Finite Group Databasebeta

← #99694

Subgroups — #99694

Order 512.

440 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23677C211yesmaximal, characteristic
2#22798C211yesmaximal, characteristic
3#9874C211yesmaximal, characteristic, complemented
4#9869C211yesmaximal, characteristic, complemented
5#7560C211yesmaximal, characteristic
6#7031C211yesmaximal, characteristic, complemented
7#7027C211yesmaximal, characteristic, complemented
8C2×C4×SD16—21no—
9C2×#433—21no—
10C4×C4⋊Q8C2211yescharacteristic
11#2293C2211yescharacteristic
12#2084C411yescharacteristic
13#2071C411yescharacteristic
14#2066—21no—
15#2055—21no—
16#2043C411yescharacteristic
17#2013C411yescharacteristic
18#1955—21no—
19#1924—21no—
20#1923—21no—
21#1842C2211yescharacteristic
22#1798—21no—
23#1771—21no—
24#1767C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×#103—21no—
28#1744C2211yescharacteristic
29#1279—21no—
30#1279—21no—
31#1279—21no—
32#1279—21no—
33#1271—21no—
34#1270—21no—
35C22×SD16—21no—
36C2×C4⋊Q8C2×C411yescharacteristic
37C2×C4×Q8—21no—
38C2×C4×D4D411yescharacteristic
39C2×#117—21no—
40C4×SD16—42no—
41C4×SD16—42no—
42#433—42no—
43#433—42no—
44C2×#106—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8D411yescharacteristic
50C2×C4⋊C8D411yescharacteristic
51C2×C4⋊C8D411yescharacteristic
52C2×C4⋊C8D411yescharacteristic
53C2×#103—21no—
54C2×#103—21no—
55C2×#103—21no—
56C2×#102—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×#97—21no—
62C2×#97—21no—
63C2×C4×C8—21no—
64C2×C4×C8—21no—
65C2×C4×C8—21no—
66C2×C4×C8—21no—
67#391—21no—
68#390C2×C411yescharacteristic
69#390D411yescharacteristic
70#389—21no—
71#389—21no—
72#389C2×C411yescharacteristic
73#389D411yescharacteristic
74#388—21no—
75#388—21no—
76#384C2×C411yescharacteristic
77C4×C4⋊C4—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4D411yescharacteristic
80C4×C22⋊C4—21no—
81C43C2311yescharacteristic
82#340—21no—
83#340—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336—21no—
89#336—21no—
90#336C2×C411yescharacteristic
91#336C2×C411yescharacteristic
92C22×Q8—21no—
93C22×D4C22⋊C411yescharacteristic
94C23×C4—21no—
95C2×SD16—42no—
96C2×SD16—42no—
97C2×SD16—42no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—41no—
105C22×C8—41no—
106C4⋊Q8—42no—
107C4×Q8—41no—
108C4×Q8—41no—
109C4×Q8—41no—
110C4×Q8—41no—
111C4×D4—21no—
112C4×D4—21no—
113C4×D4—21no—
114C4×D4—21no—
115#117—41no—
116#117—41no—
117#117—41no—
118#117—41no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4C22⋊C411yescharacteristic
128C2×C4⋊C4C22⋊C411yescharacteristic
129C2×C4⋊C4C22⋊C411yescharacteristic
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C42—21no—
136C2×C42—41no—
137C2×C42C22×C411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42#4111yescharacteristic
143C2×C42#4111yescharacteristic
144#106—42no—
145C4⋊C8—22no—
146C4⋊C8—21no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8—41no—
152C4⋊C8—42no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8—41no—
156C4⋊C8—41no—
157C4⋊C8SD1612yescomplemented
158#103—42no—
159#103—42no—
160#103—42no—
161#102—42no—
162C22⋊C8—42no—
163C22⋊C8—42no—
164C22⋊C8—81no—
165C22⋊C8—81no—
166#97—42no—
167#97—41no—
168#97—41no—
169C4×C8—22no—
170C4×C8—41no—
171C4×C8—42no—
172C4×C8—41no—
173C4×C8—41no—
174C4×C8—41no—
175C4×C8—41no—
176#95—21no—
177#95—21no—
178#95—21no—
179#95—41no—
180#95—41no—
181C24—21no—
182C2×Q8—22no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—41no—
191C2×D4—41no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4C2×C22⋊C411yescharacteristic
204C22×C4C4×D411yescharacteristic
205C22×C4C4×D411yescharacteristic
206C22×C4#12011yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4#12311yescharacteristic
210SD16—48no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—42no—
218C2×C8—41no—
219C2×C8—42no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—81no—
232C2×C8—81no—
233C2×C8—81no—
234C2×C8—81no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—81no—
253C4⋊C4—81no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—81no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—41no—
268C42—41no—
269C42—41no—
270C42—41no—
271C42—41no—
272C42—81no—
273C42—81no—
274C42C2×SD1611yescharacteristic
275C42#13511yescharacteristic
276C42#13511yescharacteristic
277C42#13511yescharacteristic
278C42AGL(1,ℤ/8)11yescharacteristic
279C42AGL(1,ℤ/8)11yescharacteristic
280C42AGL(1,ℤ/8)11yescharacteristic
281C42#13711yescharacteristic
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23#38611yescharacteristic
290Q8—42no—
291Q8—42no—
292Q8—81no—
293Q8—81no—
294D4—42no—
295D4—81no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4#41611yescharacteristic
329C2×C4#41811yescharacteristic
330C2×C4C4×SD1611yescharacteristic
331C2×C4#44011yescharacteristic
332C2×C4#44311yescharacteristic
333C2×C4#44411yescharacteristic
334C2×C4#44911yescharacteristic
335C2×C4#44911yescharacteristic
336C2×C4#45011yescharacteristic
337C2×C4#45211yescharacteristic
338C2×C4#45511yescharacteristic
339C2×C4#45611yescharacteristic
340C2×C4#46111yescharacteristic
341C2×C4#46311yescharacteristic
342C2×C4#48211yescharacteristic
343C2×C4#48211yescharacteristic
344C8—42no—
345C8—42no—
346C8—81no—
347C8—82no—
348C8—82no—
349C8—81no—
350C8—81no—
351C8—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22#187511yescharacteristic
360C22#187611yescharacteristic
361C22#187711yescharacteristic
362C22#188111yescharacteristic
363C22#188211yescharacteristic
364C22#188411yescharacteristic
365C22#189111yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—81no—
378C4—82no—
379C4—81no—
380C4—81no—
381C4—161no—
382C4#164211yescharacteristic
383C4#164611yescharacteristic
384C4#165011yescharacteristic
385C4#165411yescharacteristic
386C2—161no—
387C2#898811yescharacteristic
388C2#899611yescharacteristic
389C2#899811yescharacteristic
390C2#903611yescharacteristic
391C2#904711yescharacteristic
392C2#905111yescharacteristic
393C2#905511yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
socleC23#386
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9874C2semidirect
4. #9869C2semidirect
6. #7031C2semidirect
7. #7027C2semidirect
157. C4⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#236772
#227982
#98742
#98692
#75602
#70312
#70272