Finite Group Databasebeta

← #193357

Subgroups — #193357

Order 512.

803 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2#21994C211yesmaximal, characteristic
3#17822C211yesmaximal, characteristic
4#17734C211yesmaximal, characteristic
5C2×#1493C211yesmaximal, characteristic
6#8933C214yesmaximal, complemented
7#8931C214yesmaximal, complemented
8#7885C211yesmaximal, characteristic
9#7875C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2987—22no—
12#2985—22no—
13#2984—22no—
14#2983—22no—
15#2962C2212yes—
16#2962C2212yes—
17#2961—21no—
18#2961—21no—
19#2924—22no—
20#2920C2211yescharacteristic
21#2920C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C2×#432C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#1950C2214yes—
27#1855—22no—
28#1849—21no—
29#1838—21no—
30#1573—22no—
31#1571C2212yes—
32#1570C2212yes—
33#1569—22no—
34#1559C414yes—
35#1558C412yes—
36#1556C412yes—
37#1542C2214yes—
38#1493—22no—
39#1493C2214yes—
40C2×Q8⋊C8—22no—
41#1278C2214yes—
42#1277C2212yes—
43#1277C2212yes—
44C22×M16—21no—
45C23×C8—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533C2×C412yes—
51#533D412yes—
52#533Q812yes—
53#533C2312yes—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8Q812yes—
59C2×C4×Q8—21no—
60C2×C4×Q8D412yes—
61C2×#117—21no—
62C22×C42C2311yescharacteristic
63#446—24no—
64#446—24no—
65#446—24no—
66#446—24no—
67#446—24no—
68#446—24no—
69#446—22no—
70#446—22no—
71C8×Q8—22no—
72C8×Q8—22no—
73#433—24no—
74#433—24no—
75#432—24no—
76#432—21no—
77#432—21no—
78#432—21no—
79#432—21no—
80#432—21no—
81#432—21no—
82#432—21no—
83#432C2×C412yes—
84#432C2×C412yes—
85#432C2×C412yes—
86#432D412yes—
87#432D412yes—
88#431—24no—
89#431—21no—
90#431—21no—
91#431—21no—
92#431C2312yes—
93C2×#108—24no—
94#424—22no—
95#424—22no—
96#423—21no—
97#423—21no—
98#423—21no—
99#423C2×C412yes—
100#423D411yescharacteristic
101#423D411yescharacteristic
102#423C2312yes—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108#407—21no—
109#407D412yes—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112#405—22no—
113#405—22no—
114C4×M16—21no—
115C4×M16C2312yes—
116C2×#97—21no—
117C2×#97—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8C2312yes—
124#341—24no—
125#335—24no—
126#334—24no—
127#333C2×C418yes—
128#332C2×C418yes—
129Q8⋊C8—28no—
130C22×Q8—21no—
131C22×Q8C4⋊C412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×Q811yescharacteristic
134C23×C4#4111yescharacteristic
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—41no—
138C2×M16—82no—
139C2×M16C2×D412yes—
140C2×M16C2×D412yes—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—44no—
146C22×C8—41no—
147C22×C8—41no—
148C22×C8—41no—
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—84no—
202C4⋊C8—24no—
203C4⋊C8—24no—
204C4⋊C8—24no—
205C4⋊C8—24no—
206C4⋊C8—22no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—41no—
217C4⋊C8—41no—
218C4⋊C8C22×C414yes—
219C4⋊C8C22×C414yes—
220C4⋊C8C2×D412yes—
221C4⋊C8C2×D412yes—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230#97—24no—
231#97—24no—
232#97—22no—
233#97—22no—
234#97—22no—
235#97—22no—
236#97—22no—
237#97—22no—
238C4×C8—24no—
239C4×C8—24no—
240C4×C8—22no—
241C4×C8—22no—
242C4×C8—22no—
243C4×C8—22no—
244C4×C8—41no—
245C4×C8—41no—
246C24C22⋊Q811yescharacteristic
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—44no—
251C2×Q8—82no—
252C2×Q8C2×C4⋊C414yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4C2×C4⋊C411yescharacteristic
260C22×C4C2×C4⋊C411yescharacteristic
261C22×C4C4×D411yescharacteristic
262C22×C4C4×D411yescharacteristic
263C22×C4C4×D411yescharacteristic
264C22×C4C4×D411yescharacteristic
265C22×C4#12011yescharacteristic
266C22×C4#12011yescharacteristic
267C22×C4C22⋊Q811yescharacteristic
268C22×C4C22⋊Q811yescharacteristic
269C22×C4C22⋊Q811yescharacteristic
270C22×C4C22⋊Q811yescharacteristic
271C22×C4C22⋊Q811yescharacteristic
272C22×C4#12311yescharacteristic
273C22×C4#12311yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×Q811yescharacteristic
276C22×C4C2×#4111yescharacteristic
277M16—24no—
278M16—24no—
279M16—44no—
280M16—82no—
281M16—82no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—42no—
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—82no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4C2×C4⋊C414yes—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C42—24no—
314C42—24no—
315C42C2×C4⋊C411yescharacteristic
316C42C2×C4⋊C411yescharacteristic
317C42C4×D411yescharacteristic
318C42C4×D411yescharacteristic
319C42C4×D411yescharacteristic
320C42C4×D411yescharacteristic
321C42#12011yescharacteristic
322C42#12011yescharacteristic
323C42C22⋊Q811yescharacteristic
324C42C22⋊Q811yescharacteristic
325C42#12311yescharacteristic
326C42#12311yescharacteristic
327C42C23×C411yescharacteristic
328C42C22×D411yescharacteristic
329C42C22×D411yescharacteristic
330C42C2×#4111yescharacteristic
331C23—22no—
332C23—22no—
333C23—41no—
334C23#38511yescharacteristic
335C23#38511yescharacteristic
336C23C2×C22⋊Q811yescharacteristic
337Q8—44no—
338Q8—44no—
339Q8—82no—
340Q8—82no—
341C2×C4—22no—
342C2×C4—22no—
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—44no—
363C2×C4—82no—
364C2×C4#38511yescharacteristic
365C2×C4#38511yescharacteristic
366C2×C4#38511yescharacteristic
367C2×C4#38511yescharacteristic
368C2×C4#38511yescharacteristic
369C2×C4#38511yescharacteristic
370C2×C4C22×C4⋊C411yescharacteristic
371C2×C4C2×C4×D411yescharacteristic
372C2×C4C2×C4×D411yescharacteristic
373C2×C4C2×#12011yescharacteristic
374C2×C4C2×C22⋊Q811yescharacteristic
375C2×C4C2×#12311yescharacteristic
376C2×C4#57712yes—
377C2×C4#57812yes—
378C8—44no—
379C8—44no—
380C8—44no—
381C8—42no—
382C8—42no—
383C8—42no—
384C8—42no—
385C8—82no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—44no—
394C22#185812yes—
395C22C2×#38511yescharacteristic
396C22#308212yes—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—44no—
407C4—44no—
408C4#294712yes—
409C4#295612yes—
410C4#301011yescharacteristic
411C4#301111yescharacteristic
412C2—22no—
413C2—22no—
414C2—41no—
415C2C2×#185811yescharacteristic
416C2#2087711yescharacteristic
417C2#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
#602512
#219942
#178222
#177342
C2×#14932
#89332
#89312
#78852
#78752