Finite Group Databasebeta

← #416494

Subgroups — #416494

Order 512.

1140 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3139C211yesmaximal, characteristic
2#21312C211yesmaximal, characteristic
3#21302C211yesmaximal, characteristic
4C2×#1981C211yesmaximal, characteristic
5C2×#1966C211yesmaximal, characteristic
6C2×#1950C211yesmaximal, characteristic
7#11164C214yesmaximal
8#11128C214yesmaximal
9C4×#343C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3139—24no—
13#3139—24no—
14C2×#491—22no—
15C2×#487—21no—
16C2×#487—21no—
17C22×#108C2212yes—
18C2×#423C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C2×#419C2212yes—
21C2×#419C2212yes—
22C22×#103C2214yes—
23C22×#101C2214yes—
24C2×C4×M16C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1989—22no—
28#1988—21no—
29#1987—21no—
30#1985—22no—
31#1984—22no—
32#1981—22no—
33#1981—22no—
34#1966—24no—
35#1950—24no—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#341C2212yes—
39#1736C2214yes—
40#1729C2214yes—
41C23×Q8C2312yes—
42C22×M16C2312yes—
43C22×M16C2311yescharacteristic
44C22×M16C2311yescharacteristic
45C22×M16C2312yes—
46C23×C8C2312yes—
47#533—22no—
48#533—22no—
49#533—22no—
50#533—22no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8D414yes—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D411yescharacteristic
61C2×#125D412yes—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×#117C2312yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2312yes—
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72#491—216no—
73#487—28no—
74#487—28no—
75C2×#108—24no—
76C2×#108—44no—
77#423—22no—
78#423—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8D412yes—
84C2×C4⋊C8D412yes—
85#419—24no—
86#419—24no—
87#419—24no—
88#419—24no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—28no—
94C2×#103—28no—
95C2×#103—44no—
96C2×#101—28no—
97C2×#101—28no—
98C2×#101—44no—
99C4×M16—22no—
100C4×M16—22no—
101C4×M16—22no—
102C4×M16D414yes—
103C2×#97—21no—
104C2×#97—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C2×#95—21no—
112#343—24no—
113#343—24no—
114#341—24no—
115#340—24no—
116#340—24no—
117#336—28no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—24no—
126C22×Q8—24no—
127C22×Q8—24no—
128C22×Q8—24no—
129C22×Q8C2×D414yes—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4#4111yescharacteristic
133C23×C4#4111yescharacteristic
134C23×C4C2411yescharacteristic
135C2×M16—21no—
136C2×M16—21no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—24no—
149C2×M16—24no—
150C2×M16C2×D412yes—
151C2×M16C2×D412yes—
152C2×M16#4114yes—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—24no—
158C22×C8—22no—
159C22×C8—24no—
160C22×C8—22no—
161C22×C8C2×D414yes—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—24no—
165C4⋊Q8—24no—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174#125—24no—
175#125—24no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C22⋊Q8—42no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×Q8—42no—
184#117—22no—
185#117—22no—
186#117—22no—
187#117—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4#4114yes—
208C2×C22⋊C4—22no—
209C2×C42—22no—
210C2×C42—41no—
211C2×C42—41no—
212C2×C42—82no—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42#4112yes—
220C2×C42#4112yes—
221#108—48no—
222C4⋊C8—24no—
223C4⋊C8—24no—
224C4⋊C8—42no—
225C4⋊C8—42no—
226#103—28no—
227#103—28no—
228#103—28no—
229#103—28no—
230#103—416no—
231#101—416no—
232#97—24no—
233#97—42no—
234C4×C8—24no—
235C4×C8—42no—
236#95—41no—
237#95—41no—
238C24C2×#4111yescharacteristic
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—28no—
242C2×Q8—28no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—44no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—44no—
257C2×Q8—44no—
258C2×Q8—44no—
259C2×Q8—44no—
260C2×Q8—44no—
261C2×Q8—82no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4#12111yescharacteristic
278C22×C4#12111yescharacteristic
279C22×C4#12111yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4#12312yes—
282C22×C4#12312yes—
283C22×C4#12412yes—
284C22×C4#12412yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C2×#4111yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C4C2×#4112yes—
292C22×C4C2×#4112yes—
293C22×C4H(5,𝔽2)12yes—
294C22×C42−1+412yes—
295M16—24no—
296M16—22no—
297M16—22no—
298M16—22no—
299M16—22no—
300M16—24no—
301M16—42no—
302M16—42no—
303M16—42no—
304M16—42no—
305M16—42no—
306M16—42no—
307M16—44no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—24no—
311C2×C8—28no—
312C2×C8—24no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—44no—
320C2×C8—42no—
321C2×C8—44no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C42—42no—
341C42—42no—
342C42—82no—
343C42—82no—
344C42#12012yes—
345C42#12012yes—
346C42#12112yes—
347C42#12112yes—
348C42#12112yes—
349C42#12112yes—
350C42#12312yes—
351C42#12312yes—
352C23—21no—
353C23—21no—
354C23—22no—
355C23C2×#12111yescharacteristic
356C23C2×#12311yescharacteristic
357C23C2×#12411yescharacteristic
358C23#52912yes—
359C23#54212yes—
360Q8—48no—
361Q8—44no—
362Q8—44no—
363Q8—84no—
364Q8—84no—
365Q8—84no—
366Q8—82no—
367Q8—82no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—48no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—84no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—84no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—84no—
391C2×C4#39412yes—
392C2×C4#39412yes—
393C2×C4#39412yes—
394C2×C4#39412yes—
395C2×C4#47112yes—
396C2×C4#47312yes—
397C2×C4C2×#12011yescharacteristic
398C2×C4C2×#12111yescharacteristic
399C2×C4C2×#12111yescharacteristic
400C2×C4C2×#12311yescharacteristic
401C2×C4#52912yes—
402C2×C4#53211yescharacteristic
403C2×C4#53211yescharacteristic
404C2×C4#53511yescharacteristic
405C2×C4#53611yescharacteristic
406C2×C4#54212yes—
407C2×C4#54612yes—
408C2×C4#54612yes—
409C2×C4#54712yes—
410C2×C4#54812yes—
411C2×C4#54812yes—
412C2×C4#55012yes—
413C2×C4#55612yes—
414C2×C4#55912yes—
415C8—44no—
416C8—42no—
417C8—42no—
418C8—44no—
419C8—82no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22C2×#39411yescharacteristic
429C22#242612yes—
430C22#244712yes—
431C22#264011yescharacteristic
432C22#286111yescharacteristic
433C22C2×#47111yescharacteristic
434C22C2×#47311yescharacteristic
435C22#306912yes—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—84no—
440C4—84no—
441C4—82no—
442C4—82no—
443C4#205214yes—
444C4#242712yes—
445C4#243111yescharacteristic
446C4#243211yescharacteristic
447C4#244812yes—
448C4#254411yescharacteristic
449C4#254511yescharacteristic
450C2—21no—
451C2—21no—
452C2—22no—
453C2#1522611yescharacteristic
454C2C2×#205211yescharacteristic
455C2#2353211yescharacteristic
456C2#2531312yes—
457C2#2531412yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#31392
#213122
#213022
C2×#19812
C2×#19662
C2×#19502
#111642
#111282
C4×#3432