—1661no—
251C2×#95—1661no—
252#327—1664no—
253C22×D4—21no—
254C22×D4C2×Dic8311yescharacteristic
255C23×C4—1662no—
256C23×C4C22⋊D8311yescharacteristic
257C23×C4C22⋊D8311yescharacteristic
258C23×C4C22×D8311yescharacteristic
259C22×C8—1662no—
260#123—42no—
261#123—42no—
262#121—22no—
263#121—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4C2×Dic8311yescharacteristic
267C2×C4⋊C4C2×Dic8311yescharacteristic
268C2×C4⋊C4C2×Dic8311yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42C2×Dic8311yescharacteristic
278C2×C42C2×Dic8311yescharacteristic
279C2×C42C22⋊D8311yescharacteristic
280C2×C42C22⋊D8311yescharacteristic
281C4⋊C8—1664no—
282C22⋊C8—834no—
283C22⋊C8—1662no—
284C4×C8—1664no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—3322no—
289#95—1664no—
290#95—1662no—
291#95—1662no—
292C24—21no—
293C24C2×C22⋊D8311yescharacteristic
294C2×D4—22no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—3322no—
312C22×C4—3322no—
313C22×C4—3322no—
314C22×C4—3322no—
315C22×C4—3322no—
316C22×C4—3322no—
317C22×C4—1664no—
318C22×C4C22⋊Dic8311yescharacteristic
319C22×C4C22⋊Dic8311yescharacteristic
320C22×C4C22⋊Dic8311yescharacteristic
321C22×C4C22⋊Dic8311yescharacteristic
322C22×C4#1305057512yes—
323C22×C4#1305057812yes—
324C22×C4C22×Dic8311yescharacteristic
325C22×C4D4×D8312yes—
326C22×C4C2×C22⋊D8311yescharacteristic
327C22×C4#1305058912yes—
328C2×C8—3322no—
329C2×C8—3322no—
330C2×C8—1664no—
331C4⋊C4—22no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—22no—
349C42—22no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23C2×C22⋊Dic8311yescharacteristic
362C23#1458176412yes—
363C23C2×#1305057511yescharacteristic
364C23C2×#1305057811yescharacteristic
365C23#1458181212yes—
366D4—82no—
367D4—82no—
368D4—82no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—24no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—24no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—3324no—
397C2×C4—3324no—
398C2×C4—3324no—
399C2×C4—6642no—
400C2×C4—6642no—
401C2×C4—6642no—
402C2×C4—6642no—
403C2×C4#1458171312yes—
404C2×C4#1458171312yes—
405C2×C4#1458171912yes—
406C2×C4D4×Dic8312yes—
407C2×C4#1458176011yescharacteristic
408C2×C4#1458177311yescharacteristic
409C8—3324no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22#1912758111yescharacteristic
426C22#1912767511yescharacteristic
427C22#1912767511yescharacteristic
428C22C2×#1458171311yescharacteristic
429C22#1912769711yescharacteristic
430C22C2×#1458171911yescharacteristic
431C22#1912771811yescharacteristic
432C22#1912792812yes—
433C22#1912793512yes—
434C4—22no—
435C4—22no—
436C4—21no—
437C4—21no—
438C4—22no—
439C4—22no—
440C4—41no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—6644no—
445C2—22no—
446C2—22no—
447C2—161no—
448C2#2490594911yescharacteristic
449C2#2490595811yescharacteristic
450C2#2490602311yescharacteristic
451C2#2490604011yescharacteristic
452C2#2490776611yescharacteristic
453C2#2490789211yescharacteristic
454C2#2490792211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic83
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#2854C2
socleC23×C83C2×C22⋊C4
prime radicalC2×#121×C83C22
cosocleC23×C4×C83C23
hypercenter#2854D83
hypocenterC83#9114
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
1. C2×#19127754C2semidirect
3. #24905690C2semidirect
4. #24905341C2semidirect
17. #390×C83C4semidirect
18. #390×C83C4semidirect
19. #389×C83C4semidirect
20. #389×C83C4semidirect
233. C83#9114semidirect
241. #390Dic83semidirect
242. #390Dic83semidirect
243. #389Dic83semidirect
244. #389Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×#191277542
C2×#191274352
#249056902
#249053412
C83×#28542
#911483