Finite Group Databasebeta

← #101636

Subgroups — #101636

Order 512.

445 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 445 conjugacy classes of subgroups — too many to draw. Only its 84 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#29235C211yesmaximal, characteristic
2#23121C211yesmaximal, characteristic
3#9883C211yesmaximal, characteristic, complemented
4#7647C211yesmaximal, characteristic
5#7607C211yesmaximal, characteristic, complemented
6#7045C211yesmaximal, characteristic, complemented
7#7033C211yesmaximal, characteristic, complemented
8C2×#436—21no—
9C4×#121C2211yescharacteristic
10C2×#389C2211yescharacteristic
11#2088—21no—
12#2065—21no—
13#1924—21no—
14#1920—21no—
15#1916—21no—
16#1874C411yescharacteristic
17#1874C411yescharacteristic
18#1870—21no—
19#1870—21no—
20#1864—21no—
21#1853C411yescharacteristic
22#1853C411yescharacteristic
23#1817C2211yescharacteristic
24#1797C2211yescharacteristic
25#1797C2211yescharacteristic
26#1753—21no—
27#1749C2211yescharacteristic
28#1728C2211yescharacteristic
29#1280—21no—
30#1280—21no—
31#1271—21no—
32#1271—21no—
33#1271—21no—
34#1270—21no—
35C22×M16—21no—
36C22×M16—21no—
37C2×#121C2×C411yescharacteristic
38C2×C4×D4—21no—
39C2×C4×D4D411yescharacteristic
40C22×C4⋊C4—21no—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C42C2311yescharacteristic
44#436—42no—
45#436—42no—
46C2×#106—21no—
47C2×#106—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×#102—21no—
52C2×#102—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8C2×C411yescharacteristic
59C2×C22⋊C8C2×C411yescharacteristic
60C2×C22⋊C8D411yescharacteristic
61C2×C22⋊C8D411yescharacteristic
62C2×#97—21no—
63C2×C4×C8—21no—
64#390D411yescharacteristic
65#389—21no—
66#389—21no—
67#389—21no—
68#389—21no—
69#389D411yescharacteristic
70#388—21no—
71#386—21no—
72#380C2×C411yescharacteristic
73C4×C4⋊C4D411yescharacteristic
74C4×C22⋊C4—21no—
75C2×#95—21no—
76#340—21no—
77#340—21no—
78#340—21no—
79#340—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336D411yescharacteristic
89#336D411yescharacteristic
90C22×D4—21no—
91C22×D4C22⋊C411yescharacteristic
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4C22×C411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4#4111yescharacteristic
98C2×M16—42no—
99C2×M16—42no—
100C2×M16—42no—
101C2×M16—41no—
102C2×M16—41no—
103C2×M16—81no—
104C2×M16—81no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—41no—
112C22×C8—41no—
113#121—22no—
114#121—41no—
115C4×D4—41no—
116C4×D4—41no—
117C4×D4—41no—
118C4×D4—41no—
119C4×D4—81no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—81no—
129C2×C4⋊C4—82no—
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C4⋊C4C22⋊C411yescharacteristic
132C2×C4⋊C4C22⋊C411yescharacteristic
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C42—21no—
139C2×C42—81no—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42#4111yescharacteristic
144#106—42no—
145#106—81no—
146C4⋊C8—42no—
147C4⋊C8—81no—
148C4⋊C8—81no—
149#102—42no—
150#102—81no—
151C22⋊C8—21no—
152C22⋊C8—21no—
153C22⋊C8—21no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156C22⋊C8—41no—
157C22⋊C8—41no—
158C22⋊C8—41no—
159C22⋊C8—41no—
160C22⋊C8—81no—
161C22⋊C8—81no—
162C22⋊C8SD1612yes—
163#97—81no—
164C4×C8—42no—
165#95—21no—
166#95—21no—
167#95—21no—
168#95—41no—
169#95—41no—
170C24—21no—
171C24C4×D411yescharacteristic
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4—42no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4—81no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4C2×C22⋊C411yescharacteristic
208C22×C4C4×D411yescharacteristic
209C22×C4#12011yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4#12111yescharacteristic
212C22×C4#12311yescharacteristic
213C22×C4C2×SD1611yescharacteristic
214C22×C4#13511yescharacteristic
215C22×C4AGL(1,ℤ/8)11yescharacteristic
216C22×C4#13711yescharacteristic
217M16—84no—
218M16—82no—
219M16—82no—
220C2×C8—22no—
221C2×C8—22no—
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—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—81no—
239C2×C8—81no—
240C2×C8—81no—
241C2×C8—81no—
242C2×C8—81no—
243C2×C8—81no—
244C4⋊C4—21no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—81no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4#10112yes—
259C22⋊C4—42no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—81no—
265C22⋊C4—81no—
266C42—41no—
267C42—41no—
268C42—41no—
269C42—41no—
270C42—41no—
271C42—41no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23#38611yescharacteristic
286C23C4×SD1611yescharacteristic
287C23#44011yescharacteristic
288D4—42no—
289D4—81no—
290D4—82no—
291D4—82no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—162no—
336C2×C4C2×#10111yescharacteristic
337C2×C4#41311yescharacteristic
338C2×C4#45011yescharacteristic
339C2×C4#45211yescharacteristic
340C2×C4#46111yescharacteristic
341C2×C4#46311yescharacteristic
342C8—42no—
343C8—42no—
344C8—81no—
345C8—81no—
346C8—81no—
347C8—81no—
348C8—161no—
349C8—161no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22#187711yescharacteristic
368C22#188111yescharacteristic
369C22#189111yescharacteristic
370C22#190111yescharacteristic
371C22#190311yescharacteristic
372C22#190611yescharacteristic
373C22#191311yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—42no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—81no—
385C4—82no—
386C4—81no—
387C4—161no—
388C2—41no—
389C2—41no—
390C2—161no—
391C2#904711yescharacteristic
392C2#924911yescharacteristic
393C2#927511yescharacteristic
394C2#933211yescharacteristic
395C2#935211yescharacteristic
396C2#951811yescharacteristic
397C2#953211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#121C2×C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9883C2semidirect
5. #7607C2semidirect
6. #7045C2semidirect
7. #7033C2semidirect

Maximal subgroups

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

subgroupindex
#292352
#231212
#98832
#76472
#76072
#70452
#70332