Finite Group Databasebeta

← #100159

Subgroups — #100159

Order 512.

696 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 128 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#20523C211yesmaximal, characteristic
2#14240C211yesmaximal, characteristic
3#8707C212yesmaximal, complemented
4#8559C211yesmaximal, characteristic
5#7024C212yesmaximal, complemented
6C22×C22⋊C8—21no—
7C2×#399C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797C2212yes—
11C8×C22⋊C4—24no—
12C4×C22⋊C8—21no—
13C4×C22⋊C8C2212yes—
14C2×#323—21no—
15#1445C411yescharacteristic
16#1442C411yescharacteristic
17#1439C411yescharacteristic
18#1437—21no—
19#1436—21no—
20#1436—21no—
21#1434C2211yescharacteristic
22C22⋊#95—21no—
23#1280—21no—
24#1279—24no—
25C24×C4D411yescharacteristic
26C23×C8—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C22⋊C4—21no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4C2×C411yescharacteristic
33C22×C22⋊C4D411yescharacteristic
34C22×C42—21no—
35C22×C42D411yescharacteristic
36C22×C42C2311yescharacteristic
37C2×C4⋊C8—22no—
38C2×C22⋊C8—24no—
39C2×C22⋊C8—24no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C4×C8—22no—
44C2×C4×C8D414yes—
45#399—22no—
46#399C814yescomplemented
47C4×C4⋊C4—22no—
48C4×C4⋊C4—22no—
49C4×C22⋊C4—22no—
50C4×C22⋊C4—22no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95C2×C411yescharacteristic
62C2×#95D411yescharacteristic
63#336—22no—
64#336—22no—
65#336—22no—
66#323—42no—
67C25C22⋊C411yescharacteristic
68C23×C4—22no—
69C23×C4—21no—
70C23×C4—22no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—42no—
77C23×C4—81no—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C22×C8—24no—
85C22×C8—28no—
86C22×C8—42no—
87C22×C8—82no—
88C22×C8—82no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—81no—
96C2×C4⋊C4—81no—
97C2×C4⋊C4C2×C812yescomplemented
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—81no—
106C2×C22⋊C4—81no—
107C2×C22⋊C4—81no—
108C2×C22⋊C4C2×C812yes—
109C2×C22⋊C4M1612yes—
110C2×C22⋊C4SD1612yes—
111C2×C22⋊C4Q1612yes—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—42no—
115C2×C42—42no—
116C2×C42—81no—
117C2×C42—81no—
118C2×C42—81no—
119C2×C42C2×D412yes—
120C2×C42#4112yes—
121C4⋊C8—44no—
122C22⋊C8—28no—
123C22⋊C8—24no—
124C22⋊C8—24no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127C4×C8—44no—
128C4×C8—82no—
129#95—22no—
130#95—22no—
131#95—21no—
132#95—22no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95—41no—
137#95—41no—
138#95—42no—
139#95—81no—
140#95—81no—
141#95—81no—
142#95C2×C812yescomplemented
143C24—21no—
144C24—21no—
145C24—21no—
146C24—21no—
147C24—41no—
148C24—41no—
149C24—41no—
150C24—41no—
151C24C22⋊C812yes—
152C24#10312yes—
153C24#10412yes—
154C24C2×C22⋊C411yescharacteristic
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—24no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
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—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—84no—
211C22×C4—84no—
212C22×C4C22⋊C812yes—
213C22×C4#10312yes—
214C22×C4#10412yes—
215C22×C4C4×D412yes—
216C22×C4#12011yescharacteristic
217C22×C4#12112yes—
218C22×C4#12311yescharacteristic
219C22×C4C22×C811yescharacteristic
220C22×C4C2×M1611yescharacteristic
221C22×C4#13112yes—
222C22×C4C2×SD1611yescharacteristic
223C22×C4C2×Q1611yescharacteristic
224C22×C4#13512yes—
225C2×C8—28no—
226C2×C8—44no—
227C2×C8—44no—
228C2×C8—88no—
229C2×C8—82no—
230C2×C8—82no—
231C2×C8—82no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—84no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—84no—
242C22⋊C4—84no—
243C42—24no—
244C42—42no—
245C42—42no—
246C42—81no—
247C42—84no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23Q8⋊C814yes—
275C23#38611yescharacteristic
276C23C2×C22⋊C811yescharacteristic
277C23#40811yescharacteristic
278C23C2×#10311yescharacteristic
279C23#41611yescharacteristic
280C23C2×#10411yescharacteristic
281C23C2×#10411yescharacteristic
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—84no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4C8×D412yes—
346C2×C4#43512yes—
347C2×C4#45812yes—
348C2×C4#46512yes—
349C2×C4#46712yes—
350C8—48no—
351C8—162no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—24no—
365C22—44no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—82no—
373C22—82no—
374C22—82no—
375C22C2×Q8⋊C811yescharacteristic
376C22#147611yescharacteristic
377C22#149311yescharacteristic
378C22#149411yescharacteristic
379C22#186911yescharacteristic
380C22#189011yescharacteristic
381C22#189111yescharacteristic
382C22#189211yescharacteristic
383C22#189611yescharacteristic
384C22#191011yescharacteristic
385C22#191111yescharacteristic
386C4—22no—
387C4—22no—
388C4—22no—
389C4—22no—
390C4—22no—
391C4—41no—
392C4—42no—
393C4—42no—
394C4—42no—
395C4—82no—
396C4—82no—
397C4—82no—
398C4—161no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2—21no—
403C2—44no—
404C2#896611yescharacteristic
405C2#907111yescharacteristic
406C2#910111yescharacteristic
407C2#946911yescharacteristic
408C2#947711yescharacteristic
409C2#948311yescharacteristic
410C2#949211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ 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. #8707C2semidirect
5. #7024C2semidirect
46. #399C8semidirect
97. C2×C4⋊C4C2×C8semidirect
142. #95C2×C8semidirect

Maximal subgroups

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

subgroupindex
#205232
#142402
#87072
#85592
#70242