Finite Group Databasebeta

← #116913

Subgroups — #116913

Order 512.

647 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 647 conjugacy classes of subgroups — too many to draw. Only its 134 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#21969C211yesmaximal, characteristic, complemented
2#21851C211yesmaximal, characteristic
3#17451C212yesmaximal
4C2×#1292C211yesmaximal, characteristic, complemented
5#7990C212yesmaximal, complemented
6C2×C4×#41C2211yescharacteristic
7#2920C2212yes—
8C2×#432C2211yescharacteristic
9C2×#423C2212yes—
10C22×C4⋊C8C2211yescharacteristic
11#1985C412yes—
12#1955C412yes—
13#1950—24no—
14#1869—22no—
15#1810—22no—
16#1797—21no—
17#1797—21no—
18#1489—24no—
19#1487—22no—
20#1486—22no—
21#1477—24no—
22#1292—21no—
23#1292—21no—
24#1290—24no—
25C22×#41Q811yescharacteristic
26C22×M16—22no—
27C23×C8D412yes—
28C4×#41—22no—
29C4×#41—22no—
30C4×#41—22no—
31C4×#41—22no—
32C2×C4×Q8C2×C411yescharacteristic
33C2×C4×Q8D411yescharacteristic
34C2×C4×D4—21no—
35C2×C4×D4—21no—
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4D411yescharacteristic
38C2×#117—21no—
39C2×#117Q811yescharacteristic
40C22×C42—21no—
41C22×C42C2311yescharacteristic
42#432—22no—
43#432—22no—
44#432—22no—
45#432—22no—
46#431—22no—
47#431—22no—
48C2×#108—24no—
49#423—22no—
50#423—22no—
51#423—22no—
52#423—22no—
53#423—22no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8C2×C412yes—
59C2×C4⋊C8D414yes—
60C2×C4⋊C8D411yescharacteristic
61C2×C4⋊C8D411yescharacteristic
62#407—22no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C4×M16D414yes—
67C2×#97C2×C412yes—
68#341—24no—
69#336—22no—
70#336—22no—
71Q8⋊C8—28no—
72D4⋊C8—28no—
73C2×#41—22no—
74C2×#41—22no—
75C2×#41—44no—
76C22×Q8C4⋊C411yescharacteristic
77C22×D4—21no—
78C22×D4C4⋊C411yescharacteristic
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×Q811yescharacteristic
84C23×C4#4111yescharacteristic
85C2×M16—24no—
86C2×M16—42no—
87C2×M16—42no—
88C2×M16—82no—
89C2×M16—82no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—24no—
94C22×C8—82no—
95C22×C8C22⋊C412yes—
96C22×C8C22⋊C412yes—
97C4×Q8—22no—
98C4×Q8—22no—
99C4×Q8—22no—
100C4×Q8—22no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—41no—
106C4×D4—41no—
107C4×D4—42no—
108C4×D4—41no—
109C4×D4—42no—
110C4×D4—41no—
111#117—22no—
112#117—22no—
113#117—42no—
114#117—42no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4C4⋊C411yescharacteristic
118C2×C4⋊C4C4⋊C411yescharacteristic
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—42no—
126C2×C42—42no—
127C2×C42C22×C411yescharacteristic
128C2×C42C2×D412yes—
129C2×C42C2×D412yes—
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42#4112yes—
133C2×C42#4112yes—
134C2×C42#4111yescharacteristic
135#108—84no—
136C4⋊C8—24no—
137C4⋊C8—24no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—24no—
141C4⋊C8—42no—
142C4⋊C8—42no—
143C4⋊C8—42no—
144C4⋊C8—42no—
145C4⋊C8—42no—
146C22⋊C8—42no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—42no—
150C22⋊C8—82no—
151C22⋊C8—82no—
152#97—24no—
153#97—24no—
154C4×C8—24no—
155C24—21no—
156C24C22⋊Q811yescharacteristic
157#41—28no—
158#41—44no—
159#41—44no—
160#41—44no—
161C2×Q8—22no—
162C2×Q8—42no—
163C2×D4—24no—
164C2×D4—22no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—41no—
168C2×D4—41no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4C2×C22⋊C411yescharacteristic
192C22×C4C2×C4⋊C411yescharacteristic
193C22×C4#11711yescharacteristic
194C22×C4#12011yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4C22⋊Q812yes—
198C22×C4C22⋊Q812yes—
199C22×C4C22⋊Q811yescharacteristic
200C22×C4#12311yescharacteristic
201C22×C4#12312yes—
202C22×C4#12412yes—
203C22×C4#12411yescharacteristic
204C22×C4#12711yescharacteristic
205M16—44no—
206M16—44no—
207M16—84no—
208M16—82no—
209M16—82no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—44no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—44no—
219C2×C8—44no—
220C2×C8—82no—
221C2×C8—84no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—42no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C42—41no—
233C42—41no—
234C42—42no—
235C42—42no—
236C42C4×D412yes—
237C42C4×D412yes—
238C42#12012yes—
239C42#12112yes—
240C42#12112yes—
241C42#12112yes—
242C42#12312yes—
243C42#12612yes—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23#38011yescharacteristic
256C23#39311yescharacteristic
257C23#39911yescharacteristic
258Q8—24no—
259Q8—42no—
260D4—24no—
261D4—42no—
262D4—44no—
263D4—44no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—48no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4#38512yes—
300C2×C4#38611yescharacteristic
301C2×C4#38812yes—
302C2×C4#39011yescharacteristic
303C2×C4#39211yescharacteristic
304C2×C4#39412yes—
305C2×C4#39612yes—
306C2×C4#39711yescharacteristic
307C2×C4#47112yes—
308C2×C4#47211yescharacteristic
309C2×C4#47311yescharacteristic
310C8—42no—
311C8—42no—
312C8—82no—
313C8—84no—
314C8—84no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22#143911yescharacteristic
332C22#181511yescharacteristic
333C22#181611yescharacteristic
334C22#209112yes—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—42no—
342C4—81no—
343C4—81no—
344C4#191111yescharacteristic
345C4#191211yescharacteristic
346C4#205112yes—
347C4#205212yes—
348C2—21no—
349C2—21no—
350C2—22no—
351C2—41no—
352C2—82no—
353C2#1107211yescharacteristic
354C2#1110111yescharacteristic
355C2#1110511yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC22×#41Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21969C2semidirect
4. C2×#1292C2semidirect
5. #7990C2semidirect

Maximal subgroups

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

subgroupindex
#219692
#218512
#174512
C2×#12922
#79902