Finite Group Databasebeta

← #99715

Subgroups — #99715

Order 512.

513 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 513 conjugacy classes of subgroups — too many to draw. Only its 98 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#23730C211yesmaximal, characteristic
2#21936C211yesmaximal, characteristic
3#8710C212yesmaximal, complemented
4#7563C211yesmaximal, characteristic
5#7029C212yesmaximal, complemented
6C2×#427—21no—
7C2×#416—21no—
8C2×#408—21no—
9#2297C2211yescharacteristic
10#2285C2211yescharacteristic
11#2097—21no—
12#2096C411yescharacteristic
13#2095C411yescharacteristic
14#2094C411yescharacteristic
15#2093C411yescharacteristic
16#2092—21no—
17#1851—22no—
18#1850—22no—
19#1798—21no—
20#1797C2212yes—
21C4×C22⋊C8C2212yes—
22C2×#336C2211yescharacteristic
23#1279—22no—
24#1279—22no—
25#1272—21no—
26#1270—21no—
27C22×#41D411yescharacteristic
28C22×M16—21no—
29C23×C8—21no—
30C23×C8—21no—
31C2×#117—21no—
32C2×#117D411yescharacteristic
33C22×C42C2311yescharacteristic
34#427—42no—
35#427—42no—
36C2×#107—21no—
37C2×#107—21no—
38C2×#106—21no—
39C2×#106—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8D412yes—
43C2×C4⋊C8D412yes—
44#416—42no—
45#416—42no—
46C2×#103—21no—
47C2×#103—21no—
48C2×#102—21no—
49C2×#102—21no—
50#408—42no—
51#408—42no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C4×C8—22no—
56#391C2×C411yescharacteristic
57#391D411yescharacteristic
58#390C2×C411yescharacteristic
59#390D411yescharacteristic
60#386—22no—
61#384C2×C411yescharacteristic
62#384C2×C411yescharacteristic
63#382—21no—
64#380—22no—
65C4×C4⋊C4—22no—
66C4×C22⋊C4—22no—
67#336—22no—
68#336—22no—
69#336—21no—
70#336—22no—
71#336C2×C411yescharacteristic
72#336C2×C411yescharacteristic
73C2×#41—21no—
74C2×#41—21no—
75C2×#41—41no—
76C2×#41—41no—
77C2×#41—41no—
78C2×#41—41no—
79C2×#41SD1612yes—
80C2×#41Q1612yes—
81C22×Q8C22⋊C411yescharacteristic
82C22×D4—21no—
83C22×D4C22⋊C411yescharacteristic
84C23×C4—21no—
85C23×C4C2×D411yescharacteristic
86C23×C4#4112yes—
87C2×M16—42no—
88C2×M16—81no—
89C2×M16—81no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—41no—
97C22×C8—41no—
98C22×C8—41no—
99C22×C8—41no—
100C22×C8—42no—
101C22×C8—42no—
102C22×C8—81no—
103C22×C8—81no—
104#117—41no—
105#117—41no—
106#117—41no—
107#117—41no—
108#117—41no—
109#117—41no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4C22⋊C411yescharacteristic
115C2×C4⋊C4C22⋊C411yescharacteristic
116C2×C22⋊C4—22no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—41no—
120C2×C42—21no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42—41no—
124C2×C42C22×C411yescharacteristic
125C2×C42C2×D412yes—
126C2×C42C2×D411yescharacteristic
127#107—42no—
128#107—42no—
129#106—42no—
130#106—42no—
131C4⋊C8—42no—
132C4⋊C8—42no—
133C4⋊C8—81no—
134C4⋊C8—81no—
135#103—42no—
136#103—42no—
137#102—42no—
138#102—42no—
139C22⋊C8—24no—
140C22⋊C8—42no—
141C22⋊C8—44no—
142C22⋊C8—42no—
143C22⋊C8—42no—
144C4×C8—42no—
145C4×C8—42no—
146#95—22no—
147#95—22no—
148#95—42no—
149C24—21no—
150C24#12311yescharacteristic
151#41—42no—
152#41—42no—
153#41—42no—
154#41—42no—
155#41—81no—
156#41—81no—
157#41—81no—
158#41—81no—
159C2×Q8—21no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8#10312yes—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—21no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—81no—
169C2×D4—81no—
170C2×D4#10312yes—
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—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4C2×C22⋊C411yescharacteristic
191C22×C4C4×D412yes—
192C22×C4#12011yescharacteristic
193C22×C4#12112yes—
194C22×C4C2×SD1611yescharacteristic
195C22×C4C2×Q1611yescharacteristic
196C22×C4AGL(1,ℤ/8)11yescharacteristic
197C22×C4#13711yescharacteristic
198M16—84no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—44no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—82no—
217C2×C8—82no—
218C2×C8—84no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—82no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C42—22no—
235C42—22no—
236C42—41no—
237C42—41no—
238C42—41no—
239C42—41no—
240C42—81no—
241C42—81no—
242C23—22no—
243C23—22no—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—42no—
248C23—42no—
249C23—41no—
250C23—41no—
251C23#38611yescharacteristic
252C23#48311yescharacteristic
253C23#48411yescharacteristic
254Q8—42no—
255Q8—81no—
256D4—42no—
257D4—81no—
258D4—82no—
259D4—82no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4C2×#10311yescharacteristic
293C2×C4#41711yescharacteristic
294C2×C4#44312yes—
295C2×C4#44412yes—
296C2×C4#45011yescharacteristic
297C2×C4#45111yescharacteristic
298C2×C4#45311yescharacteristic
299C2×C4#45311yescharacteristic
300C2×C4#45811yescharacteristic
301C2×C4#45811yescharacteristic
302C8—42no—
303C8—42no—
304C8—84no—
305C8—82no—
306C8—82no—
307C22—22no—
308C22—22no—
309C22—21no—
310C22—21no—
311C22—41no—
312C22—42no—
313C22—41no—
314C22—44no—
315C22—42no—
316C22—81no—
317C22—82no—
318C22—81no—
319C22#187211yescharacteristic
320C22#187612yes—
321C22#188112yes—
322C22#188711yescharacteristic
323C22#189011yescharacteristic
324C4—21no—
325C4—21no—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—42no—
330C4—81no—
331C4—81no—
332C4—161no—
333C4—161no—
334C4#161311yescharacteristic
335C4#161311yescharacteristic
336C4#161411yescharacteristic
337C4#161411yescharacteristic
338C2—41no—
339C2—41no—
340C2—82no—
341C2#899312yes—
342C2#900711yescharacteristic
343C2#902111yescharacteristic
344C2#903812yes—
345C2#906911yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8710C2semidirect
5. #7029C2semidirect

Maximal subgroups

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

subgroupindex
#237302
#219362
#87102
#75632
#70292