Finite Group Databasebeta

← #106982

Subgroups — #106982

Order 512.

806 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2984C211yesmaximal, characteristic
2#20735C211yesmaximal, characteristic
3C2×#1491C211yesmaximal, characteristic
4#8089C211yesmaximal, characteristic
5#7972C211yesmaximal, characteristic
6#7926C211yesmaximal, characteristic
7#7918C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2984—22no—
10#2984—22no—
11#2984—22no—
12#2984C414yescomplemented
13C2×#446—22no—
14C2×#446—21no—
15C2×C8×Q8—21no—
16C2×#432—21no—
17C2×#432C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431—21no—
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23#2283C2211yescharacteristic
24#1823C412yes—
25#1805—22no—
26#1787C412yes—
27C4×#103—22no—
28#1491—22no—
29#1491—21no—
30#1491—21no—
31C2×Q8⋊C8C414yescomplemented
32#1302—22no—
33#1288—22no—
34#1282—21no—
35#1282—21no—
36#1281—22no—
37C23×Q8C2×C411yescharacteristic
38C22×M16—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8C2×C412yes—
46C2×C4×Q8C2×C412yes—
47C2×C4×Q8D412yes—
48C2×C4×Q8D412yes—
49C2×C4×Q8D411yescharacteristic
50C2×C4×Q8D411yescharacteristic
51C2×C4×Q8Q811yescharacteristic
52C2×C4×Q8Q811yescharacteristic
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C42—21no—
56C22×C42C2311yescharacteristic
57#446—28no—
58#446—24no—
59#446—24no—
60#446—28no—
61C8×Q8—24no—
62C8×Q8—24no—
63#432—24no—
64#432—21no—
65#432—21no—
66#432—24no—
67#432—21no—
68#432—21no—
69#432—21no—
70#432—21no—
71#432C2×C412yes—
72#432D412yes—
73#431—24no—
74#431—21no—
75#431—21no—
76#431—24no—
77#431—21no—
78#431—21no—
79#431—21no—
80#431—21no—
81#431C2×C412yes—
82#431D412yes—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2×C412yes—
89C2×#103—24no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8C2×C412yes—
93C2×C22⋊C8D411yescharacteristic
94C2×C22⋊C8D411yescharacteristic
95C4×M16—21no—
96C4×M16—21no—
97C4×M16—21no—
98C4×M16—21no—
99C2×#97—22no—
100C2×#97—21no—
101C2×#97—21no—
102C2×#97D412yes—
103C2×C4×C8—22no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2×C412yes—
107#383—21no—
108#382—21no—
109#381—21no—
110#380C2×C412yes—
111C4×C4⋊C4—21no—
112C4×C22⋊C4—21no—
113#376—21no—
114#340—24no—
115Q8⋊C8—24no—
116Q8⋊C8—24no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—44no—
121C22×Q8C4212yes—
122C22×Q8C22⋊C412yes—
123C22×Q8C4⋊C411yescharacteristic
124C22×Q8C4⋊C411yescharacteristic
125C23×C4—21no—
126C23×C4—21no—
127C23×C4C22×C411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4#4111yescharacteristic
130C2×M16—24no—
131C2×M16—42no—
132C2×M16—41no—
133C2×M16—41no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—21no—
145C4×Q8—22no—
146C4×Q8—21no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—21no—
150C4×Q8—21no—
151C4×Q8—21no—
152C4×Q8—21no—
153C4×Q8—22no—
154C4×Q8—21no—
155C4×Q8—21no—
156C4×Q8C22⋊C412yes—
157C4×Q8C22⋊C412yes—
158C4×Q8C22⋊C412yes—
159C4×Q8C22⋊C412yes—
160C4×Q8C4⋊C412yes—
161C4×Q8C4⋊C412yes—
162C4×Q8C4⋊C412yes—
163C4×Q8C4⋊C412yes—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C4212yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C4⋊C411yescharacteristic
179C2×C4⋊C4C4⋊C411yescharacteristic
180C2×C22⋊C4—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—41no—
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×Q811yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42#4111yescharacteristic
202C4⋊C8—28no—
203C4⋊C8—28no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212#103—48no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—21no—
216C22⋊C8—21no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8—41no—
220C22⋊C8—41no—
221C22⋊C8C22⋊C412yes—
222C22⋊C8C22⋊C412yes—
223#97—28no—
224#97—22no—
225#97—41no—
226#97—41no—
227#97—41no—
228#97—41no—
229#97C22⋊C414yes—
230C4×C8—28no—
231C4×C8—22no—
232C4×C8—22no—
233C4×C8—22no—
234C4×C8—22no—
235C4×C8—41no—
236C4×C8—41no—
237#95—22no—
238#95—41no—
239#95—41no—
240#95—41no—
241C24C4×D411yescharacteristic
242C2×Q8—24no—
243C2×Q8—22no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—44no—
253C2×Q8—44no—
254C2×Q8—44no—
255C2×Q8#9512yes—
256C2×Q8#9512yes—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4C2×C4211yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C4⋊C411yescharacteristic
277C22×C4#11711yescharacteristic
278C22×C4C4×D411yescharacteristic
279C22×C4C4×D411yescharacteristic
280C22×C4C4×D411yescharacteristic
281C22×C4C4×D411yescharacteristic
282C22×C4C4×D411yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4#12011yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4C22⋊Q811yescharacteristic
288C22×C4C22⋊Q811yescharacteristic
289C22×C4#12311yescharacteristic
290C22×C4#12311yescharacteristic
291M16—44no—
292M16—44no—
293C2×C8—24no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—24no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—44no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—44no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—82no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—44no—
319C4⋊C4#9512yes—
320C4⋊C4#9512yes—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—82no—
332C42C2×C22⋊C411yescharacteristic
333C42C2×C22⋊C411yescharacteristic
334C42C2×C22⋊C411yescharacteristic
335C42C2×C4⋊C411yescharacteristic
336C42C2×C4⋊C411yescharacteristic
337C42#11711yescharacteristic
338C42C4×D411yescharacteristic
339C42C4×D411yescharacteristic
340C42#12011yescharacteristic
341C42#12011yescharacteristic
342C42#12111yescharacteristic
343C42#12111yescharacteristic
344C42C22⋊Q811yescharacteristic
345C42C22⋊Q811yescharacteristic
346C42#12311yescharacteristic
347C42#12311yescharacteristic
348C23—41no—
349C23—41no—
350C23C4×C22⋊C411yescharacteristic
351C23#38511yescharacteristic
352C23#38611yescharacteristic
353C23#44312yes—
354C23#44412yes—
355Q8—24no—
356Q8—22no—
357Q8—22no—
358Q8—22no—
359Q8—44no—
360Q8—44no—
361Q8—42no—
362Q8#34414yes—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4C2×#9511yescharacteristic
400C2×C4C4×C22⋊C411yescharacteristic
401C2×C4#37911yescharacteristic
402C2×C4#38011yescharacteristic
403C2×C4#38011yescharacteristic
404C2×C4#38111yescharacteristic
405C2×C4#38511yescharacteristic
406C2×C4#38511yescharacteristic
407C2×C4#38511yescharacteristic
408C2×C4#38611yescharacteristic
409C2×C4#38611yescharacteristic
410C2×C4#38611yescharacteristic
411C8—42no—
412C8—42no—
413C8—44no—
414C8—44no—
415C8—82no—
416C8—82no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—42no—
424C22—42no—
425C22C22⋊#9511yescharacteristic
426C22#176411yescharacteristic
427C22#176511yescharacteristic
428C22#186612yes—
429C22#187612yes—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—24no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—42no—
443C4—42no—
444C4—162no—
445C4C2×#34412yes—
446C4#190512yes—
447C2—22no—
448C2—22no—
449C2—21no—
450C2—21no—
451C2#1073711yescharacteristic
452C2#1078611yescharacteristic
453C2#1079111yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2984C4semidirect
31. C2×Q8⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29842
#207352
C2×#14912
#80892
#79722
#79262
#79182