Finite Group Databasebeta

← #121492

Subgroups — #121492

Order 512.

700 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21359C211yesmaximal, characteristic
2#21170C211yesmaximal, characteristic
3#14150C211yesmaximal, characteristic
4#8026C211yesmaximal, characteristic
5C2×#1286C211yesmaximal, characteristic
6#7901C211yesmaximal, characteristic
7#7892C211yesmaximal, characteristic
8C2×#431C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#428—21no—
11C2×#423C2211yescharacteristic
12C2×#389—21no—
13#2285C2211yescharacteristic
14C2×#382C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17#1949—22no—
18#1948—21no—
19#1948—21no—
20#1948—21no—
21#1947—21no—
22#1853—21no—
23#1853—21no—
24C4×#106—22no—
25#1736—21no—
26#1444—21no—
27#1442—21no—
28#1436—22no—
29C4⋊#95—21no—
30#1294—22no—
31#1286C414yescomplemented
32#1279—22no—
33#1279—22no—
34C22×M16—21no—
35C2×#117—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4D411yescharacteristic
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C42—21no—
43C22×C42—21no—
44C22×C42C2311yescharacteristic
45#431—21no—
46#431—21no—
47#431—21no—
48#431—21no—
49#431—21no—
50#431C2×C412yes—
51#431Q812yes—
52#431Q812yes—
53#428—42no—
54#428—42no—
55C2×#107—22no—
56C2×#106—22no—
57C2×#106—22no—
58C2×#106—22no—
59#423—21no—
60#423—21no—
61#423C2×C412yes—
62#423Q812yes—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8D412yes—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C4×C8—21no—
70C2×C4×C8D412yes—
71#389—42no—
72#389—42no—
73#384D411yescharacteristic
74#384D411yescharacteristic
75#382—21no—
76#382—21no—
77#382—21no—
78#382C2×C412yes—
79#382Q812yes—
80#380—21no—
81#380—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95D412yes—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—41no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—41no—
120#117—41no—
121#117—41no—
122#117—41no—
123#117—41no—
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—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4SD1612yes—
156C2×C4⋊C4SD1612yes—
157C2×C22⋊C4—41no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42—41no—
167C2×C42—82no—
168C2×C42C22×C411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×Q811yescharacteristic
173C2×C42C2×Q811yescharacteristic
174C2×C42C2×Q811yescharacteristic
175C2×C42C2×Q811yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180#107—44no—
181#106—44no—
182#106—44no—
183#106—44no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—41no—
191C4⋊C8C4⋊C414yescomplemented
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—41no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—41no—
199C4×C8C4⋊C414yescomplemented
200#95—22no—
201#95—22no—
202#95—22no—
203#95—44no—
204#95—44no—
205#95—41no—
206#95—41no—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—82no—
212#95C4⋊C414yes—
213C24#12011yescharacteristic
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4C2×C4⋊C411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4C4×D411yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12011yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4C22⋊Q811yescharacteristic
264C22×C4#12311yescharacteristic
265C22×C4#12711yescharacteristic
266C22×C4C4⋊Q811yescharacteristic
267C22×C4C4⋊Q811yescharacteristic
268C22×C4C4⋊Q811yescharacteristic
269C22×C4C2×SD1611yescharacteristic
270C22×C4C2×SD1611yescharacteristic
271C22×C4#13511yescharacteristic
272C22×C4#13511yescharacteristic
273C22×C4AGL(1,ℤ/8)11yescharacteristic
274C22×C4AGL(1,ℤ/8)11yescharacteristic
275C22×C4#13711yescharacteristic
276C22×C4#13711yescharacteristic
277M16—42no—
278M16—42no—
279M16—42no—
280M16—42no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—81no—
289C2×C8—81no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—81no—
317C4⋊C4—81no—
318C4⋊C4—88no—
319C22⋊C4—81no—
320C22⋊C4—81no—
321C42—82no—
322C42—82no—
323C42—81no—
324C42—81no—
325C42—81no—
326C42—81no—
327C42—81no—
328C42—81no—
329C42C2×C4⋊C411yescharacteristic
330C42C2×C4⋊C411yescharacteristic
331C42C4×D411yescharacteristic
332C42C4×Q811yescharacteristic
333C42C22⋊Q811yescharacteristic
334C42C22⋊Q811yescharacteristic
335C42C22⋊Q811yescharacteristic
336C42C22⋊Q811yescharacteristic
337C42#12311yescharacteristic
338C42#12311yescharacteristic
339C42#12511yescharacteristic
340C42#12511yescharacteristic
341C42#12511yescharacteristic
342C42C4⋊Q811yescharacteristic
343C42C4⋊Q811yescharacteristic
344C42C4⋊Q811yescharacteristic
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23#38511yescharacteristic
350C23#39211yescharacteristic
351C23#39511yescharacteristic
352C23#44811yescharacteristic
353C23#44911yescharacteristic
354C23#45011yescharacteristic
355C23#45211yescharacteristic
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4#38411yescharacteristic
410C2×C4#38511yescharacteristic
411C2×C4#38511yescharacteristic
412C2×C4#38911yescharacteristic
413C2×C4#38911yescharacteristic
414C2×C4#38911yescharacteristic
415C2×C4#39511yescharacteristic
416C2×C4#39511yescharacteristic
417C2×C4#39811yescharacteristic
418C2×C4#39811yescharacteristic
419C2×C4#39811yescharacteristic
420C2×C4#39911yescharacteristic
421C2×C4C4×SD1611yescharacteristic
422C2×C4C4×SD1611yescharacteristic
423C2×C4#44111yescharacteristic
424C2×C4#44211yescharacteristic
425C2×C4#44811yescharacteristic
426C2×C4#44911yescharacteristic
427C2×C4#45011yescharacteristic
428C2×C4#45211yescharacteristic
429C2×C4#46511yescharacteristic
430C2×C4#46511yescharacteristic
431C2×C4#46811yescharacteristic
432C2×C4#46811yescharacteristic
433C2×C4#47511yescharacteristic
434C2×C4#47511yescharacteristic
435C2×C4#47611yescharacteristic
436C2×C4#47611yescharacteristic
437C2×C4#47811yescharacteristic
438C2×C4#47811yescharacteristic
439C2×C4#47911yescharacteristic
440C2×C4#47911yescharacteristic
441C2×C4#49811yescharacteristic
442C2×C4#49911yescharacteristic
443C2×C4#50111yescharacteristic
444C2×C4#50111yescharacteristic
445C8—44no—
446C8—44no—
447C8—82no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22#144411yescharacteristic
461C22#186411yescharacteristic
462C22#186511yescharacteristic
463C22#200011yescharacteristic
464C22#200011yescharacteristic
465C22#200811yescharacteristic
466C22#200911yescharacteristic
467C22#201011yescharacteristic
468C22#201011yescharacteristic
469C22#205811yescharacteristic
470C22#205911yescharacteristic
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—84no—
480C4—82no—
481C4—82no—
482C4—82no—
483C4—82no—
484C4—161no—
485C4—161no—
486C4#186411yescharacteristic
487C4#186511yescharacteristic
488C4#194111yescharacteristic
489C4#194311yescharacteristic
490C4#205811yescharacteristic
491C4#205911yescharacteristic
492C4#208811yescharacteristic
493C4#208811yescharacteristic
494C2—21no—
495C2—21no—
496C2—21no—
497C2—21no—
498C2#999811yescharacteristic
499C2#999911yescharacteristic
500C2#1000011yescharacteristic
501C2#1000111yescharacteristic
502C2#1132711yescharacteristic
503C2#1133311yescharacteristic
504C2#1133511yescharacteristic
subgroupG/H
center, socleC23#392
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
31. #1286C4semidirect
191. C4⋊C8C4⋊C4semidirect
199. C4×C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#213592
#211702
#141502
#80262
C2×#12862
#79012
#78922