Finite Group Databasebeta

← #101194

Subgroups — #101194

Order 512.

648 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 104 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#21912C211yesmaximal, characteristic, complemented
2#15163C211yesmaximal, characteristic
3#8558C211yesmaximal, characteristic, complemented
4#7640C212yesmaximal
5#7039C212yesmaximal, complemented
6C2×#398C411yescharacteristic
7C2×#396—21no—
8C2×#394C411yescharacteristic
9C2×#390C2211yescharacteristic
10C2×#388—21no—
11C2×C4×C4⋊C4C2211yescharacteristic
12#1916—24no—
13#1817—21no—
14#1797C2212yes—
15#1797C2212yes—
16C4×C22⋊C8—21no—
17C2×#324—21no—
18#1445C411yescharacteristic
19#1439—21no—
20#1438C411yescharacteristic
21#1437—21no—
22C4⋊#95C2211yescharacteristic
23#1280—21no—
24#1270—24no—
25C23×D4C2×C411yescharacteristic
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4D411yescharacteristic
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C42D411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42C2311yescharacteristic
40C2×C4⋊C8—22no—
41C2×C4⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C4×C8—22no—
45#398—42no—
46#396—42no—
47#396—42no—
48#394—42no—
49#390—22no—
50#390—22no—
51#388—44no—
52C4×C4⋊C4—22no—
53C4×C4⋊C4—22no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95C2×C411yescharacteristic
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95D411yescharacteristic
62#336—24no—
63#336—22no—
64#336—22no—
65#336D414yes—
66#324—42no—
67C25—21no—
68C22×D4—22no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4—42no—
72C22×D4—81no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C22×C8—24no—
84C22×C8—42no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—81no—
100C2×C4⋊C4D812yes—
101C2×C4⋊C4SD1612yes—
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—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—81no—
121C2×C22⋊C4—81no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—21no—
128C2×C42—81no—
129C2×C42—81no—
130C2×C42C2×D412yes—
131C2×C42#4112yes—
132C4⋊C8—82no—
133C4⋊C8—82no—
134C22⋊C8—42no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138C4×C8—82no—
139#95—22no—
140#95—22no—
141#95—41no—
142#95—41no—
143#95—42no—
144#95—42no—
145#95—42no—
146#95—42no—
147C24—22no—
148C24—41no—
149C24—41no—
150C24—41no—
151C24—41no—
152C24—41no—
153C24—41no—
154C24—41no—
155C24—41no—
156C24—41no—
157C24—41no—
158C24—41no—
159C24—41no—
160C24—41no—
161C24—41no—
162C24C2×C22⋊C411yescharacteristic
163C2×D4—24no—
164C2×D4—42no—
165C2×D4—82no—
166C2×D4—82no—
167C2×D4—84no—
168C2×D4—84no—
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—21no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
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—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4#10012yes—
218C22×C4#10212yes—
219C22×C4#10212yes—
220C22×C4C4×D412yes—
221C22×C4#12011yescharacteristic
222C22×C4#12112yes—
223C22×C4#12311yescharacteristic
224C22×C4C2×D811yescharacteristic
225C22×C4C2×SD1611yescharacteristic
226C22×C4#13512yes—
227C22×C4#13512yes—
228C22×C4AGL(1,ℤ/8)12yes—
229C2×C8—44no—
230C2×C8—44no—
231C2×C8—44no—
232C2×C8—82no—
233C2×C8—82no—
234C2×C8—82no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—44no—
238C4⋊C4—82no—
239C4⋊C4—84no—
240C4⋊C4—82no—
241C4⋊C4—82no—
242C22⋊C4—48no—
243C22⋊C4—84no—
244C22⋊C4—84no—
245C22⋊C4—84no—
246C22⋊C4—82no—
247C22⋊C4—82no—
248C22⋊C4—82no—
249C22⋊C4—82no—
250C42—42no—
251C42—42no—
252C42—42no—
253C42—42no—
254C42—42no—
255C42—81no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
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—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23#38611yescharacteristic
296C23C2×#10011yescharacteristic
297C23#41311yescharacteristic
298C23C2×#10211yescharacteristic
299C23#41611yescharacteristic
300C23#41611yescharacteristic
301C23#41811yescharacteristic
302D4—84no—
303D4—84no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
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—41no—
340C2×C4—41no—
341C2×C4—41no—
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—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4#44912yes—
364C2×C4#46512yes—
365C2×C4#46612yes—
366C2×C4#48212yes—
367C8—84no—
368C8—162no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22—82no—
385C22—82no—
386C22—82no—
387C22—82no—
388C22—161no—
389C22—161no—
390C22—161no—
391C22—161no—
392C22—161no—
393C22—161no—
394C22—161no—
395C22—161no—
396C22#154011yescharacteristic
397C22#154311yescharacteristic
398C22#154511yescharacteristic
399C22#155311yescharacteristic
400C22#187511yescharacteristic
401C22#188411yescharacteristic
402C22#189111yescharacteristic
403C22#189311yescharacteristic
404C22#190011yescharacteristic
405C22#191211yescharacteristic
406C22#191311yescharacteristic
407C4—22no—
408C4—22no—
409C4—22no—
410C4—22no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—82no—
416C4—81no—
417C4—161no—
418C4—161no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—21no—
423C2—162no—
424C2#903611yescharacteristic
425C2#913411yescharacteristic
426C2#920611yescharacteristic
427C2#938411yescharacteristic
428C2#944811yescharacteristic
429C2#948611yescharacteristic
430C2#951511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×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
1. #21912C2semidirect
3. #8558C2semidirect
5. #7039C2semidirect

Maximal subgroups

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

subgroupindex
#219122
#151632
#85582
#76402
#70392