Finite Group Databasebeta

← #120091

Subgroups — #120091

Order 512.

696 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21170C211yesmaximal, characteristic
2#13954C211yesmaximal, characteristic
3C2×#1303C211yesmaximal, characteristic
4#8026C211yesmaximal, characteristic
5#8014C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7#7892C211yesmaximal, characteristic
8C2×#431C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#428—21no—
11C2×#423C2211yescharacteristic
12C2×#389—21no—
13C2×#389—21no—
14#2285C2211yescharacteristic
15C2×#382C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18#1949—22no—
19#1948C412yescomplemented
20#1947C412yescomplemented
21#1736—21no—
22#1444—22no—
23C4⋊#95—21no—
24C4⋊#95—21no—
25#1303—22no—
26#1294—21no—
27#1294—21no—
28#1294—22no—
29#1286—22no—
30#1279—22no—
31C22×M16—21no—
32C2×#117—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4D411yescharacteristic
38C22×C4⋊C4D411yescharacteristic
39C22×C42—21no—
40C22×C42—21no—
41C22×C42C2311yescharacteristic
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431—21no—
48#431Q812yes—
49#431Q812yes—
50#428—42no—
51#428—42no—
52C2×#107—22no—
53C2×#106—22no—
54#423—21no—
55#423—21no—
56#423D412yes—
57#423Q812yes—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8C2×C412yes—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C4×C8—21no—
65C2×C4×C8—21no—
66#389—42no—
67#389—42no—
68#389—42no—
69#389—42no—
70#384C2×C411yescharacteristic
71#384C2×C411yescharacteristic
72#382—21no—
73#382—21no—
74#382—21no—
75#382—21no—
76#382—21no—
77#380—21no—
78#380—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4C2×C412yes—
88C4×C4⋊C4D412yes—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95D411yescharacteristic
94C2×#95D411yescharacteristic
95C2×#95D412yes—
96#340—22no—
97#340—22no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—41no—
109C2×M16—41no—
110C2×M16—41no—
111C2×M16—41no—
112C22×C8—22no—
113C22×C8—41no—
114C22×C8—41no—
115#117—41no—
116#117—41no—
117#117—41no—
118#117—41no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—82no—
146C2×C4⋊C4—82no—
147C2×C4⋊C4C22⋊C412yes—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4D812yes—
150C2×C4⋊C4SD1612yes—
151C2×C4⋊C4SD1612yes—
152C2×C4⋊C4Q1612yes—
153C2×C22⋊C4—41no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—41no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42—82no—
164C2×C42C22×C411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×Q811yescharacteristic
168C2×C42C2×Q811yescharacteristic
169C2×C42C2×Q811yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176#107—44no—
177#106—44no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185C4⋊C8C4⋊C414yescomplemented
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190C4×C8—22no—
191C4×C8—22no—
192C4×C8—41no—
193C4×C8—41no—
194#95—22no—
195#95—22no—
196#95—21no—
197#95—21no—
198#95—22no—
199#95—21no—
200#95—21no—
201#95—22no—
202#95—44no—
203#95—44no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—41no—
208C24#12011yescharacteristic
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4C22⋊Q811yescharacteristic
255C22×C4C22⋊Q811yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4#12311yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12311yescharacteristic
260C22×C4#12411yescharacteristic
261C22×C4C4⋊Q811yescharacteristic
262C22×C4C4⋊Q811yescharacteristic
263C22×C4C4⋊Q811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×SD1611yescharacteristic
266C22×C4C2×SD1611yescharacteristic
267C22×C4C2×Q1611yescharacteristic
268C22×C4AGL(1,ℤ/8)11yescharacteristic
269C22×C4AGL(1,ℤ/8)11yescharacteristic
270C22×C4#13711yescharacteristic
271C22×C4#13711yescharacteristic
272M16—42no—
273M16—42no—
274M16—42no—
275M16—42no—
276C2×C8—24no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—81no—
282C2×C8—81no—
283C2×C8—81no—
284C2×C8—81no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—81no—
298C4⋊C4—81no—
299C4⋊C4—81no—
300C4⋊C4—81no—
301C4⋊C4—88no—
302C4⋊C4—88no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C4⋊C4#10214yes—
308C4⋊C4#10314yes—
309C22⋊C4—81no—
310C22⋊C4—81no—
311C42—82no—
312C42—82no—
313C42—81no—
314C42—81no—
315C42—81no—
316C42—81no—
317C42—81no—
318C42—81no—
319C42C2×C4⋊C411yescharacteristic
320C42#11711yescharacteristic
321C42C4×Q811yescharacteristic
322C42C4×Q811yescharacteristic
323C42#12111yescharacteristic
324C42C22⋊Q811yescharacteristic
325C42C22⋊Q811yescharacteristic
326C42C22⋊Q811yescharacteristic
327C42C22⋊Q811yescharacteristic
328C42C22⋊Q811yescharacteristic
329C42#12511yescharacteristic
330C42#12511yescharacteristic
331C42#12511yescharacteristic
332C42#12511yescharacteristic
333C42#12611yescharacteristic
334C42#12611yescharacteristic
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23#38611yescharacteristic
340C23#39411yescharacteristic
341C23#39511yescharacteristic
342C23#44711yescharacteristic
343C23#44811yescharacteristic
344C23#45011yescharacteristic
345C23#45111yescharacteristic
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4#38011yescharacteristic
400C2×C4#38711yescharacteristic
401C2×C4#38711yescharacteristic
402C2×C4#38911yescharacteristic
403C2×C4#38911yescharacteristic
404C2×C4#39111yescharacteristic
405C2×C4#39611yescharacteristic
406C2×C4#39611yescharacteristic
407C2×C4#39811yescharacteristic
408C2×C4#39811yescharacteristic
409C2×C4#39811yescharacteristic
410C2×C4#40011yescharacteristic
411C2×C4C2×#10211yescharacteristic
412C2×C4C2×#10311yescharacteristic
413C2×C4#41711yescharacteristic
414C2×C4#41711yescharacteristic
415C2×C4#45911yescharacteristic
416C2×C4#46011yescharacteristic
417C2×C4#46111yescharacteristic
418C2×C4#46211yescharacteristic
419C2×C4#47411yescharacteristic
420C2×C4#47511yescharacteristic
421C2×C4#47611yescharacteristic
422C2×C4#47711yescharacteristic
423C2×C4#48011yescharacteristic
424C2×C4#48111yescharacteristic
425C2×C4#48311yescharacteristic
426C2×C4#48411yescharacteristic
427C2×C4#48611yescharacteristic
428C2×C4#48711yescharacteristic
429C2×C4#49011yescharacteristic
430C2×C4#49111yescharacteristic
431C2×C4#49811yescharacteristic
432C2×C4#50011yescharacteristic
433C2×C4#50111yescharacteristic
434C2×C4#50111yescharacteristic
435C8—44no—
436C8—82no—
437C8—82no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22#144111yescharacteristic
451C22#187211yescharacteristic
452C22#187411yescharacteristic
453C22#190411yescharacteristic
454C22#190511yescharacteristic
455C22#190611yescharacteristic
456C22#190611yescharacteristic
457C22#204111yescharacteristic
458C22#204311yescharacteristic
459C22#205711yescharacteristic
460C22#205811yescharacteristic
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—44no—
470C4—44no—
471C4—82no—
472C4—82no—
473C4—162no—
474C4—161no—
475C4—161no—
476C4#180211yescharacteristic
477C4#180311yescharacteristic
478C4#199011yescharacteristic
479C4#199211yescharacteristic
480C4#208311yescharacteristic
481C4#208411yescharacteristic
482C4#209911yescharacteristic
483C4#210011yescharacteristic
484C2—21no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2#917511yescharacteristic
489C2#917611yescharacteristic
490C2#917911yescharacteristic
491C2#918011yescharacteristic
492C2#1120711yescharacteristic
493C2#1121311yescharacteristic
494C2#1121511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #1948C4semidirect
20. #1947C4semidirect
185. C4⋊C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#211702
#139542
C2×#13032
#80262
#80142
C2×#12862
#78922