Finite Group Databasebeta

← #101027

Subgroups — #101027

Order 512.

846 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 112 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#21934C211yesmaximal, characteristic
2#15178C211yesmaximal, characteristic
3#8557C211yesmaximal, characteristic
4#8340C212yesmaximal, complemented
5#7039C212yesmaximal, complemented
6C22×C4⋊Q8C2211yescharacteristic
7C22×C22⋊Q8—21no—
8C2×#407—21no—
9C2×#391—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1851—24no—
12#1797—21no—
13#1797C2212yes—
14#1797C2212yes—
15C2×#332—24no—
16C2×#323—21no—
17#1444—21no—
18#1444C412yes—
19#1439—21no—
20#1439C412yes—
21C4⋊#95C2211yescharacteristic
22#1280—21no—
23C23×Q8—21no—
24C24×C4D411yescharacteristic
25C22×M16—22no—
26C2×C4⋊Q8—22no—
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8—44no—
29C2×C22⋊Q8—44no—
30C2×C22⋊Q8—44no—
31C2×C22⋊Q8—42no—
32C2×C22⋊Q8—42no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4C2×C412yes—
37C22×C22⋊C4—22no—
38C22×C22⋊C4C2×C411yescharacteristic
39C22×C22⋊C4D411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42C2311yescharacteristic
42C2×C4⋊C8—22no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8D414yes—
45#407—42no—
46#407—42no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50#391—44no—
51C4×C22⋊C4—22no—
52C4×C22⋊C4—22no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—21no—
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C412yes—
58C2×#95D411yescharacteristic
59#336—22no—
60#336—22no—
61#336—22no—
62#332—48no—
63#323—42no—
64C25C22⋊C411yescharacteristic
65C22×Q8—24no—
66C22×Q8—42no—
67C22×Q8—42no—
68C22×Q8—42no—
69C22×Q8—41no—
70C22×Q8—41no—
71C22×Q8—41no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8—41no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4D812yes—
87C23×C4SD1612yes—
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C2×M16—44no—
92C2×M16—82no—
93C2×M16—82no—
94C22×C8—24no—
95C22×C8—42no—
96C4⋊Q8—28no—
97C4⋊Q8—44no—
98C4⋊Q8—48no—
99C22⋊Q8—416no—
100C22⋊Q8—416no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—44no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—44no—
107C2×C4⋊C4—44no—
108C2×C4⋊C4—44no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—44no—
120C2×C22⋊C4—44no—
121C2×C22⋊C4—44no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—42no—
125C2×C42—42no—
126C2×C42C2×D412yes—
127C2×C42#4112yes—
128C4⋊C8—44no—
129C4⋊C8—44no—
130C4⋊C8—82no—
131C22⋊C8—44no—
132C22⋊C8—42no—
133C22⋊C8—42no—
134C22⋊C8—42no—
135C22⋊C8—42no—
136#95—22no—
137#95—22no—
138#95—42no—
139#95—44no—
140#95—44no—
141#95—41no—
142#95—41no—
143C24—21no—
144C24—21no—
145C24—21no—
146C24—21no—
147C24—21no—
148C24—21no—
149C24—41no—
150C24—41no—
151C24—41no—
152C24—41no—
153C24#10212yes—
154C24C2×C22⋊C411yescharacteristic
155C2×Q8—44no—
156C2×Q8—44no—
157C2×Q8—44no—
158C2×Q8—44no—
159C2×Q8—44no—
160C2×Q8—44no—
161C2×Q8—82no—
162C2×Q8—82no—
163C2×Q8—82no—
164C2×Q8—82no—
165C2×Q8—82no—
166C2×Q8—81no—
167C2×Q8—81no—
168C2×Q8—82no—
169C2×Q8—81no—
170C2×Q8—81no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—82no—
215C22×C4—82no—
216C22×C4—81no—
217C22×C4—81no—
218C22×C4#10112yes—
219C22×C4#10212yes—
220C22×C4C4×D412yes—
221C22×C4#12011yescharacteristic
222C22×C4#12112yes—
223C22×C4#12311yescharacteristic
224C22×C4C2×D811yescharacteristic
225C22×C4C2×SD1611yescharacteristic
226C22×C4AGL(1,ℤ/8)11yescharacteristic
227C22×C4AGL(1,ℤ/8)11yescharacteristic
228C22×C4#13711yescharacteristic
229C22×C4#13712yes—
230C22×C4#13711yescharacteristic
231M16—88no—
232C2×C8—44no—
233C2×C8—44no—
234C2×C8—44no—
235C2×C8—82no—
236C2×C8—82no—
237C2×C8—82no—
238C4⋊C4—28no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—44no—
242C4⋊C4—48no—
243C4⋊C4—416no—
244C22⋊C4—44no—
245C22⋊C4—416no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C42—24no—
249C42—42no—
250C42—42no—
251C42—81no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23—81no—
277C23—81no—
278C23—81no—
279C23#38611yescharacteristic
280C23C2×#10111yescharacteristic
281C23#41311yescharacteristic
282C23C2×#10211yescharacteristic
283C23#41711yescharacteristic
284C23#41711yescharacteristic
285C23#41911yescharacteristic
286Q8—84no—
287Q8—84no—
288Q8—84no—
289Q8—84no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—41no—
323C2×C4—44no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—81no—
339C2×C4#44711yescharacteristic
340C2×C4#44811yescharacteristic
341C2×C4#45312yes—
342C2×C4B(4,𝔽2)12yes—
343C2×C4#46812yes—
344C2×C4#47012yes—
345C2×C4#48011yescharacteristic
346C2×C4#48111yescharacteristic
347C8—84no—
348C8—162no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—44no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22#154611yescharacteristic
373C22#154711yescharacteristic
374C22#155911yescharacteristic
375C22#156111yescharacteristic
376C22#187411yescharacteristic
377C22#188711yescharacteristic
378C22#188911yescharacteristic
379C22#189411yescharacteristic
380C22#190111yescharacteristic
381C22#190911yescharacteristic
382C22#191511yescharacteristic
383C4—21no—
384C4—21no—
385C4—22no—
386C4—22no—
387C4—42no—
388C4—42no—
389C4—42no—
390C4—84no—
391C4—84no—
392C4—81no—
393C4#159512yes—
394C4#159612yes—
395C2—21no—
396C2—21no—
397C2—21no—
398C2—21no—
399C2—82no—
400C2#901711yescharacteristic
401C2#913911yescharacteristic
402C2#919511yescharacteristic
403C2#940311yescharacteristic
404C2#945511yescharacteristic
405C2#947111yescharacteristic
406C2#950611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊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
4. #8340C2semidirect
5. #7039C2semidirect

Maximal subgroups

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

subgroupindex
#219342
#151782
#85572
#83402
#70392