Finite Group Databasebeta

← #117600

Subgroups — #117600

Order 512.

768 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21948C211yesmaximal, characteristic
2C2×#1826C211yesmaximal, characteristic
3#20699C211yesmaximal, characteristic
4#20685C211yesmaximal, characteristic
5#8051C211yesmaximal, characteristic
6#7925C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C22×#117C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1855—22no—
16#1855—22no—
17#1826—22no—
18#1826—22no—
19#1825—22no—
20#1824C411yescharacteristic, complemented
21#1824C411yescharacteristic, complemented
22#1823C411yescharacteristic, complemented
23#1823C411yescharacteristic, complemented
24#1818—22no—
25#1817—22no—
26#1810—22no—
27#1735—22no—
28C2×#339—22no—
29#1729—22no—
30#1297—22no—
31#1282—22no—
32#1282—22no—
33C24×C4C2311yescharacteristic
34C22×M16D412yes—
35C22×M16D412yes—
36C23×C8—21no—
37C23×C8D412yes—
38C2×#117—21no—
39C2×#117—21no—
40C2×#117—21no—
41C2×#117—21no—
42C2×#117—21no—
43C2×#117—21no—
44C2×#117—21no—
45C2×#117—21no—
46C2×#117—21no—
47C2×#117—21no—
48C2×#117—21no—
49C2×#117D412yes—
50C2×#117Q812yes—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C42—21no—
56C22×C42—21no—
57C2×#108—24no—
58#407—21no—
59#407—21no—
60#407—21no—
61#407—21no—
62#407—21no—
63#407—21no—
64#407—21no—
65#407—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8C2×C411yescharacteristic
80C2×C22⋊C8C2×C411yescharacteristic
81C2×C22⋊C8D412yes—
82C2×C22⋊C8D412yes—
83#381—21no—
84#380—21no—
85#380C2×C411yescharacteristic
86#380C2×C411yescharacteristic
87#379—21no—
88C4×C22⋊C4—21no—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—21no—
91C4×C22⋊C4—21no—
92C4×C22⋊C4—21no—
93C4×C22⋊C4D412yes—
94C2×#95—21no—
95#341—24no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#339—28no—
101#336—28no—
102C25#4111yescharacteristic
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—44no—
106C23×C4—41no—
107C23×C4C22×C411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×Q811yescharacteristic
116C23×C4#4111yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C23×C4#4111yescharacteristic
120C23×C4#4111yescharacteristic
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—44no—
126C2×M16—44no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—44no—
133C22×C8—44no—
134C22×C8—41no—
135C22×C8—41no—
136C22×C8C22⋊C412yes—
137C22×C8C22⋊C412yes—
138#117—22no—
139#117—22no—
140#117—22no—
141#117—22no—
142#117—22no—
143#117—22no—
144#117—22no—
145#117—22no—
146#117—41no—
147#117—41no—
148#117—41no—
149#117—41no—
150#117—41no—
151#117—41no—
152#117—41no—
153#117—41no—
154#117—44no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4C4⋊C412yes—
168C2×C4⋊C4C4⋊C412yes—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—41no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42—41no—
190C2×C42—44no—
191C2×C42—41no—
192C2×C42—41no—
193#108—84no—
194C22⋊C8—21no—
195C22⋊C8—21no—
196C22⋊C8—21no—
197C22⋊C8—21no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206C22⋊C8—41no—
207C22⋊C8—41no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—41no—
211C22⋊C8—41no—
212#95—21no—
213#95—21no—
214#95—22no—
215#95—41no—
216#95—41no—
217C24—22no—
218C24—22no—
219C24—42no—
220C24—81no—
221C24C4×D411yescharacteristic
222C24#12111yescharacteristic
223C24C22⋊Q811yescharacteristic
224C24#12311yescharacteristic
225C24#12411yescharacteristic
226C24#12411yescharacteristic
227C24#12611yescharacteristic
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C4⋊C411yescharacteristic
259C22×C4#11711yescharacteristic
260C22×C4C4×D411yescharacteristic
261C22×C4C4×D411yescharacteristic
262C22×C4C4×D411yescharacteristic
263C22×C4#12011yescharacteristic
264C22×C4#12011yescharacteristic
265C22×C4#12011yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4#12111yescharacteristic
273C22×C4C22⋊Q811yescharacteristic
274C22×C4C22⋊Q811yescharacteristic
275C22×C4C22⋊Q811yescharacteristic
276C22×C4C22⋊Q811yescharacteristic
277C22×C4C22⋊Q811yescharacteristic
278C22×C4#12311yescharacteristic
279C22×C4#12311yescharacteristic
280C22×C4#12311yescharacteristic
281C22×C4#12311yescharacteristic
282C22×C4#12411yescharacteristic
283C22×C4#12611yescharacteristic
284C22×C4#12711yescharacteristic
285M16—44no—
286M16—44no—
287M16—44no—
288M16—82no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—44no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—44no—
295C2×C8—44no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—82no—
300C2×C8—82no—
301C2×C8—82no—
302C2×C8—82no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C22⋊C4—24no—
314C22⋊C4—44no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C42—44no—
325C42—42no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—82no—
330C42—82no—
331C23—22no—
332C23—22no—
333C23—21no—
334C23—21no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—21no—
341C23—21no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—84no—
354C23—84no—
355C23#38511yescharacteristic
356C23#38811yescharacteristic
357C23#39011yescharacteristic
358C23#39311yescharacteristic
359C23#39411yescharacteristic
360C23#39611yescharacteristic
361C23#39711yescharacteristic
362C23#44312yes—
363C23#44412yes—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—88no—
401C2×C4—82no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4#38011yescharacteristic
411C2×C4#38511yescharacteristic
412C2×C4#38611yescharacteristic
413C2×C4#38811yescharacteristic
414C2×C4#39211yescharacteristic
415C2×C4#39411yescharacteristic
416C2×C4#39611yescharacteristic
417C2×C4#39911yescharacteristic
418C2×C4#45814yes—
419C8—42no—
420C8—42no—
421C8—82no—
422C8—82no—
423C8—82no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—42no—
442C22—42no—
443C22—44no—
444C22—42no—
445C22—42no—
446C22—44no—
447C22—81no—
448C22—81no—
449C22—82no—
450C22—82no—
451C22#143911yescharacteristic
452C22#186612yes—
453C22#192912yes—
454C22#197411yescharacteristic
455C22#197511yescharacteristic
456C22#203012yes—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—84no—
464C4—82no—
465C4—82no—
466C4—161no—
467C4—161no—
468C4#189012yes—
469C4#202212yes—
470C2—21no—
471C2—21no—
472C2—22no—
473C2—22no—
474C2—22no—
475C2—41no—
476C2—41no—
477C2—41no—
478C2#1107011yescharacteristic
479C2#1112511yescharacteristic
480C2#1113111yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
socleC22#1439
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #1824C4semidirect
21. #1824C4semidirect
22. #1823C4semidirect
23. #1823C4semidirect

Maximal subgroups

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

subgroupindex
#219482
C2×#18262
#206992
#206852
#80512
#79252
C2×#12822