Finite Group Databasebeta

← #106904

Subgroups — #106904

Order 512.

772 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2#20740C211yesmaximal, characteristic
3#17449C211yesmaximal, characteristic
4#8118C211yesmaximal, characteristic
5#8109C211yesmaximal, characteristic
6#8094C211yesmaximal, characteristic
7#7998C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2984—22no—
10#2984—22no—
11#2983—22no—
12#2983—21no—
13#2982—21no—
14Q8×M16—22no—
15Q8×M16C414yescomplemented
16#2920—22no—
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21C2×#432C2211yescharacteristic
22#2285C2211yescharacteristic
23#1823C412yes—
24#1803—22no—
25#1787C412yes—
26#1763—22no—
27#1491—22no—
28#1487—21no—
29#1487—21no—
30#1479C414yescomplemented
31#1307—22no—
32#1305—22no—
33#1303—22no—
34#1291—22no—
35C23×Q8C2×C411yescharacteristic
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8C2×C412yes—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8D412yes—
45C2×C4×Q8D412yes—
46C2×C4×Q8D411yescharacteristic
47C2×C4×Q8D411yescharacteristic
48C2×C4×Q8Q811yescharacteristic
49C2×C4×Q8Q811yescharacteristic
50C2×#117—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55#446—28no—
56#446—24no—
57#446—24no—
58#446—28no—
59C8×Q8—24no—
60C8×Q8—24no—
61#432—24no—
62#432—24no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432—21no—
68#432—21no—
69#432—21no—
70#432—21no—
71#432—21no—
72#432—21no—
73#432—21no—
74#432D412yes—
75#431—24no—
76#431—24no—
77#431—21no—
78#431—21no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431C2×C412yes—
83#431C2×C412yes—
84#431D412yes—
85#423—24no—
86#423—21no—
87#423—21no—
88#423C2×C412yes—
89#423D412yes—
90C2×C4⋊C8—21no—
91C2×#103—24no—
92#407—21no—
93#407—21no—
94#407—21no—
95#407D412yes—
96C2×C22⋊C8C2×C412yes—
97C4×M16—24no—
98C4×M16—21no—
99C4×M16—21no—
100C4×M16—21no—
101C4×M16C2×C412yes—
102C2×#97—21no—
103#384—21no—
104#382—21no—
105#380—21no—
106#380C2×C412yes—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109#340—24no—
110Q8⋊C8—24no—
111Q8⋊C8—24no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—44no—
116C22×Q8C4212yes—
117C22×Q8C22⋊C412yes—
118C22×Q8C4⋊C411yescharacteristic
119C22×Q8C4⋊C411yescharacteristic
120C23×C4—21no—
121C23×C4—21no—
122C23×C4C22×C411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4#4111yescharacteristic
125C2×M16—22no—
126C2×M16—41no—
127C2×M16—41no—
128C2×M16C22⋊C414yes—
129C22×C8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—21no—
135C4×Q8—22no—
136C4×Q8—21no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—21no—
140C4×Q8—21no—
141C4×Q8—21no—
142C4×Q8—21no—
143C4×Q8—22no—
144C4×Q8—21no—
145C4×Q8—21no—
146C4×Q8C22⋊C412yes—
147C4×Q8C22⋊C412yes—
148C4×Q8C22⋊C412yes—
149C4×Q8C22⋊C412yes—
150C4×Q8C4⋊C412yes—
151C4×Q8C4⋊C412yes—
152C4×Q8C4⋊C412yes—
153C4×Q8C4⋊C412yes—
154#117—82no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4C4212yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C4⋊C411yescharacteristic
172C2×C4⋊C4C4⋊C411yescharacteristic
173C2×C22⋊C4—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—41no—
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×Q811yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C4⋊C8—28no—
196C4⋊C8—28no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206C4⋊C8C22⋊C414yes—
207#103—48no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—41no—
217C22⋊C8—41no—
218#97—28no—
219#97—22no—
220#97—22no—
221#97—41no—
222#97—41no—
223#97—41no—
224#97—41no—
225C4×C8—28no—
226C4×C8—22no—
227C4×C8—22no—
228C4×C8—22no—
229C4×C8—22no—
230#95—22no—
231#95—41no—
232C24C4×D411yescharacteristic
233C2×Q8—24no—
234C2×Q8—22no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—44no—
246C2×Q8#9512yes—
247C2×Q8#9512yes—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4C2×C4211yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C4⋊C411yescharacteristic
268C22×C4#11711yescharacteristic
269C22×C4C4×D411yescharacteristic
270C22×C4C4×D411yescharacteristic
271C22×C4C4×D411yescharacteristic
272C22×C4C4×D411yescharacteristic
273C22×C4C4×D411yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4#12111yescharacteristic
277C22×C4#12111yescharacteristic
278C22×C4C22⋊Q811yescharacteristic
279C22×C4C22⋊Q811yescharacteristic
280C22×C4#12311yescharacteristic
281C22×C4#12311yescharacteristic
282M16—44no—
283M16—44no—
284M16—81no—
285M16—81no—
286M16—82no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—44no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—44no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4#9512yes—
309C4⋊C4#9512yes—
310C22⋊C4—82no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—82no—
320C42C2×C22⋊C411yescharacteristic
321C42C2×C22⋊C411yescharacteristic
322C42C2×C22⋊C411yescharacteristic
323C42C2×C4⋊C411yescharacteristic
324C42C2×C4⋊C411yescharacteristic
325C42#11711yescharacteristic
326C42C4×D411yescharacteristic
327C42C4×D411yescharacteristic
328C42#12011yescharacteristic
329C42#12011yescharacteristic
330C42#12111yescharacteristic
331C42#12111yescharacteristic
332C42C22⋊Q811yescharacteristic
333C42C22⋊Q811yescharacteristic
334C42#12311yescharacteristic
335C42#12311yescharacteristic
336C23—21no—
337C23—21no—
338C23—41no—
339C23—41no—
340C23C4×C22⋊C411yescharacteristic
341C23#38511yescharacteristic
342C23#38611yescharacteristic
343Q8—24no—
344Q8—22no—
345Q8—22no—
346Q8—22no—
347Q8—44no—
348Q8—44no—
349Q8—42no—
350Q8#34414yes—
351C2×C4—24no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—82no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4C2×#9511yescharacteristic
388C2×C4C4×C22⋊C411yescharacteristic
389C2×C4#37911yescharacteristic
390C2×C4#38011yescharacteristic
391C2×C4#38011yescharacteristic
392C2×C4#38111yescharacteristic
393C2×C4#38511yescharacteristic
394C2×C4#38511yescharacteristic
395C2×C4#38511yescharacteristic
396C2×C4#38611yescharacteristic
397C2×C4#38611yescharacteristic
398C2×C4#38611yescharacteristic
399C8—44no—
400C8—44no—
401C8—82no—
402C8—82no—
403C8—82no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—42no—
413C22—42no—
414C22C22⋊#9511yescharacteristic
415C22#190211yescharacteristic
416C22#190311yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—24no—
426C4—21no—
427C4—21no—
428C4—42no—
429C4—42no—
430C4—42no—
431C4—162no—
432C4C2×#34412yes—
433C4#179612yes—
434C2—21no—
435C2—21no—
436C2—41no—
437C2—41no—
438C2#1073711yescharacteristic
439C2#1077711yescharacteristic
440C2#1078411yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. Q8×M16C4semidirect
30. #1479C4semidirect

Maximal subgroups

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

subgroupindex
#602482
#207402
#174492
#81182
#81092
#80942
#79982