Finite Group Databasebeta

← #125103

Subgroups — #125103

Order 512.

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

normal subgroups only

This group has 606 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#15084C211yesmaximal, characteristic
2#8560C211yesmaximal, characteristic, complemented
3C4×#324C211yesmaximal, characteristic, complemented
4#7123C212yesmaximal, complemented
5#7103C212yesmaximal
6C2×#399—21no—
7C2×#390C411yescharacteristic
8C2×#388—21no—
9C2×#388—21no—
10C2×C4×C4⋊C4C2211yescharacteristic
11#1797—21no—
12C4×C22⋊C8—21no—
13C2×#324—21no—
14C2×#324—21no—
15#1444C411yescharacteristic
16#1444C2211yescharacteristic
17#1439—21no—
18#1438C411yescharacteristic
19#1438C2211yescharacteristic
20#1434C411yescharacteristic
21#1280C2212yes—
22#1280C2212yes—
23C23×D4C2×C411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4D411yescharacteristic
28C22×C4⋊C4D411yescharacteristic
29C22×C22⋊C4—21no—
30C22×C22⋊C4—21no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C42C2×C411yescharacteristic
36C22×C42D411yescharacteristic
37C22×C42D411yescharacteristic
38C2×C4⋊C8—41no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C4×C8—41no—
42#399—42no—
43#399—42no—
44#390—41no—
45#390—41no—
46#388—44no—
47#388—44no—
48C4×C4⋊C4—41no—
49C4×C4⋊C4—41no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95C2311yescharacteristic
58#336—41no—
59#336—41no—
60#324—24no—
61#324—42no—
62C25—21no—
63C22×D4—22no—
64C22×D4—41no—
65C22×D4—41no—
66C22×D4—42no—
67C22×D4—81no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C22×C8—42no—
79C22×C8—42no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—81no—
92C2×C4⋊C4—81no—
93C2×C4⋊C4—81no—
94C2×C4⋊C4D812yes—
95C2×C4⋊C4SD1612yes—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—81no—
115C2×C22⋊C4—81no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—22no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—41no—
124C2×C42—42no—
125C2×C42—42no—
126C4⋊C8—82no—
127C22⋊C8—24no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130C22⋊C8—42no—
131C4×C8—82no—
132#95—24no—
133#95—22no—
134#95—41no—
135#95—41no—
136#95—41no—
137#95—42no—
138#95—42no—
139#95—42no—
140C24—22no—
141C24—41no—
142C24—41no—
143C24—41no—
144C24—41no—
145C24—41no—
146C24—41no—
147C24—41no—
148C24—41no—
149C24—41no—
150C24—41no—
151C24—41no—
152C24—41no—
153C24—41no—
154C24—41no—
155C24C2×C22⋊C411yescharacteristic
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—82no—
159C2×D4—84no—
160C2×D4—82no—
161C2×D4—84no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
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—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4#9911yescharacteristic
214C22×C4#9911yescharacteristic
215C22×C4#10212yes—
216C22×C4#10212yes—
217C22×C4#10412yes—
218C22×C4#13112yes—
219C22×C4C2×D811yescharacteristic
220C22×C4C2×SD1611yescharacteristic
221C22×C4#13512yes—
222C2×C8—44no—
223C2×C8—82no—
224C2×C8—82no—
225C2×C8—82no—
226C2×C8—82no—
227C2×C8—82no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—44no—
231C4⋊C4—84no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C4⋊C4—82no—
235C22⋊C4—48no—
236C22⋊C4—48no—
237C22⋊C4—82no—
238C22⋊C4—84no—
239C22⋊C4—84no—
240C22⋊C4—82no—
241C22⋊C4—82no—
242C22⋊C4—82no—
243C42—42no—
244C42—42no—
245C42—42no—
246C42—42no—
247C42—81no—
248C42—81no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—81no—
261C23—81no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23—81no—
267C23—81no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23—81no—
277C23—81no—
278C23—81no—
279C23—81no—
280C23—81no—
281C23—81no—
282C23—81no—
283C23—81no—
284C23—81no—
285C23—81no—
286C23—81no—
287C23—81no—
288C23#40811yescharacteristic
289C23C2×#9911yescharacteristic
290C23#41311yescharacteristic
291C23C2×#10211yescharacteristic
292C23#41611yescharacteristic
293C23C2×#10411yescharacteristic
294C23#42111yescharacteristic
295D4—84no—
296D4—84no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4#35312yes—
360C8—84no—
361C8—162no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—82no—
375C22—82no—
376C22—82no—
377C22—82no—
378C22—82no—
379C22—82no—
380C22—82no—
381C22—161no—
382C22—161no—
383C22—161no—
384C22—161no—
385C22—161no—
386C22—161no—
387C22—161no—
388C22—161no—
389C22#146811yescharacteristic
390C22#148211yescharacteristic
391C22#149211yescharacteristic
392C22#150111yescharacteristic
393C22#151911yescharacteristic
394C22#152911yescharacteristic
395C22#154011yescharacteristic
396C22#212011yescharacteristic
397C22#212211yescharacteristic
398C22C2×#35311yescharacteristic
399C22#213311yescharacteristic
400C4—22no—
401C4—42no—
402C4—41no—
403C4—41no—
404C4—42no—
405C4—41no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—82no—
410C4—82no—
411C4—161no—
412C2—21no—
413C2—21no—
414C2—21no—
415C2—21no—
416C2—162no—
417C2#765811yescharacteristic
418C2#1154211yescharacteristic
419C2#1154911yescharacteristic
420C2#1172611yescharacteristic
421C2#1173811yescharacteristic
422C2#1187711yescharacteristic
423C2#1187911yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8560C2semidirect
3. C4×#324C2semidirect
4. #7123C2semidirect

Maximal subgroups

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

subgroupindex
#150842
#85602
C4×#3242
#71232
#71032