Finite Group Databasebeta

← #193351

Subgroups — #193351

Order 512.

816 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60192C211yesmaximal, characteristic
2#21370C211yesmaximal, characteristic
3#17820C211yesmaximal, characteristic
4#17718C211yesmaximal, characteristic
5#17479C211yesmaximal, characteristic
6#8933C214yesmaximal, complemented
7#8931C214yesmaximal, complemented
8#7888C211yesmaximal, characteristic
9#7880C211yesmaximal, 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×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2905—21no—
22C2×#423C2211yescharacteristic
23#2873—21no—
24C2×C4×M16C2211yescharacteristic
25#1950C2212yes—
26#1950C2212yes—
27#1855—21no—
28#1855—21no—
29C4×#108—22no—
30#1574—22no—
31#1571C2212yes—
32#1570C2212yes—
33#1569—22no—
34#1554C414yes—
35#1548C412yes—
36#1546C412yes—
37#1542C2214yes—
38#1497—22no—
39#1493C2212yes—
40#1493C2212yes—
41#1479—22no—
42#1278C2214yes—
43#1277C2214yes—
44C22×M16—21no—
45C22×M16D411yescharacteristic
46C22×M16D411yescharacteristic
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533C2×C412yes—
52#533D412yes—
53#533Q812yes—
54#533C2312yes—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#125—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8Q812yes—
60C2×C4×Q8—21no—
61C2×C4×Q8D412yes—
62C2×#117—21no—
63C22×C42C2311yescharacteristic
64#446—24no—
65#446—28no—
66#446—28no—
67#446—24no—
68C8×Q8—24no—
69C8×Q8—24no—
70#432—21no—
71#432—21no—
72#432—21no—
73#432C2×C412yes—
74#431—21no—
75#431—21no—
76#431—21no—
77#431—21no—
78#431—21no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431C2×C412yes—
84#431C2×C412yes—
85#431C2312yes—
86C2×#108—22no—
87C2×#108—22no—
88#424—24no—
89#424—24no—
90#423—24no—
91#423—21no—
92#423—21no—
93#423C2×C412yes—
94#423D412yes—
95#423D412yes—
96#423C2311yescharacteristic
97#423C2311yescharacteristic
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2312yes—
102#407—21no—
103#407—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106#405—24no—
107#405—24no—
108C4×M16—24no—
109C4×M16—21no—
110C4×M16—21no—
111C4×M16—21no—
112C4×M16—21no—
113C4×M16D412yes—
114C4×M16C2312yes—
115C2×#97—21no—
116C2×C4×C8—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—42no—
141C2×M16—44no—
142C2×M16C2×D412yes—
143C2×M16C2×D412yes—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8C4⋊C412yes—
154C4⋊Q8C4⋊C412yes—
155C4⋊Q8C22×C412yes—
156C4⋊Q8C22×C412yes—
157#125—22no—
158#125—22no—
159#125—22no—
160#125—22no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C22⋊Q8C4⋊C414yes—
168C22⋊Q8C2×Q814yes—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8C4⋊C414yes—
176C4×Q8C2×D414yes—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4C4⋊C412yes—
187C2×C22⋊C4—22no—
188C2×C42—22no—
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×Q811yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42C2411yescharacteristic
201#108—44no—
202#108—44no—
203C4⋊C8—28no—
204C4⋊C8—28no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—22no—
214C4⋊C8—41no—
215C4⋊C8—41no—
216C4⋊C8C22×C414yes—
217C4⋊C8C22×C414yes—
218C4⋊C8C2×D414yes—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—41no—
224C22⋊C8—41no—
225C22⋊C8—41no—
226C22⋊C8—41no—
227#97—28no—
228#97—22no—
229#97—22no—
230#97—22no—
231#97—41no—
232C4×C8—28no—
233C4×C8—22no—
234C4×C8—22no—
235C4×C8—22no—
236C4×C8—22no—
237C4×C8—22no—
238C4×C8—22no—
239C4×C8—22no—
240C4×C8—41no—
241C24C22⋊Q811yescharacteristic
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—44no—
246C2×Q8—82no—
247C2×Q8C2×C4⋊C414yes—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4C2×C4⋊C411yescharacteristic
255C22×C4C2×C4⋊C411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4C4×D411yescharacteristic
258C22×C4C4×D411yescharacteristic
259C22×C4C4×D411yescharacteristic
260C22×C4#12011yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4C22⋊Q811yescharacteristic
264C22×C4C22⋊Q811yescharacteristic
265C22×C4C22⋊Q811yescharacteristic
266C22×C4C22⋊Q811yescharacteristic
267C22×C4#12311yescharacteristic
268C22×C4#12311yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×Q811yescharacteristic
271C22×C4C2×#4111yescharacteristic
272M16—24no—
273M16—22no—
274M16—22no—
275M16—22no—
276M16—42no—
277M16—44no—
278M16—44no—
279M16—42no—
280M16—42no—
281M16—82no—
282M16#13714yes—
283C2×C8—24no—
284C2×C8—22no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—22no—
288C2×C8—24no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—44no—
293C2×C8—44no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4C2×C4⋊C414yes—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C42—24no—
312C42—24no—
313C42C2×C4⋊C411yescharacteristic
314C42C2×C4⋊C411yescharacteristic
315C42C4×D411yescharacteristic
316C42C4×D411yescharacteristic
317C42C4×D411yescharacteristic
318C42C4×D411yescharacteristic
319C42#12011yescharacteristic
320C42#12011yescharacteristic
321C42C22⋊Q811yescharacteristic
322C42C22⋊Q811yescharacteristic
323C42#12311yescharacteristic
324C42#12311yescharacteristic
325C42C23×C411yescharacteristic
326C42C22×D411yescharacteristic
327C42C22×D411yescharacteristic
328C42C2×#4111yescharacteristic
329C23—22no—
330C23—41no—
331C23—41no—
332C23#38511yescharacteristic
333C23#38511yescharacteristic
334C23C2×C22⋊Q811yescharacteristic
335Q8—44no—
336Q8—44no—
337Q8—44no—
338Q8—82no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4#38511yescharacteristic
363C2×C4#38511yescharacteristic
364C2×C4#38511yescharacteristic
365C2×C4#38511yescharacteristic
366C2×C4#38511yescharacteristic
367C2×C4#38511yescharacteristic
368C2×C4C22×C4⋊C411yescharacteristic
369C2×C4C2×C4×D411yescharacteristic
370C2×C4C2×C4×D411yescharacteristic
371C2×C4C2×#12011yescharacteristic
372C2×C4C2×C22⋊Q811yescharacteristic
373C2×C4C2×#12311yescharacteristic
374C2×C4C2×#13712yes—
375C2×C4#57512yes—
376C8—24no—
377C8—44no—
378C8—44no—
379C8—42no—
380C8—44no—
381C8—82no—
382C8—81no—
383C8—81no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—82no—
393C22C2×#38511yescharacteristic
394C22#307812yes—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—44no—
404C4—44no—
405C4#185812yes—
406C4C4×#13712yes—
407C4#295712yes—
408C4#300411yescharacteristic
409C4#300511yescharacteristic
410C2—22no—
411C2—41no—
412C2—41no—
413C2C2×#185811yescharacteristic
414C2#2087411yescharacteristic
415C2#2088611yescharacteristic
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
#213702
#178202
#177182
#174792
#89332
#89312
#78882
#78802