Finite Group Databasebeta

← #35299272

Subgroups — #35299272

Order 21248.

500 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#13050574C211yesmaximal, characteristic, complemented
2#24908383C211yesmaximal, characteristic
3C4×#14581716C211yesmaximal, characteristic, complemented
4#24905914C211yesmaximal, characteristic
5#24905365C211yesmaximal, characteristic, complemented
6#24905299C211yesmaximal, characteristic, complemented
7C83×#3103C211yesmaximal, characteristic
8#19127832—22no—
9#19127832—21no—
10Q8×C83⋊C8—21no—
11#19127725—21no—
12#19127725C2211yescharacteristic
13C2×#14581719—21no—
14C2×#14581719—21no—
15C2×#14581716—21no—
16C4×#13050574—21no—
17C4×#13050574C2211yescharacteristic
18C2×C4×C83⋊C8C2211yescharacteristic
19C4×C4⋊Dic83C2211yescharacteristic
20#19127463—21no—
21D4⋊(C83⋊C8)—22no—
22#532×C83C2211yescharacteristic
23C2×C4×Q8×C83C2211yescharacteristic
24#481×C83—21no—
25#475×C83C411yescharacteristic, complemented
26#475×C83C411yescharacteristic, complemented
27#460×C83C411yescharacteristic, complemented
28#460×C83C411yescharacteristic, complemented
29#448×C83—21no—
30C4×SD16×C83—21no—
31C4×SD16×C83—21no—
32#431×C83C2211yescharacteristic
33C2×#13050574—21no—
34C2×#13050574D411yescharacteristic
35C2×#13050574D411yescharacteristic
36C22×C83⋊C8—21no—
37C2×C4⋊Dic83—21no—
38C2×C4×Dic83—21no—
39C4⋊(C83⋊C8)—22no—
40C4⋊(C83⋊C8)—22no—
41C4⋊(C83⋊C8)—21no—
42C4⋊(C83⋊C8)D412yes—
43#14581719—42no—
44#14581719—42no—
45#14581716—24no—
46#14581703—22no—
47#14581703—21no—
48C4×C83⋊C8—22no—
49C4×C83⋊C8—21no—
50C4×C83⋊C8—21no—
51C4×C83⋊C8—21no—
52C83⋊#95—21no—
53C2×#41×C83—21no—
54C22×Q8×C83D411yescharacteristic
55C2×SD16×C83—21no—
56C2×SD16×C83—21no—
57C4⋊Q8×C83C2×C411yescharacteristic
58#127×C83C2×C411yescharacteristic
59#124×C83—21no—
60#121×C83—21no—
61#121×C83—21no—
62C4×Q8×C83—21no—
63C4×Q8×C83—21no—
64C4×Q8×C83C2×C411yescharacteristic
65C4×Q8×C83C2×C411yescharacteristic
66C4×Q8×C83D411yescharacteristic
67C4×Q8×C83D411yescharacteristic
68C4×D4×C83—21no—
69C4×D4×C83—21no—
70C2×C4⋊C4×C83—21no—
71C2×C4⋊C4×C83D411yescharacteristic
72C2×C42×C83—21no—
73C2×C42×C83C2311yescharacteristic
74#106×C83—21no—
75#106×C83—21no—
76C4⋊C8×C83C2×C411yescharacteristic
77C4⋊C8×C83C2×C411yescharacteristic
78#103×C83—21no—
79#103×C83—21no—
80#102×C83—21no—
81#102×C83—21no—
82C22⋊C8×C83—21no—
83C4×C8×C83—21no—
84C22×Dic83—41no—
85#13050574—21no—
86#13050574—21no—
87#13050574—21no—
88#13050574—41no—
89#13050574—41no—
90#13050574SD1612yescomplemented
91C2×C83⋊C8—21no—
92C2×C83⋊C8—22no—
93C2×C83⋊C8—22no—
94C2×C83⋊C8—21no—
95C2×C83⋊C8—41no—
96C2×C83⋊C8—41no—
97C4⋊Dic83—22no—
98C4⋊Dic83—41no—
99C4×Dic83—41no—
100C4×Dic83—41no—
101#41×C83—41no—
102#41×C83—41no—
103#41×C83—41no—
104#41×C83—41no—
105C2×Q8×C83—21no—
106C2×Q8×C83—22no—
107C2×Q8×C83—21no—
108C2×Q8×C83—41no—
109C2×Q8×C83C22⋊C411yescharacteristic
110C2×Q8×C83C22⋊C411yescharacteristic
111C2×D4×C83—21no—
112C2×D4×C83—21no—
113C2×D4×C83—21no—
114C2×D4×C83—41no—
115C22×C4×C83—21no—
116C22×C4×C83—21no—
117C22×C4×C83—41no—
118C22×C4×C83C2×D411yescharacteristic
119C22×C4×C83C2×D411yescharacteristic
120C22×C4×C83#4111yescharacteristic
121SD16×C83—42no—
122SD16×C83—42no—
123C2×C8×C83—21no—
124C2×C8×C83—21no—
125C4⋊C4×C83—21no—
126C4⋊C4×C83—21no—
127C4⋊C4×C83—21no—
128C4⋊C4×C83—21no—
129C4⋊C4×C83—21no—
130C4⋊C4×C83—21no—
131C4⋊C4×C83—21no—
132C4⋊C4×C83C22⋊C411yescharacteristic
133C4⋊C4×C83C22⋊C411yescharacteristic
134C22⋊C4×C83—41no—
135C22⋊C4×C83—41no—
136C42×C83—21no—
137C42×C83—21no—
138C42×C83—21no—
139C42×C83—21no—
140C42×C83C22×C411yescharacteristic
141C42×C83C2×D411yescharacteristic
142C42×C83C2×D411yescharacteristic
143C42×C83#4111yescharacteristic
144C23×C83—41no—
145C23×C83#12111yescharacteristic
146Q8×C83—21no—
147Q8×C83—21no—
148Q8×C83—21no—
149Q8×C83—42no—
150Q8×C83—41no—
151Q8×C83—41no—
152Q8×C83#10412yescomplemented
153D4×C83—42no—
154D4×C83—41no—
155D4×C83—41no—
156D4×C83—41no—
157D4×C83—41no—
158D4×C83—81no—
159D4×C83—81no—
160C2×C4×C83—21no—
161C2×C4×C83—21no—
162C2×C4×C83—21no—
163C2×C4×C83—21no—
164C2×C4×C83—21no—
165C2×C4×C83—21no—
166C2×C4×C83—41no—
167C2×C4×C83—42no—
168C2×C4×C83—41no—
169C2×C4×C83—81no—
170C2×C4×C83—81no—
171C2×C4×C83—81no—
172C2×C4×C83C2×C22⋊C411yescharacteristic
173C2×C4×C83C4×D411yescharacteristic
174C2×C4×C83C4×D411yescharacteristic
175C2×C4×C83#12011yescharacteristic
176C2×C4×C83#12111yescharacteristic
177C2×C4×C83#12311yescharacteristic
178C2×C4×C83C2×SD1611yescharacteristic
179C2×C4×C83#13511yescharacteristic
180C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C83#13711yescharacteristic
182C2×Dic83—42no—
183C2×Dic83—81no—
184C2×Dic83—81no—
185C8×C83—41no—
186C8×C83—41no—
187C83⋊C8—22no—
188C83⋊C8—42no—
189C83⋊C8—41no—
190C83⋊C8—42no—
191C22×C83—21no—
192C22×C83—21no—
193C22×C83—81no—
194C22×C83—81no—
195C22×C83—81no—
196C22×C83#38611yescharacteristic
197C22×C83#46011yescharacteristic
198C22×C83#46411yescharacteristic
199C4×C83—21no—
200C4×C83—22no—
201C4×C83—21no—
202C4×C83—21no—
203C4×C83—41no—
204C4×C83—41no—
205C4×C83—41no—
206C4×C83—81no—
207C4×C83C2×#10411yescharacteristic
208C4×C83#42111yescharacteristic
209C4×C83C4×SD1611yescharacteristic
210C4×C83#44011yescharacteristic
211C4×C83#44811yescharacteristic
212C4×C83#44911yescharacteristic
213Dic83—82no—
214#8992—831nomaximal, Sylow
215C2×C83—21no—
216C2×C83—21no—
217C2×C83—161no—
218C2×C83#187111yescharacteristic
219C2×C83#188011yescharacteristic
220C2×C83#188111yescharacteristic
221#3103D8311yescharacteristic
222Q8×M16—831no—
223#1980—831no—
224C4×#103—831no—
225#1492—831no—
226#1303—831no—
227#1278—831no—
228C83#899211yescharacteristic, complemented, Sylow
229#532C2×D8311yescharacteristic
230C2×C4×Q8C2×D8311yescharacteristic
231#481—21no—
232#475Dic8311yescharacteristic, complemented
233#475Dic8311yescharacteristic, complemented
234#460Dic8311yescharacteristic, complemented
235#460Dic8311yescharacteristic, complemented
236#448—21no—
237#446—1662no—
238#446—1661no—
239C8×Q8—1661no—
240C4×SD16—21no—
241C4×SD16—21no—
242#431C2×D8311yescharacteristic
243#423—831no—
244#423—1661no—
245C2×#104—1661no—
246C2×#104—1661no—
247C2×#103—1661no—
248C4×M16—831no—
249C4×M16—1661no—
250C2×C4×C8—831no—
251C4×C4⋊C4—831no—
252#340—1661no—
253D4⋊C8—1662no—
254C2×#41—21no—
255C22×Q8C22⋊D8311yescharacteristic
256C2×SD16—21no—
257C2×SD16—21no—
258C2×M16—831no—
259C2×M16—831no—
260C2×M16—1661no—
261C22×C8—1661no—
262C4⋊Q8C2×Dic8311yescharacteristic
263#127C2×Dic8311yescharacteristic
264#124—21no—
265#121—21no—
266#121—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C2×Dic8311yescharacteristic
270C4×Q8C2×Dic8311yescharacteristic
271C4×Q8C22⋊D8311yescharacteristic
272C4×Q8C22⋊D8311yescharacteristic
273C4×D4—21no—
274C4×D4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—1661no—
277C2×C4⋊C4C22⋊D8311yescharacteristic
278C2×C42—21no—
279C2×C42—1661no—
280C2×C42C22×D8311yescharacteristic
281#106—21no—
282#106—21no—
283C4⋊C8—832no—
284C4⋊C8—1662no—
285C4⋊C8—1662no—
286C4⋊C8—1661no—
287C4⋊C8C2×Dic8311yescharacteristic
288C4⋊C8C2×Dic8311yescharacteristic
289#104—3322no—
290#104—3322no—
291#103—21no—
292#103—21no—
293#103—1664no—
294#102—21no—
295#102—21no—
296C22⋊C8—21no—
297#97—1662no—
298#97—1661no—
299C4×C8—21no—
300C4×C8—1662no—
301C4×C8—1661no—
302C4×C8—1661no—
303C4×C8—1661no—
304#95—1661no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×Q8—21no—
312C2×Q8—41no—
313C2×Q8C22⋊Dic8311yescharacteristic
314C2×Q8C22⋊Dic8311yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—3321no—
323C22×C4D4×D8311yescharacteristic
324C22×C4C2×C22⋊D8311yescharacteristic
325C22×C4#1305058911yescharacteristic
326SD16—42no—
327SD16—42no—
328M16—3321no—
329M16—3321no—
330M16—832no—
331M16—1661no—
332M16—1661no—
333M16—1661no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—3321no—
337C2×C8—3321no—
338C2×C8—1662no—
339C2×C8—1662no—
340C2×C8—1661no—
341C2×C8—1661no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—3321no—
350C4⋊C4—1662no—
351C4⋊C4C22⋊Dic8311yescharacteristic
352C4⋊C4C22⋊Dic8311yescharacteristic
353C22⋊C4—41no—
354C22⋊C4—41no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—3321no—
360C42—3321no—
361C42C22×Dic8311yescharacteristic
362C42D4×D8311yescharacteristic
363C42C2×C22⋊D8311yescharacteristic
364C42#1305058911yescharacteristic
365C23—41no—
366C23#1458176411yescharacteristic
367Q8—21no—
368Q8—21no—
369Q8—21no—
370Q8—42no—
371Q8—41no—
372Q8—41no—
373Q8#1458171912yescomplemented
374D4—42no—
375D4—41no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—81no—
380D4—81no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—3322no—
394C2×C4—6641no—
395C2×C4—6641no—
396C2×C4C2×C22⋊Dic8311yescharacteristic
397C2×C4D4×Dic8311yescharacteristic
398C2×C4D4×Dic8311yescharacteristic
399C2×C4#1458176011yescharacteristic
400C2×C4#1458176411yescharacteristic
401C2×C4#1458177311yescharacteristic
402C2×C4SD16×D8311yescharacteristic
403C2×C4#1458181311yescharacteristic
404C2×C4#1458181911yescharacteristic
405C2×C4#1458182111yescharacteristic
406C8—41no—
407C8—41no—
408C8—3322no—
409C8—3322no—
410C8—3321no—
411C8—1662no—
412C22—21no—
413C22—21no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22#1912758111yescharacteristic
418C22#1912789911yescharacteristic
419C22#1912792111yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—81no—
428C4—6642no—
429C4C2×#1458171911yescharacteristic
430C4#1912771811yescharacteristic
431C4SD16×Dic8311yescharacteristic
432C4#1912780211yescharacteristic
433C4#1912783811yescharacteristic
434C4#1912784511yescharacteristic
435C2—21no—
436C2—21no—
437C2—161no—
438C2#2490776611yescharacteristic
439C2#2490782011yescharacteristic
440C2#2490782711yescharacteristic
subgroupG/H
centerC2×C4#14581760
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#3103C2
socleC22×C83#386
prime radical#532×C83C22
cosocleC2×C42×C83C23
hypercenter#3103D83
hypocenterC83#8992
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
1. Q8×#13050574C2semidirect
3. C4×#14581716C2semidirect
5. #24905365C2semidirect
6. #24905299C2semidirect
25. #475×C83C4semidirect
26. #475×C83C4semidirect
27. #460×C83C4semidirect
28. #460×C83C4semidirect
90. #13050574SD16semidirect
152. Q8×C83#104semidirect
228. C83#8992semidirect
232. #475Dic83semidirect
233. #475Dic83semidirect
234. #460Dic83semidirect
235. #460Dic83semidirect
373. Q8#14581719semidirect

Maximal subgroups

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

subgroupindex
Q8×#130505742
#249083832
C4×#145817162
#249059142
#249053652
#249052992
C83×#31032
#899283