Finite Group Databasebeta

← #101199

Subgroups — #101199

Order 512.

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

normal subgroups only

This group has 620 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#21934C211yesmaximal, characteristic
2#15165C211yesmaximal, characteristic
3#8557C211yesmaximal, characteristic
4#7641C212yesmaximal, complemented
5#7039C212yesmaximal, complemented
6C2×#407—21no—
7C2×#396—21no—
8C2×#389C2211yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10#1851—24no—
11#1797—21no—
12#1797C2212yes—
13#1797C2212yes—
14C2×#323—21no—
15#1444C411yescharacteristic
16#1442C411yescharacteristic
17#1441—21no—
18#1439C411yescharacteristic
19#1439C411yescharacteristic
20#1436—21no—
21C4⋊#95C2211yescharacteristic
22C22⋊#95—21no—
23#1280—21no—
24#1271—24no—
25C24×C4D411yescharacteristic
26C22×M16—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4C2×C411yescharacteristic
35C22×C22⋊C4D411yescharacteristic
36C22×C42D411yescharacteristic
37C22×C42C2311yescharacteristic
38C2×C4⋊C8—22no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—22no—
41#407—42no—
42#407—42no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46#396—44no—
47#389—22no—
48#389—22no—
49C4×C22⋊C4—22no—
50C4×C22⋊C4—22no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
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—22no—
63#336—22no—
64#336D414yes—
65#323—42no—
66C25C22⋊C411yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—81no—
76C23×C4—81no—
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4SD1612yes—
81C23×C4Q1612yes—
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C2×M16—44no—
86C2×M16—82no—
87C2×M16—82no—
88C22×C8—24no—
89C22×C8—42no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—81no—
102C2×C4⋊C4—81no—
103C2×C4⋊C4—81no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—81no—
116C2×C22⋊C4—81no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—81no—
120C2×C42—81no—
121C2×C42C2×D412yes—
122C2×C42#4112yes—
123C4⋊C8—82no—
124C4⋊C8—82no—
125C4⋊C8—82no—
126C22⋊C8—44no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130C22⋊C8—42no—
131#95—22no—
132#95—22no—
133#95—41no—
134#95—41no—
135#95—42no—
136#95—42no—
137#95—41no—
138#95—41no—
139#95—41no—
140#95—41no—
141#95—81no—
142#95—81no—
143#95—81no—
144C24—21no—
145C24—21no—
146C24—21no—
147C24—21no—
148C24—21no—
149C24—21no—
150C24—41no—
151C24—41no—
152C24—41no—
153C24—41no—
154C24#10312yes—
155C24C2×C22⋊C411yescharacteristic
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
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—41no—
184C22×C4—41no—
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—81no—
212C22×C4—81no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4—82no—
216C22×C4—82no—
217C22×C4#10112yes—
218C22×C4#10312yes—
219C22×C4C4×D412yes—
220C22×C4#12011yescharacteristic
221C22×C4#12112yes—
222C22×C4#12311yescharacteristic
223C22×C4C2×SD1611yescharacteristic
224C22×C4C2×Q1611yescharacteristic
225C22×C4AGL(1,ℤ/8)11yescharacteristic
226C22×C4AGL(1,ℤ/8)12yes—
227C22×C4AGL(1,ℤ/8)11yescharacteristic
228C22×C4#13711yescharacteristic
229C22×C4#13711yescharacteristic
230M16—88no—
231C2×C8—44no—
232C2×C8—44no—
233C2×C8—44no—
234C2×C8—82no—
235C2×C8—82no—
236C2×C8—82no—
237C4⋊C4—24no—
238C4⋊C4—42no—
239C4⋊C4—82no—
240C4⋊C4—84no—
241C4⋊C4—84no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C22⋊C4—44no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—84no—
248C22⋊C4—84no—
249C42—42no—
250C42—42no—
251C42—42no—
252C42—81no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—81no—
275C23—81no—
276C23—81no—
277C23—81no—
278C23—81no—
279C23—81no—
280C23#38611yescharacteristic
281C23C2×#10111yescharacteristic
282C23#41311yescharacteristic
283C23C2×#10311yescharacteristic
284C23#41711yescharacteristic
285C23#41711yescharacteristic
286C23#41811yescharacteristic
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
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#45011yescharacteristic
353C2×C4#45111yescharacteristic
354C2×C4#46812yes—
355C2×C4#46912yes—
356C2×C4#48311yescharacteristic
357C2×C4#48411yescharacteristic
358C8—84no—
359C8—162no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—44no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—82no—
377C22—82no—
378C22—82no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22#154911yescharacteristic
384C22#155011yescharacteristic
385C22#155711yescharacteristic
386C22#156111yescharacteristic
387C22#187211yescharacteristic
388C22#188411yescharacteristic
389C22#189111yescharacteristic
390C22#189511yescharacteristic
391C22#190111yescharacteristic
392C22#191111yescharacteristic
393C22#191311yescharacteristic
394C4—22no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—82no—
404C4—81no—
405C4—82no—
406C4—161no—
407C4—161no—
408C2—21no—
409C2—21no—
410C2—21no—
411C2—21no—
412C2—82no—
413C2#903311yescharacteristic
414C2#913711yescharacteristic
415C2#919911yescharacteristic
416C2#938611yescharacteristic
417C2#944911yescharacteristic
418C2#948911yescharacteristic
419C2#951811yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊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
4. #7641C2semidirect
5. #7039C2semidirect

Maximal subgroups

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

subgroupindex
#219342
#151652
#85572
#76412
#70392