Finite Group Databasebeta

← #193354

Subgroups — #193354

Order 512.

825 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60192C211yesmaximal, characteristic
2#21371C211yesmaximal, characteristic
3#17820C211yesmaximal, characteristic
4#17734C211yesmaximal, characteristic
5#17501C211yesmaximal, characteristic
6#8933C214yesmaximal, complemented
7#8931C214yesmaximal, complemented
8#7887C211yesmaximal, characteristic
9#7876C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11Q8×M16—22no—
12Q8×M16—22no—
13#2962—22no—
14#2962C2214yes—
15#2961—22no—
16#2960—22no—
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19C2×#432C2211yescharacteristic
20#2905—21no—
21C2×#423C2211yescharacteristic
22#2873—21no—
23C2×C4×M16C2211yescharacteristic
24C2×C4×M16C2211yescharacteristic
25#1950C2212yes—
26#1950C2212yes—
27#1855—21no—
28#1855—21no—
29#1778—22no—
30#1574—22no—
31#1571C2212yes—
32#1570C2212yes—
33#1569—22no—
34#1559C414yes—
35#1558C412yes—
36#1556C412yes—
37#1542C2214yes—
38#1493C2212yes—
39#1493C2212yes—
40#1491—22no—
41#1485—22no—
42#1278C2214yes—
43#1277C2214yes—
44C22×M16—21no—
45C22×M16—21no—
46C22×M16D411yescharacteristic
47C22×M16D411yescharacteristic
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533C2×C412yes—
53#533D412yes—
54#533Q812yes—
55#533C2312yes—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#125—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8Q812yes—
61C2×C4×Q8—21no—
62C2×C4×Q8D412yes—
63C2×#117—21no—
64C22×C42C2311yescharacteristic
65#446—24no—
66#446—28no—
67#446—28no—
68#446—24no—
69C8×Q8—24no—
70C8×Q8—24no—
71#432—21no—
72#432—21no—
73#432—21no—
74#432—21no—
75#432—21no—
76#432C2×C412yes—
77#432C2×C412yes—
78#432C2×C412yes—
79#431—21no—
80#431—21no—
81#431—21no—
82#431C2312yes—
83C2×#108—22no—
84C2×#108—22no—
85#424—24no—
86#424—24no—
87#423—24no—
88#423—21no—
89#423—21no—
90#423C2×C412yes—
91#423D412yes—
92#423D412yes—
93#423C2311yescharacteristic
94#423C2311yescharacteristic
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98#407—21no—
99#407—21no—
100C2×C22⋊C8—21no—
101#405—24no—
102#405—24no—
103C4×M16—24no—
104C4×M16—21no—
105C4×M16—21no—
106C4×M16—21no—
107C4×M16—21no—
108C4×M16—21no—
109C4×M16—21no—
110C4×M16—21no—
111C4×M16—21no—
112C4×M16D412yes—
113C4×M16C2312yes—
114C2×#97—21no—
115C2×#97—21no—
116C2×#97—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2312yes—
119#341—22no—
120#341—22no—
121#335—24no—
122#334—24no—
123#333C2×C418yes—
124#332C2×C418yes—
125Q8⋊C8—24no—
126Q8⋊C8—24no—
127C22×Q8—21no—
128C22×Q8C4⋊C412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×Q811yescharacteristic
131C23×C4#4111yescharacteristic
132C2×M16—21no—
133C2×M16—24no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—21no—
137C2×M16—22no—
138C2×M16—21no—
139C2×M16—21no—
140C2×M16—44no—
141C2×M16—42no—
142C2×M16—44no—
143C2×M16—41no—
144C2×M16—41no—
145C2×M16C2×D412yes—
146C2×M16C2×D412yes—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8C4⋊C412yes—
157C4⋊Q8C4⋊C412yes—
158C4⋊Q8C22×C412yes—
159C4⋊Q8C22×C412yes—
160#125—22no—
161#125—22no—
162#125—22no—
163#125—22no—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C22⋊Q8C4⋊C414yes—
171C22⋊Q8C2×Q814yes—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8C4⋊C414yes—
179C4×Q8C2×D414yes—
180#117—22no—
181#117—22no—
182#117—22no—
183#117—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C4⋊C412yes—
190C2×C22⋊C4—22no—
191C2×C42—22no—
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×Q811yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42C2411yescharacteristic
204#108—44no—
205#108—44no—
206C4⋊C8—28no—
207C4⋊C8—28no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—22no—
217C4⋊C8C22×C414yes—
218C4⋊C8C22×C414yes—
219C4⋊C8C2×D414yes—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—41no—
225C22⋊C8—41no—
226#97—28no—
227#97—22no—
228#97—22no—
229#97—22no—
230#97—22no—
231#97—22no—
232#97—41no—
233#97—41no—
234#97—41no—
235C4×C8—28no—
236C4×C8—22no—
237C4×C8—22no—
238C4×C8—22no—
239C4×C8—22no—
240C4×C8—22no—
241C4×C8—41no—
242C24C22⋊Q811yescharacteristic
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—82no—
247C2×Q8—82no—
248C2×Q8C2×C4⋊C414yes—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—81no—
255C22×C4C2×C4⋊C411yescharacteristic
256C22×C4C2×C4⋊C411yescharacteristic
257C22×C4C4×D411yescharacteristic
258C22×C4C4×D411yescharacteristic
259C22×C4C4×D411yescharacteristic
260C22×C4C4×D411yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4#12011yescharacteristic
263C22×C4C22⋊Q811yescharacteristic
264C22×C4C22⋊Q811yescharacteristic
265C22×C4C22⋊Q811yescharacteristic
266C22×C4C22⋊Q811yescharacteristic
267C22×C4C22⋊Q811yescharacteristic
268C22×C4#12311yescharacteristic
269C22×C4#12311yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C22×Q811yescharacteristic
272C22×C4C2×#4111yescharacteristic
273M16—24no—
274M16—22no—
275M16—22no—
276M16—22no—
277M16—42no—
278M16—44no—
279M16—44no—
280M16—44no—
281M16—44no—
282M16—42no—
283M16—42no—
284M16—82no—
285M16#13714yes—
286C2×C8—24no—
287C2×C8—22no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—22no—
292C2×C8—24no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—44no—
299C2×C8—41no—
300C2×C8—41no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4C2×C4⋊C414yes—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C42—24no—
316C42—24no—
317C42C2×C4⋊C411yescharacteristic
318C42C2×C4⋊C411yescharacteristic
319C42C4×D411yescharacteristic
320C42C4×D411yescharacteristic
321C42C4×D411yescharacteristic
322C42C4×D411yescharacteristic
323C42#12011yescharacteristic
324C42#12011yescharacteristic
325C42C22⋊Q811yescharacteristic
326C42C22⋊Q811yescharacteristic
327C42#12311yescharacteristic
328C42#12311yescharacteristic
329C42C23×C411yescharacteristic
330C42C22×D411yescharacteristic
331C42C22×D411yescharacteristic
332C42C2×#4111yescharacteristic
333C23—22no—
334C23—41no—
335C23—41no—
336C23#38511yescharacteristic
337C23#38511yescharacteristic
338C23C2×C22⋊Q811yescharacteristic
339Q8—44no—
340Q8—44no—
341Q8—44no—
342Q8—82no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4#38511yescharacteristic
367C2×C4#38511yescharacteristic
368C2×C4#38511yescharacteristic
369C2×C4#38511yescharacteristic
370C2×C4#38511yescharacteristic
371C2×C4#38511yescharacteristic
372C2×C4C22×C4⋊C411yescharacteristic
373C2×C4C2×C4×D411yescharacteristic
374C2×C4C2×C4×D411yescharacteristic
375C2×C4C2×#12011yescharacteristic
376C2×C4C2×C22⋊Q811yescharacteristic
377C2×C4C2×#12311yescharacteristic
378C2×C4C2×#13712yes—
379C2×C4#57512yes—
380C8—24no—
381C8—44no—
382C8—44no—
383C8—44no—
384C8—42no—
385C8—44no—
386C8—81no—
387C8—81no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—82no—
397C22C2×#38511yescharacteristic
398C22#307812yes—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—44no—
409C4—44no—
410C4C4×#13712yes—
411C4#295612yes—
412C4#300411yescharacteristic
413C4#300511yescharacteristic
414C2—22no—
415C2—41no—
416C2—41no—
417C2#2086011yescharacteristic
418C2#2087411yescharacteristic
419C2#2088511yescharacteristic
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
#601922
#213712
#178202
#177342
#175012
#89332
#89312
#78872
#78762