Finite Group Databasebeta

← #34183112

Subgroups — #34183112

Order 20224.

540 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773408C211yesmaximal, characteristic
2D4×#11929608C211yesmaximal, characteristic
3#23764343C211yesmaximal, characteristic, complemented
4#23761856C211yesmaximal, characteristic, complemented
5#23761303C211yesmaximal, characteristic, complemented
6#23761237C211yesmaximal, characteristic, complemented
7C4×#102×C79C211yesmaximal, characteristic
8C22×#11929608—21no—
9#19099120C2211yescharacteristic
10#19098893—21no—
11#19098853C411yescharacteristic, complemented
12#19098853C411yescharacteristic, complemented
13#19098768C411yescharacteristic, complemented
14#19098768C411yescharacteristic, complemented
15#19098714—21no—
16C4×#11929612—21no—
17C4×#11929612—21no—
18#19098641—21no—
19#19098635—22no—
20D4×C79⋊C8—21no—
21#19098616C2211yescharacteristic
22#19098587C2211yescharacteristic
23#19098527—21no—
24C4×#11929608C2211yescharacteristic
25#19098323—21no—
26#19098323—21no—
27#19098295—22no—
28C2×C4×D4×C79C2211yescharacteristic
29C2×#102×C79—21no—
30C2×C4×C8×C79C2211yescharacteristic
31C4×C4⋊C4×C79C2211yescharacteristic
32#340×C79—21no—
33C2×#11929612—21no—
34C2×#11929612—21no—
35C2×#11929608—22no—
36C2×#11929608—21no—
37C2×#11929608—42no—
38C2×#11929608—42no—
39C2×#11929608D411yescharacteristic
40C2×#11929608D411yescharacteristic
41C22×C79⋊C8—22no—
42#14579161C2×C411yescharacteristic
43#14579161C2×C411yescharacteristic
44#14579153—21no—
45#14579140—21no—
46#14579140—21no—
47C4×Dic158—21no—
48C4×Dic158—21no—
49#14579120—21no—
50#14579092—21no—
51#14579092—21no—
52C4⋊(C79⋊C8)—21no—
53C4⋊(C79⋊C8)C2×C411yescharacteristic
54C4⋊(C79⋊C8)C2×C411yescharacteristic
55C4⋊(C79⋊C8)D412yes—
56#14579084—21no—
57#14579084—21no—
58#14579082—21no—
59#14579082—21no—
60C22⋊(C79⋊C8)—22no—
61C22⋊(C79⋊C8)—22no—
62C22⋊(C79⋊C8)—21no—
63#14579072—21no—
64C4×C79⋊C8—21no—
65C4×C79⋊C8—21no—
66C22×D4×C79D411yescharacteristic
67C23×C4×C79—21no—
68C22×C8×C79—21no—
69C4×D4×C79—21no—
70C4×D4×C79—21no—
71C4×D4×C79C2×C411yescharacteristic
72C4×D4×C79C2×C411yescharacteristic
73C4×D4×C79D411yescharacteristic
74C4×D4×C79D411yescharacteristic
75C2×C4⋊C4×C79—21no—
76C2×C4⋊C4×C79D411yescharacteristic
77C2×C22⋊C4×C79—21no—
78C2×C42×C79—21no—
79C2×C42×C79C2311yescharacteristic
80#102×C79—24no—
81C4×C8×C79—21no—
82C4×C8×C79—21no—
83#95×C79—21no—
84C2×Dic158—21no—
85C2×Dic158—21no—
86#11929612—42no—
87#11929612—42no—
88#11929608—21no—
89#11929608—21no—
90#11929608—21no—
91#11929608—44no—
92#11929608—41no—
93#11929608—41no—
94#11929608SD1612yescomplemented
95C2×C79⋊C8—21no—
96C2×C79⋊C8—21no—
97C2×C79⋊C8—21no—
98C2×C79⋊C8—21no—
99C2×C79⋊C8—22no—
100C2×C79⋊C8—44no—
101C4⋊Dic79—21no—
102C4⋊Dic79—21no—
103C4⋊Dic79—21no—
104C4⋊Dic79—21no—
105Dic79⋊C4—41no—
106Dic79⋊C4—41no—
107C22⋊Dic79—41no—
108C4×Dic79—41no—
109C24×C79—21no—
110C2×D4×C79—22no—
111C2×D4×C79—21no—
112C2×D4×C79—41no—
113C2×D4×C79C22⋊C411yescharacteristic
114C2×D4×C79C22⋊C411yescharacteristic
115C22×C4×C79—21no—
116C22×C4×C79—21no—
117C22×C4×C79—21no—
118C22×C4×C79—21no—
119C22×C4×C79—21no—
120C22×C4×C79—41no—
121C22×C4×C79—41no—
122C22×C4×C79—41no—
123C22×C4×C79—41no—
124C22×C4×C79—41no—
125C22×C4×C79C2×D411yescharacteristic
126C22×C4×C79C2×D411yescharacteristic
127C22×C4×C79#4111yescharacteristic
128C2×C8×C79—22no—
129C2×C8×C79—41no—
130C2×C8×C79—41no—
131C4⋊C4×C79—21no—
132C4⋊C4×C79—22no—
133C4⋊C4×C79—41no—
134C4⋊C4×C79C22⋊C411yescharacteristic
135C4⋊C4×C79C22⋊C411yescharacteristic
136C22⋊C4×C79—21no—
137C22⋊C4×C79—21no—
138C22⋊C4×C79—21no—
139C22⋊C4×C79—21no—
140C42×C79—41no—
141C42×C79—41no—
142C42×C79C22×C411yescharacteristic
143C42×C79C2×D411yescharacteristic
144C42×C79C2×D411yescharacteristic
145C42×C79#4111yescharacteristic
146C23×C79—21no—
147C23×C79—21no—
148C23×C79—42no—
149C23×C79—41no—
150C23×C79—41no—
151C23×C79—81no—
152C23×C79#12111yescharacteristic
153D4×C79—21no—
154D4×C79—21no—
155D4×C79—21no—
156D4×C79—42no—
157D4×C79#10412yes—
158C2×C4×C79—21no—
159C2×C4×C79—21no—
160C2×C4×C79—21no—
161C2×C4×C79—21no—
162C2×C4×C79—41no—
163C2×C4×C79—41no—
164C2×C4×C79—41no—
165C2×C4×C79—41no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
168C2×C4×C79—41no—
169C2×C4×C79—41no—
170C2×C4×C79—41no—
171C2×C4×C79—41no—
172C2×C4×C79—41no—
173C2×C4×C79—42no—
174C2×C4×C79—81no—
175C2×C4×C79—81no—
176C2×C4×C79C2×C22⋊C411yescharacteristic
177C2×C4×C79C4×D411yescharacteristic
178C2×C4×C79C4×D411yescharacteristic
179C2×C4×C79#12011yescharacteristic
180C2×C4×C79#12111yescharacteristic
181C2×C4×C79#12311yescharacteristic
182C2×C4×C79C2×SD1611yescharacteristic
183C2×C4×C79#13511yescharacteristic
184C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C79#13711yescharacteristic
186C2×Dic79—41no—
187C2×Dic79—41no—
188Dic158—42no—
189Dic158—41no—
190Dic158—41no—
191C8×C79—42no—
192C79⋊C8—22no—
193C79⋊C8—41no—
194C79⋊C8—41no—
195C79⋊C8—42no—
196C79⋊C8—41no—
197C22×C79—21no—
198C22×C79—21no—
199C22×C79—22no—
200C22×C79—41no—
201C22×C79—41no—
202C22×C79—41no—
203C22×C79—41no—
204C22×C79—41no—
205C22×C79—82no—
206C22×C79—82no—
207C22×C79#38611yescharacteristic
208C22×C79#46111yescharacteristic
209C22×C79#46311yescharacteristic
210C4×C79—21no—
211C4×C79—21no—
212C4×C79—21no—
213C4×C79—41no—
214C4×C79—41no—
215C4×C79—82no—
216C4×C79C2×#10411yescharacteristic
217C4×C79#42111yescharacteristic
218C4×C79C4×SD1611yescharacteristic
219C4×C79#44011yescharacteristic
220C4×C79#45011yescharacteristic
221C4×C79#45211yescharacteristic
222Dic79—81no—
223Dic79—81no—
224#8989—791nomaximal, Sylow
225C2×C79—21no—
226C2×C79—21no—
227C2×C79—42no—
228C2×C79—81no—
229C2×C79#187111yescharacteristic
230C2×C79#187711yescharacteristic
231C2×C79#188111yescharacteristic
232#3101—791no—
233D4×M16—791no—
234#1986—791no—
235C4×#102D7911yescharacteristic
236#1493—791no—
237#1303—791no—
238#1278—791no—
239C79#898911yescharacteristic, complemented, Sylow
240C22×M16—1581no—
241#533—791no—
242C2×C4×D4C2×D7911yescharacteristic
243#483—1581no—
244#476—791no—
245#476—791no—
246#461—791no—
247#461—791no—
248#450—1581no—
249C4×SD16—1581no—
250C4×SD16—1581no—
251#436—1581no—
252#435—1582no—
253C8×D4—1581no—
254#431—791no—
255#423—791no—
256C2×#102—21no—
257#407—1581no—
258C4×M16—791no—
259C2×C4×C8C2×D7911yescharacteristic
260C4×C4⋊C4C2×D7911yescharacteristic
261#340—21no—
262#339—1581no—
263#339—1581no—
264Q8⋊C8—1582no—
265C22×D4C22⋊D7911yescharacteristic
266C23×C4—21no—
267C2×SD16—1581no—
268C2×SD16—1581no—
269C2×M16—3162no—
270C2×M16—3162no—
271C2×M16—791no—
272C2×M16—791no—
273C2×M16—1582no—
274C2×M16—1581no—
275C22×C8—21no—
276C22×C8—1582no—
277C4⋊Q8—791no—
278C4⋊Q8—791no—
279#125—1581no—
280C22⋊Q8—1581no—
281C22⋊Q8—1581no—
282C4×Q8—1581no—
283C4×Q8—1581no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4C4×D7911yescharacteristic
287C4×D4C4×D7911yescharacteristic
288C4×D4D31611yescharacteristic
289C4×D4D31611yescharacteristic
290#117—1581no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4C22⋊D7911yescharacteristic
293C2×C22⋊C4—21no—
294C2×C42—21no—
295C2×C42C22×D7911yescharacteristic
296#106—1581no—
297#106—1581no—
298C4⋊C8—791no—
299C4⋊C8—791no—
300C4⋊C8—792no—
301C4⋊C8—1581no—
302#103—1581no—
303#103—1581no—
304#102—24no—
305#102—1581no—
306#102—1581no—
307C22⋊C8—1582no—
308C22⋊C8—1582no—
309C22⋊C8—1581no—
310#97—1581no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—1581no—
314C4×C8—1581no—
315#95—21no—
316C24—21no—
317C2×Q8—1581no—
318C2×Q8—1581no—
319C2×D4—22no—
320C2×D4—21no—
321C2×D4—41no—
322C2×D4(C2×D79)⋊C411yescharacteristic
323C2×D4(C2×D79)⋊C411yescharacteristic
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4D4×D7911yescharacteristic
335C22×C4C2×C22⋊D7911yescharacteristic
336C22×C4#1192962311yescharacteristic
337SD16—3162no—
338SD16—3162no—
339M16—3161no—
340M16—3161no—
341M16—3164no—
342M16—792no—
343M16—1581no—
344M16—1581no—
345M16—1581no—
346C2×C8—22no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—3164no—
350C2×C8—1581no—
351C2×C8—1581no—
352C2×C8—1581no—
353C2×C8—1582no—
354C2×C8—1581no—
355C4⋊C4—21no—
356C4⋊C4—22no—
357C4⋊C4—41no—
358C4⋊C4—3161no—
359C4⋊C4—3161no—
360C4⋊C4—1581no—
361C4⋊C4—1581no—
362C4⋊C4—1581no—
363C4⋊C4—1581no—
364C4⋊C4(C2×D79)⋊C411yescharacteristic
365C4⋊C4(C2×D79)⋊C411yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—21no—
369C22⋊C4—21no—
370C22⋊C4—3161no—
371C42—41no—
372C42—41no—
373C42—3161no—
374C42C2×C4×D7911yescharacteristic
375C42D4×D7911yescharacteristic
376C42C2×D31611yescharacteristic
377C42#1192962311yescharacteristic
378C23—21no—
379C23—21no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—81no—
384C23#1457913311yescharacteristic
385Q8—3162no—
386Q8—3161no—
387Q8—3161no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—42no—
392D4#1457908912yes—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—3161no—
412C2×C4—3161no—
413C2×C4C2×(C2×D79)⋊C411yescharacteristic
414C2×C4(C22⋊D79)⋊C411yescharacteristic
415C2×C4(C22⋊D79)⋊C411yescharacteristic
416C2×C4#1457913111yescharacteristic
417C2×C4#1457913311yescharacteristic
418C2×C4#1457914511yescharacteristic
419C2×C4SD16×D7911yescharacteristic
420C2×C4#1457918011yescharacteristic
421C2×C4#1457918511yescharacteristic
422C2×C4#1457919011yescharacteristic
423C8—42no—
424C8—3162no—
425C8—3161no—
426C8—3161no—
427C8—3161no—
428C8—1582no—
429C22—21no—
430C22—21no—
431C22—22no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—82no—
438C22—82no—
439C22#1909844011yescharacteristic
440C22#1909876611yescharacteristic
441C22#1909877611yescharacteristic
442C4—21no—
443C4—21no—
444C4—21no—
445C4—41no—
446C4—41no—
447C4—82no—
448C4—6321no—
449C4—6321no—
450C4C2×#1457908911yescharacteristic
451C4#1909858111yescharacteristic
452C4#1909865411yescharacteristic
453C4#1909866011yescharacteristic
454C4#1909871511yescharacteristic
455C4#1909872611yescharacteristic
456C2—21no—
457C2—21no—
458C2—42no—
459C2—81no—
460C2#2376370011yescharacteristic
461C2#2376373211yescharacteristic
462C2#2376376011yescharacteristic
subgroupG/H
centerC2×C4#14579131
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC4×#102×C79C2
socleC22×C79#386
prime radicalC22×D4×C79D4
cosocleC2×C42×C79C23
hypercenterC4×#102D79
hypocenterC79#8989
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
3. #23764343C2semidirect
4. #23761856C2semidirect
5. #23761303C2semidirect
6. #23761237C2semidirect
11. #19098853C4semidirect
12. #19098853C4semidirect
13. #19098768C4semidirect
14. #19098768C4semidirect
94. #11929608SD16semidirect
239. C79#8989semidirect

Maximal subgroups

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

subgroupindex
#237734082
D4×#119296082
#237643432
#237618562
#237613032
#237612372
C4×#102×C792
#898979