Finite Group Databasebeta

← #128813

Subgroups — #128813

Order 512.

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

normal subgroups only

This group has 708 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
1C4×#399C211yesmaximal, characteristic
2C2×#1352C212yesmaximal
3#10801C211yesmaximal, characteristic
4#8152C211yesmaximal, characteristic
5#7250C212yesmaximal
6C2×#399C2211yescharacteristic
7C2×C4×C4⋊C4—21no—
8C2×C4×C4⋊C4C411yescharacteristic, complemented
9C2×C4×C22⋊C4—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1749—21no—
12C2×#342—21no—
13C2×#342C2212yes—
14C2×#323C2212yes—
15#1445—21no—
16#1445C411yescharacteristic, complemented
17#1444C411yescharacteristic, complemented
18#1439—21no—
19#1439C2211yescharacteristic
20C22⋊#95—21no—
21C4×#95C411yescharacteristic, complemented
22#1352C418yescomplemented
23#1280—21no—
24C24×C4D411yescharacteristic
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C22⋊C4—21no—
30C22×C22⋊C4—22no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4D411yescharacteristic
33C22×C22⋊C4C2311yescharacteristic
34C22×C42—21no—
35C22×C42—21no—
36C22×C42C2×C411yescharacteristic
37C22×C42D411yescharacteristic
38C2×C22⋊C8—22no—
39C2×#97—41no—
40#399—22no—
41#399C2×C414yes—
42C4×C4⋊C4—44no—
43C4×C4⋊C4—41no—
44C4×C4⋊C4—41no—
45C4×C22⋊C4—41no—
46C4×C22⋊C4—41no—
47C4×C22⋊C4—44no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—41no—
55C2×#95—41no—
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95D411yescharacteristic
60#342—22no—
61#342—42no—
62#342C2×C414yes—
63#336—41no—
64#323—22no—
65#323C2×C414yes—
66C25C2×D411yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—41no—
75C23×C4—42no—
76C23×C4—42no—
77C23×C4—42no—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C22×C8—42no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4C4212yes—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4C22×C412yes—
117C2×C22⋊C4#4112yes—
118C2×C42—24no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—42no—
124C2×C42—42no—
125C2×C42—42no—
126C2×C42—42no—
127C2×C42—42no—
128C2×C42—42no—
129C2×C42—41no—
130C2×C42—41no—
131C22⋊C8—24no—
132C22⋊C8—42no—
133#97—82no—
134#95—21no—
135#95—41no—
136#95—41no—
137#95—41no—
138#95—42no—
139#95—42no—
140#95—42no—
141#95—42no—
142#95—42no—
143#95—42no—
144#95—82no—
145#95—82no—
146#95C4212yes—
147C24—21no—
148C24—21no—
149C24—21no—
150C24—21no—
151C24—41no—
152C24—41no—
153C24—41no—
154C24—41no—
155C24C2×C22⋊C411yescharacteristic
156C24C4×D412yes—
157C24AGL(1,ℤ/8)11yescharacteristic
158C24#13711yescharacteristic
159C24#13712yes—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—48no—
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—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—44no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—84no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4#9911yescharacteristic
231C22×C4#9911yescharacteristic
232C22×C4C2×C4211yescharacteristic
233C22×C4#11711yescharacteristic
234C22×C4C4×D412yes—
235C2×C8—44no—
236C2×C8—82no—
237C2×C8—82no—
238C4⋊C4—42no—
239C4⋊C4—48no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—48no—
243C4⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—48no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—44no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—82no—
254C22⋊C4—82no—
255C42—48no—
256C42—42no—
257C42—42no—
258C42—42no—
259C42—81no—
260C42—82no—
261C42—82no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—22no—
268C23—21no—
269C23—21no—
270C23—22no—
271C23—21no—
272C23—21no—
273C23—22no—
274C23—42no—
275C23—41no—
276C23—41no—
277C23—42no—
278C23—42no—
279C23—41no—
280C23—42no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—81no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23C4×C22⋊C411yescharacteristic
293C23C2×#9911yescharacteristic
294C23#41011yescharacteristic
295C23#41711yescharacteristic
296C23#41911yescharacteristic
297C23#42111yescharacteristic
298C23#42111yescharacteristic
299C23#44012yes—
300C23#44112yes—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—24no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—84no—
346C2×C4—84no—
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—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4—162no—
368C2×C4—162no—
369C2×C4—162no—
370C2×C4—162no—
371C2×C4#35212yes—
372C2×C4#35312yes—
373C8—84no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—22no—
387C22—22no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—42no—
397C22—42no—
398C22—82no—
399C22—81no—
400C22—81no—
401C22—82no—
402C22#151111yescharacteristic
403C22#151211yescharacteristic
404C22#151311yescharacteristic
405C22#151511yescharacteristic
406C22C4×#9911yescharacteristic
407C22#175911yescharacteristic
408C22#176311yescharacteristic
409C22C2×#35211yescharacteristic
410C22#212011yescharacteristic
411C22C2×#35311yescharacteristic
412C22#212611yescharacteristic
413C4—22no—
414C4—22no—
415C4—44no—
416C4—41no—
417C4—41no—
418C4—42no—
419C4—82no—
420C4—81no—
421C4—81no—
422C4—82no—
423C4—81no—
424C4—82no—
425C4—164no—
426C2—21no—
427C2—21no—
428C2—21no—
429C2—21no—
430C2—42no—
431C2—42no—
432C2#821711yescharacteristic
433C2#1180911yescharacteristic
434C2#1181011yescharacteristic
435C2#1182111yescharacteristic
436C2#1182211yescharacteristic
437C2C4×#35211yescharacteristic
438C2C4×#35311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C2×C4×C4⋊C4C4semidirect
16. #1445C4semidirect
17. #1444C4semidirect
21. C4×#95C4semidirect
22. #1352C4semidirect

Maximal subgroups

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

subgroupindex
C4×#3992
C2×#13522
#108012
#81522
#72502