Finite Group Databasebeta

← #193348

Subgroups — #193348

Order 512.

848 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×#2962C211yesmaximal, characteristic
2#21370C211yesmaximal, characteristic
3#17807C211yesmaximal, characteristic
4#17718C211yesmaximal, characteristic
5#17501C211yesmaximal, characteristic
6#8933C214yesmaximal, complemented
7#8931C214yesmaximal, complemented
8#7887C211yesmaximal, characteristic
9#7873C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2962—22no—
12#2962—22no—
13#2962—22no—
14#2962C2214yes—
15C2×#446—21no—
16C2×#446—22no—
17C2×C8×Q8—21no—
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#424—21no—
22C22×C4⋊C8C2211yescharacteristic
23C2×#405—21no—
24C2×C4×M16C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26#1950C2212yes—
27#1950C2212yes—
28#1855—21no—
29#1855—21no—
30C4×#108—22no—
31#1573—22no—
32#1571C2212yes—
33#1570C2212yes—
34#1565—21no—
35#1565—21no—
36#1554C414yes—
37#1548C411yescharacteristic
38#1548C411yescharacteristic
39#1546C411yescharacteristic
40#1546C411yescharacteristic
41#1542C2214yes—
42#1493C2212yes—
43#1493C2212yes—
44#1491—22no—
45#1485—22no—
46#1278C2214yes—
47#1277C2214yes—
48C22×M16—21no—
49C22×M16D411yescharacteristic
50C22×M16D411yescharacteristic
51C23×C8—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533C2×C412yes—
57#533D412yes—
58#533Q812yes—
59#533C2312yes—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8Q812yes—
65C2×C4×Q8—21no—
66C2×C4×Q8D412yes—
67C2×#117—21no—
68C22×C42C2311yescharacteristic
69#446—28no—
70#446—24no—
71#446—28no—
72#446—24no—
73C8×Q8—24no—
74C8×Q8—24no—
75#432—21no—
76#432—21no—
77#432—21no—
78#432C2×C412yes—
79#431—21no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431C2×C411yescharacteristic
85#431C2×C411yescharacteristic
86#431C2×C412yes—
87#431C2312yes—
88C2×#108—22no—
89C2×#108—22no—
90#424—24no—
91#424—24no—
92#423C2×C411yescharacteristic
93#423C2×C411yescharacteristic
94#423C2311yescharacteristic
95#423C2311yescharacteristic
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8D411yescharacteristic
102C2×C4⋊C8D411yescharacteristic
103C2×C4⋊C8D412yes—
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2312yes—
106#407—21no—
107#407—21no—
108C2×C22⋊C8—21no—
109#405—24no—
110#405—24no—
111C4×M16—21no—
112C4×M16—21no—
113C4×M16—21no—
114C4×M16—21no—
115C4×M16—21no—
116C4×M16—21no—
117C4×M16—21no—
118C4×M16—21no—
119C4×M16D412yes—
120C4×M16C2312yes—
121C2×#97—22no—
122C2×#97—21no—
123C2×#97—21no—
124C2×C4×C8—22no—
125C2×C4×C8—21no—
126C2×C4×C8—21no—
127C2×C4×C8C2312yes—
128#341—22no—
129#341—22no—
130#335—22no—
131#335—22no—
132#334—22no—
133#334—22no—
134#333C2×C414yes—
135#333C2×C414yes—
136#332C2×C414yes—
137#332C2×C414yes—
138Q8⋊C8—24no—
139Q8⋊C8—24no—
140C22×Q8—21no—
141C22×Q8C4⋊C412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×Q811yescharacteristic
144C23×C4#4111yescharacteristic
145C2×M16—21no—
146C2×M16—24no—
147C2×M16—24no—
148C2×M16—21no—
149C2×M16—21no—
150C2×M16—21no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16—42no—
154C2×M16—42no—
155C2×M16—41no—
156C2×M16—41no—
157C2×M16C2×D412yes—
158C2×M16C2×D412yes—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—44no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8C4⋊C412yes—
173C4⋊Q8C4⋊C412yes—
174C4⋊Q8C22×C412yes—
175C4⋊Q8C22×C412yes—
176#125—22no—
177#125—22no—
178#125—22no—
179#125—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8C4⋊C414yes—
187C22⋊Q8C2×Q814yes—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8C4⋊C414yes—
195C4×Q8C2×D414yes—
196#117—22no—
197#117—22no—
198#117—22no—
199#117—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4C4⋊C412yes—
206C2×C22⋊C4—22no—
207C2×C42—22no—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×Q811yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42C2411yescharacteristic
220#108—44no—
221#108—44no—
222C4⋊C8—28no—
223C4⋊C8—28no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—22no—
235C4⋊C8C22×C412yes—
236C4⋊C8C22×C412yes—
237C4⋊C8C22×C412yes—
238C4⋊C8C22×C412yes—
239C4⋊C8C2×D412yes—
240C4⋊C8C2×D412yes—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246C22⋊C8—41no—
247C22⋊C8—41no—
248#97—28no—
249#97—22no—
250#97—22no—
251#97—22no—
252#97—41no—
253#97—41no—
254#97—41no—
255C4×C8—28no—
256C4×C8—22no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—22no—
260C4×C8—22no—
261C4×C8—22no—
262C4×C8—22no—
263C4×C8—22no—
264C4×C8—41no—
265C24C22⋊Q811yescharacteristic
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—82no—
272C2×Q8C2×C4⋊C414yes—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—81no—
280C22×C4C2×C4⋊C411yescharacteristic
281C22×C4C2×C4⋊C411yescharacteristic
282C22×C4C4×D411yescharacteristic
283C22×C4C4×D411yescharacteristic
284C22×C4C4×D411yescharacteristic
285C22×C4C4×D411yescharacteristic
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4C22⋊Q811yescharacteristic
289C22×C4C22⋊Q811yescharacteristic
290C22×C4C22⋊Q811yescharacteristic
291C22×C4C22⋊Q811yescharacteristic
292C22×C4C22⋊Q811yescharacteristic
293C22×C4#12311yescharacteristic
294C22×C4#12311yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×Q811yescharacteristic
297C22×C4C2×#4111yescharacteristic
298M16—24no—
299M16—22no—
300M16—22no—
301M16—22no—
302M16—44no—
303M16—44no—
304M16—42no—
305M16—42no—
306M16—42no—
307M16—42no—
308M16—42no—
309M16#13714yes—
310C2×C8—24no—
311C2×C8—22no—
312C2×C8—24no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—44no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—44no—
329C2×C8—41no—
330C2×C8—41no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4C2×C4⋊C414yes—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C42—24no—
346C42—24no—
347C42C2×C4⋊C411yescharacteristic
348C42C2×C4⋊C411yescharacteristic
349C42C4×D411yescharacteristic
350C42C4×D411yescharacteristic
351C42C4×D411yescharacteristic
352C42C4×D411yescharacteristic
353C42#12011yescharacteristic
354C42#12011yescharacteristic
355C42C22⋊Q811yescharacteristic
356C42C22⋊Q811yescharacteristic
357C42#12311yescharacteristic
358C42#12311yescharacteristic
359C42C23×C411yescharacteristic
360C42C22×D411yescharacteristic
361C42C22×D411yescharacteristic
362C42C2×#4111yescharacteristic
363C23—22no—
364C23—22no—
365C23—41no—
366C23#38511yescharacteristic
367C23#38511yescharacteristic
368C23C2×C22⋊Q811yescharacteristic
369Q8—44no—
370Q8—44no—
371Q8—42no—
372Q8—42no—
373Q8—82no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4#38511yescharacteristic
399C2×C4#38511yescharacteristic
400C2×C4#38511yescharacteristic
401C2×C4#38511yescharacteristic
402C2×C4#38511yescharacteristic
403C2×C4#38511yescharacteristic
404C2×C4#44312yes—
405C2×C4#44412yes—
406C2×C4C22×C4⋊C411yescharacteristic
407C2×C4C2×C4×D411yescharacteristic
408C2×C4C2×C4×D411yescharacteristic
409C2×C4C2×#12011yescharacteristic
410C2×C4C2×C22⋊Q811yescharacteristic
411C2×C4C2×#12311yescharacteristic
412C2×C4C2×#13712yes—
413C2×C4#57512yes—
414C8—24no—
415C8—44no—
416C8—42no—
417C8—42no—
418C8—42no—
419C8—42no—
420C8—42no—
421C8—42no—
422C8—42no—
423C8—81no—
424C8—81no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—44no—
433C22#186612yes—
434C22C2×#38511yescharacteristic
435C22#307812yes—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—44no—
445C4—44no—
446C4#186612yes—
447C4C4×#13712yes—
448C4C2×#44311yescharacteristic
449C4C2×#44411yescharacteristic
450C4#300411yescharacteristic
451C4#300511yescharacteristic
452C2—22no—
453C2—22no—
454C2—41no—
455C2#2086011yescharacteristic
456C2#2087411yescharacteristic
457C2C2×#186611yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC24C22⋊Q8
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8933C2semidirect
7. #8931C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29622
#213702
#178072
#177182
#175012
#89332
#89312
#78872
#78732