Finite Group Databasebeta

← #97140

Subgroups — #97140

Order 512.

768 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 138 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#35173C211yesmaximal, characteristic
2#11202C211yesmaximal, characteristic
3#9131C212yesmaximal
4#8558C211yesmaximal, characteristic
5#7668C212yesmaximal
6C2×#533—21no—
7C22×C4⋊C8C2211yescharacteristic
8#2857—21no—
9#2857C2212yes—
10#2659—21no—
11#2659C412yes—
12#2584—24no—
13#2567—22no—
14#2546—22no—
15#2316—21no—
16#2316C412yes—
17C2×C4×C4⋊C4C2211yescharacteristic
18#1797C2212yes—
19C2×#336—21no—
20C2×#336—21no—
21C2×#336C2211yescharacteristic
22C2×#332—22no—
23C2×#324—21no—
24#1296—22no—
25#1280—21no—
26#1271—24no—
27C23×C8—21no—
28C23×C8—21no—
29#533—42no—
30#533—42no—
31#533—42no—
32#533—42no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8D412yes—
37C2×#125—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—22no—
42C2×#117—22no—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C42D411yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42C2311yescharacteristic
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—42no—
51C2×C22⋊C8—22no—
52#398—24no—
53#398—24no—
54#395—24no—
55#395—24no—
56#391—22no—
57#391—22no—
58#389—24no—
59#389—24no—
60#389—22no—
61#389—22no—
62#389—24no—
63#389—24no—
64#389—22no—
65#389D414yes—
66#384—21no—
67#384D412yes—
68#380—22no—
69#380—22no—
70#380—22no—
71#380C2×C414yes—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—22no—
74C2×#95C2×C411yescharacteristic
75C2×#95D411yescharacteristic
76#336—22no—
77#336—22no—
78#336—42no—
79#336—42no—
80#332—44no—
81#324—42no—
82C22×Q8—22no—
83C22×Q8—22no—
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—41no—
94C22×C8—41no—
95C22×C8—41no—
96C22×C8—41no—
97C22×C8—41no—
98C22×C8—41no—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—41no—
105C22×C8—42no—
106C4⋊Q8—42no—
107C4⋊Q8—42no—
108C4⋊Q8—42no—
109C4⋊Q8—42no—
110C4⋊Q8—42no—
111C4⋊Q8—42no—
112C4⋊Q8—81no—
113C4⋊Q8—81no—
114#125—44no—
115#125—44no—
116C22⋊Q8—44no—
117C22⋊Q8—44no—
118C22⋊Q8—42no—
119C22⋊Q8—42no—
120C22⋊Q8—42no—
121C22⋊Q8—42no—
122C4×Q8—44no—
123C4×Q8—44no—
124C4×Q8—44no—
125C4×Q8—44no—
126#117—44no—
127#117—44no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4C22⋊C414yes—
146C2×C4⋊C4C22⋊C414yes—
147C2×C4⋊C4SD1612yes—
148C2×C4⋊C4Q1612yes—
149C2×C22⋊C4—24no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—24no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42D812yes—
162C2×C42SD1612yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C4⋊C8—24no—
166C4⋊C8—42no—
167C4⋊C8—42no—
168C4⋊C8—42no—
169C22⋊C8—42no—
170C22⋊C8—42no—
171#95—24no—
172#95—24no—
173#95—22no—
174#95—24no—
175#95—22no—
176#95—24no—
177C24C2×C22⋊C411yescharacteristic
178C2×Q8—44no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—82no—
186C2×Q8—82no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—24no—
204C22×C4—24no—
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#10112yes—
216C22×C4#10112yes—
217C22×C4#10212yes—
218C22×C4#10212yes—
219C22×C4#10312yes—
220C22×C4#10312yes—
221C22×C4C2×C22⋊C412yes—
222C22×C4#12011yescharacteristic
223C22×C4#12012yes—
224C22×C4#12011yescharacteristic
225C22×C4C2×D811yescharacteristic
226C22×C4C2×SD1611yescharacteristic
227C22×C4C2×SD1611yescharacteristic
228C22×C4C2×Q1611yescharacteristic
229C22×C4AGL(1,ℤ/8)11yescharacteristic
230C22×C4AGL(1,ℤ/8)11yescharacteristic
231C22×C4#13712yes—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—81no—
239C2×C8—81no—
240C2×C8—81no—
241C2×C8—81no—
242C2×C8—81no—
243C2×C8—81no—
244C2×C8—81no—
245C2×C8—81no—
246C2×C8—82no—
247C2×C8—82no—
248C2×C8—82no—
249C4⋊C4—28no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—48no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—42no—
260C4⋊C4—44no—
261C4⋊C4—42no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—42no—
277C42—48no—
278C42—41no—
279C42—41no—
280C42—81no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23#37911yescharacteristic
286C23C2×#10111yescharacteristic
287C23C2×#10111yescharacteristic
288C23C2×#10211yescharacteristic
289C23C2×#10311yescharacteristic
290C23#41811yescharacteristic
291C23#41911yescharacteristic
292Q8—88no—
293Q8—82no—
294Q8—82no—
295Q8—82no—
296Q8—82no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4#33214yescomplemented
345C2×C4#44711yescharacteristic
346C2×C4#44711yescharacteristic
347C2×C4#44812yes—
348C2×C4#45011yescharacteristic
349C2×C4#45011yescharacteristic
350C2×C4#45112yes—
351C2×C4#45411yescharacteristic
352C2×C4#45611yescharacteristic
353C2×C4B(4,𝔽2)11yescharacteristic
354C2×C4#45811yescharacteristic
355C8—82no—
356C8—82no—
357C8—162no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22C2×#33211yescharacteristic
371C22#154611yescharacteristic
372C22#154911yescharacteristic
373C22#155611yescharacteristic
374C22#178411yescharacteristic
375C22#178611yescharacteristic
376C22#178711yescharacteristic
377C22#179011yescharacteristic
378C22#179111yescharacteristic
379C22#179311yescharacteristic
380C22#179411yescharacteristic
381C4—24no—
382C4—22no—
383C4—22no—
384C4—21no—
385C4—21no—
386C4—42no—
387C4—42no—
388C4—42no—
389C4—42no—
390C4—84no—
391C4—84no—
392C4#159811yescharacteristic
393C4#160411yescharacteristic
394C4#161011yescharacteristic
395C4#161611yescharacteristic
396C2—21no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2#834011yescharacteristic
401C2#835111yescharacteristic
402C2#835611yescharacteristic
403C2#840411yescharacteristic
404C2#840911yescharacteristic
405C2#846011yescharacteristic
406C2#853711yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊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
344. C2×C4#332semidirect

Maximal subgroups

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

subgroupindex
#351732
#112022
#91312
#85582
#76682