Finite Group Databasebeta

← #134345

Subgroups — #134345

Order 512.

696 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 96 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#14039C211yesmaximal, characteristic
2C2×#1326C212yesmaximal
3#8559C211yesmaximal, characteristic
4#8557C211yesmaximal, characteristic
5#7187C212yesmaximal
6C23×C4⋊C4C411yescharacteristic
7C2×#387—21no—
8C2×#385C2211yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11#1797—21no—
12C2×#323C2212yes—
13C2×#323C2212yes—
14#1442—21no—
15#1441—21no—
16#1436—21no—
17C22⋊#95C411yescharacteristic
18C22⋊#95C411yescharacteristic
19C22⋊#95C411yescharacteristic
20C22⋊#95C2211yescharacteristic
21#1326—24no—
22#1280—21no—
23#1280—21no—
24C24×C4C2×C411yescharacteristic
25C24×C4C2×C411yescharacteristic
26C24×C4C2×C411yescharacteristic
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—44no—
31C22×C4⋊C4—41no—
32C22×C4⋊C4—41no—
33C22×C22⋊C4C2×C411yescharacteristic
34C22×C22⋊C4C2×C411yescharacteristic
35C22×C22⋊C4C2×C411yescharacteristic
36C22×C22⋊C4C2311yescharacteristic
37C22×C42D411yescharacteristic
38C22×C42D411yescharacteristic
39C2×C4⋊C8—41no—
40C2×C4⋊C8—41no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43#387—44no—
44#385—21no—
45#385—21no—
46#385D412yes—
47#385Q812yes—
48C4×C22⋊C4—41no—
49C4×C22⋊C4—41no—
50C2×#95—21no—
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×#95D411yescharacteristic
60C2×#95D411yescharacteristic
61#336—41no—
62#336—41no—
63#323—22no—
64#323—22no—
65#323—22no—
66#323—22no—
67C25C22×C411yescharacteristic
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—22no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—22no—
77C23×C4—41no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4—42no—
82C23×C4—41no—
83C23×C4—41no—
84C23×C4—41no—
85C23×C4—41no—
86C23×C4—81no—
87C23×C4—81no—
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C4⋊C412yes—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C22×C8—42no—
96C22×C8—42no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—44no—
107C2×C4⋊C4—44no—
108C2×C4⋊C4—44no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—44no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4C4⋊C412yes—
124C2×C22⋊C4C2×D411yescharacteristic
125C2×C22⋊C4C2×Q811yescharacteristic
126C2×C22⋊C4#4111yescharacteristic
127C2×C22⋊C4#4111yescharacteristic
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—42no—
133C2×C42—81no—
134C4⋊C8—82no—
135C4⋊C8—82no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138C22⋊C8—42no—
139C22⋊C8—42no—
140#95—22no—
141#95—22no—
142#95—41no—
143#95—41no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95—42no—
149#95—81no—
150#95—81no—
151#95—81no—
152C24—21no—
153C24—21no—
154C24—21no—
155C24—21no—
156C24—41no—
157C24—41no—
158C24—41no—
159C24—41no—
160C24C2×C22⋊C411yescharacteristic
161C24C2×C4⋊C411yescharacteristic
162C24#11711yescharacteristic
163C24#13112yes—
164C24#13112yes—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
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—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—48no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
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—41no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—81no—
235C22×C4—81no—
236C22×C4—84no—
237C22×C4—84no—
238C22×C4—84no—
239C22×C4—84no—
240C22×C4—82no—
241C22×C4—82no—
242C22×C4#9911yescharacteristic
243C22×C4#9911yescharacteristic
244C22×C4#10112yes—
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4C22⋊Q811yescharacteristic
248C22×C4C22⋊Q811yescharacteristic
249C2×C8—82no—
250C2×C8—82no—
251C2×C8—82no—
252C2×C8—82no—
253C2×C8—82no—
254C2×C8—82no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—48no—
259C4⋊C4—44no—
260C22⋊C4—24no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—44no—
266C22⋊C4—42no—
267C22⋊C4—82no—
268C42—42no—
269C42—42no—
270C42—81no—
271C42—81no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—41no—
285C23—41no—
286C23—42no—
287C23—41no—
288C23—41no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23#38011yescharacteristic
303C23#40811yescharacteristic
304C23#40811yescharacteristic
305C23C2×#9911yescharacteristic
306C23#41011yescharacteristic
307C23C2×#10111yescharacteristic
308C23#41311yescharacteristic
309C23#42412yes—
310C23#43312yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—24no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—84no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—84no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—84no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—84no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4AΣL(1,𝔽16)12yes—
384C8—162no—
385C8—162no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—22no—
399C22—22no—
400C22—42no—
401C22—42no—
402C22—44no—
403C22—42no—
404C22—42no—
405C22—44no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22#146811yescharacteristic
415C22#147011yescharacteristic
416C22#147211yescharacteristic
417C22#147311yescharacteristic
418C22#179811yescharacteristic
419C22#179911yescharacteristic
420C22#180111yescharacteristic
421C22C2×AΣL(1,𝔽16)11yescharacteristic
422C22#212011yescharacteristic
423C22#212111yescharacteristic
424C22#212311yescharacteristic
425C4—22no—
426C4—41no—
427C4—42no—
428C4—41no—
429C4—44no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—82no—
434C4—82no—
435C4—81no—
436C4—81no—
437C4—161no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—42no—
443C2—42no—
444C2#856211yescharacteristic
445C2#1154511yescharacteristic
446C2#1154811yescharacteristic
447C2#1157511yescharacteristic
448C2#1157711yescharacteristic
449C2#1279311yescharacteristic
450C2#1279611yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutator, prime radicalC25C22×C4
Frattini, cosocleC22×C22⋊C4C23
LCSG ▷ C25 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#140392
C2×#13262
#85592
#85572
#71872