Finite Group Databasebeta

← #99700

Subgroups — #99700

Order 512.

427 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 86 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#23817C211yesmaximal, characteristic
2#23347C211yesmaximal, characteristic
3#9877C211yesmaximal, characteristic, complemented
4#9647C211yesmaximal, characteristic, complemented
5C4×#329C211yesmaximal, characteristic
6#7030C211yesmaximal, characteristic, complemented
7#7030C211yesmaximal, characteristic, complemented
8C4×C4⋊Q8C2211yescharacteristic
9C4×#124C2211yescharacteristic
10#2101—21no—
11#2072C411yescharacteristic
12#2071C411yescharacteristic
13#2042—21no—
14#2024C411yescharacteristic
15#2023C411yescharacteristic
16#1984—21no—
17#1923—21no—
18#1923—21no—
19#1870—21no—
20#1851—21no—
21#1851—21no—
22#1831C2211yescharacteristic
23#1831C2211yescharacteristic
24C4×#103—21no—
25C4×#102—21no—
26#1744C2211yescharacteristic
27C4×#97C2211yescharacteristic
28C4×#97C2211yescharacteristic
29C2×#329—21no—
30C2×Q8⋊C8—21no—
31C2×Q8⋊C8—21no—
32#1295—21no—
33#1279—21no—
34#1279—21no—
35C22×M16—21no—
36C2×C4⋊Q8C2×C411yescharacteristic
37C2×#124D411yescharacteristic
38C2×C4×Q8—21no—
39C2×C4×Q8D411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4⋊C8—21no—
42C2×C4⋊C8D411yescharacteristic
43C2×C4⋊C8D411yescharacteristic
44C2×C4⋊C8D411yescharacteristic
45C2×C4⋊C8D411yescharacteristic
46C2×#103—21no—
47C2×#103—21no—
48C2×#102—21no—
49C2×#102—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×#97—21no—
54C2×#97—21no—
55C2×#97—21no—
56C2×#97—21no—
57C2×C4×C8—21no—
58C2×C4×C8—21no—
59C2×C4×C8—21no—
60C2×C4×C8—21no—
61#391—21no—
62#391C2×C411yescharacteristic
63#390C2×C411yescharacteristic
64#389—21no—
65#389—21no—
66#388—21no—
67#388—21no—
68#384C2×C411yescharacteristic
69#382D411yescharacteristic
70C4×C4⋊C4—21no—
71C4×C4⋊C4—21no—
72C4×C22⋊C4—21no—
73C4×C22⋊C4—21no—
74C43C2311yescharacteristic
75#340—21no—
76#340—21no—
77#340—21no—
78#340—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336C2×C411yescharacteristic
87#336C2×C411yescharacteristic
88#329—24no—
89Q8⋊C8—42no—
90Q8⋊C8—42no—
91C22×Q8—21no—
92C22×Q8C22⋊C411yescharacteristic
93C22×D4C22⋊C411yescharacteristic
94C23×C4—21no—
95C2×M16—81no—
96C2×M16—81no—
97C2×M16—81no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—41no—
105C22×C8—41no—
106C4⋊Q8—42no—
107#124—22no—
108#124—41no—
109C4×Q8—41no—
110C4×Q8—41no—
111C4×Q8—41no—
112C4×Q8—41no—
113C4×Q8—41no—
114C4×Q8—41no—
115C4×D4—41no—
116C4×D4—41no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4C22⋊C411yescharacteristic
125C2×C4⋊C4C22⋊C411yescharacteristic
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C42—21no—
132C2×C42—41no—
133C2×C42C22×C411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42#4111yescharacteristic
139C2×C42#4111yescharacteristic
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—81no—
145#103—42no—
146#103—42no—
147#102—42no—
148#102—42no—
149C22⋊C8—81no—
150C22⋊C8—81no—
151C22⋊C8—81no—
152#97—22no—
153#97—22no—
154#97—22no—
155#97—22no—
156#97—41no—
157#97—41no—
158#97—41no—
159#97—41no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—41no—
163C4×C8—41no—
164C4×C8—41no—
165C4×C8—41no—
166C4×C8—81no—
167#95—21no—
168#95—21no—
169#95—21no—
170#95—21no—
171#95—41no—
172#95—41no—
173C24—21no—
174C2×Q8—21no—
175C2×Q8—22no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8#10412yes—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—41no—
187C2×D4—41no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
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×C4C2×C22⋊C411yescharacteristic
200C22×C4C4×D411yescharacteristic
201C22×C4C4×D411yescharacteristic
202C22×C4#12011yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4#12311yescharacteristic
206M16—84no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—81no—
244C4⋊C4—81no—
245C4⋊C4—81no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—81no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—41no—
266C42—41no—
267C42—41no—
268C42—81no—
269C42—81no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23#38611yescharacteristic
278Q8—42no—
279Q8—42no—
280Q8—42no—
281Q8—81no—
282Q8—81no—
283Q8—81no—
284D4—42no—
285D4—81no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4C2×#10411yescharacteristic
321C2×C4#42111yescharacteristic
322C2×C4#44311yescharacteristic
323C2×C4#44311yescharacteristic
324C2×C4#44411yescharacteristic
325C2×C4#44411yescharacteristic
326C2×C4#45311yescharacteristic
327C2×C4#45411yescharacteristic
328C2×C4#45511yescharacteristic
329C2×C4#45511yescharacteristic
330C2×C4#45511yescharacteristic
331C2×C4#45611yescharacteristic
332C8—44no—
333C8—44no—
334C8—81no—
335C8—81no—
336C8—81no—
337C8—81no—
338C8—81no—
339C8—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22#187111yescharacteristic
348C22#187611yescharacteristic
349C22#187611yescharacteristic
350C22#188211yescharacteristic
351C22#188311yescharacteristic
352C22#188511yescharacteristic
353C22#189111yescharacteristic
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—42no—
363C4—41no—
364C4—81no—
365C4—82no—
366C4—81no—
367C4—161no—
368C4#165911yescharacteristic
369C4#166111yescharacteristic
370C4#166211yescharacteristic
371C4#166411yescharacteristic
372C2—161no—
373C2#897011yescharacteristic
374C2#897211yescharacteristic
375C2#899911yescharacteristic
376C2#902911yescharacteristic
377C2#905111yescharacteristic
378C2#905211yescharacteristic
379C2#906711yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
socleC23#386
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9877C2semidirect
4. #9647C2semidirect
6. #7030C2semidirect
7. #7030C2semidirect

Maximal subgroups

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

subgroupindex
#238172
#233472
#98772
#96472
C4×#3292
#70302