Finite Group Databasebeta

← #101349

Subgroups — #101349

Order 512.

454 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 90 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#27650C211yesmaximal, characteristic
2#22329C211yesmaximal, characteristic
3#8716C211yesmaximal, characteristic, complemented
4#8709C211yesmaximal, characteristic, complemented
5#8575C211yesmaximal, characteristic
6#7044C211yesmaximal, characteristic, complemented
7#7024C211yesmaximal, characteristic, complemented
8C2×#482—21no—
9C4×#121C2211yescharacteristic
10#2285C2211yescharacteristic
11#2084C411yescharacteristic, complemented
12#2083C411yescharacteristic, complemented
13#2071C411yescharacteristic, complemented
14#2071C411yescharacteristic, complemented
15#1954—21no—
16#1949—21no—
17#1923—21no—
18#1916—21no—
19#1869—21no—
20#1869—21no—
21#1825—21no—
22#1817C2211yescharacteristic
23#1804—21no—
24#1797C2211yescharacteristic
25#1797C2211yescharacteristic
26C8×C22⋊C4—21no—
27C4×C22⋊C8C2211yescharacteristic
28C2×#327—21no—
29#1434C2211yescharacteristic
30#1294—21no—
31#1294—21no—
32#1294—21no—
33#1279—21no—
34#1279—21no—
35C23×C8—21no—
36C2×#121D411yescharacteristic
37C2×C4×D4—21no—
38C2×C4×D4D411yescharacteristic
39C2×#117—21no—
40C22×C42C2311yescharacteristic
41#482—42no—
42#482—42no—
43C2×#107—21no—
44C2×#106—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×#102—21no—
49C2×#102—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8D411yescharacteristic
57C2×C22⋊C8D411yescharacteristic
58C2×C4×C8—21no—
59C2×C4×C8D411yescharacteristic
60C2×C4×C8D411yescharacteristic
61#390C2×C411yescharacteristic
62#389C2×C411yescharacteristic
63#388—21no—
64#386—21no—
65#384C2×C411yescharacteristic
66#384C2×C411yescharacteristic
67#382—21no—
68#380—21no—
69#380—21no—
70#380D411yescharacteristic
71C4×C4⋊C4—21no—
72C4×C4⋊C4—21no—
73C4×C4⋊C4D411yescharacteristic
74C4×C22⋊C4—21no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—21no—
78#340—21no—
79#340—21no—
80#340—21no—
81#340—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336C2×C411yescharacteristic
89#336C2×C411yescharacteristic
90#327—42no—
91C22×D4—21no—
92C22×D4C22⋊C411yescharacteristic
93C23×C4—21no—
94C23×C4—41no—
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4#4111yescharacteristic
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—42no—
105C22×C8—41no—
106C22×C8—81no—
107C22×C8—81no—
108#121—21no—
109#121—21no—
110#121D812yes—
111#121SD1612yes—
112C4×D4—41no—
113C4×D4—41no—
114C4×D4—81no—
115C4×D4—81no—
116#117—41no—
117#117—41no—
118#117—41no—
119#117—41no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4C22⋊C411yescharacteristic
128C2×C4⋊C4C22⋊C411yescharacteristic
129C2×C4⋊C4C22⋊C411yescharacteristic
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—41no—
136C2×C42—21no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42—41no—
140C2×C42C22×C411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42#4111yescharacteristic
144#107—42no—
145#106—42no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—81no—
151#102—42no—
152#102—42no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155C22⋊C8—41no—
156C22⋊C8—41no—
157C22⋊C8—41no—
158C22⋊C8—41no—
159C22⋊C8—41no—
160C22⋊C8—41no—
161C22⋊C8—81no—
162C22⋊C8—81no—
163C4×C8—41no—
164C4×C8—41no—
165C4×C8—81no—
166#95—21no—
167#95—21no—
168#95—21no—
169#95—22no—
170#95—41no—
171#95—41no—
172#95—41no—
173#95—41no—
174#95—41no—
175#95—41no—
176#95—41no—
177C24—21no—
178C24#12111yescharacteristic
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—21no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—81no—
185C2×D4—81no—
186C2×D4#10212yes—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
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—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—81no—
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C4×D411yescharacteristic
217C22×C4C4×D411yescharacteristic
218C22×C4#12011yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12311yescharacteristic
221C22×C4C2×D811yescharacteristic
222C22×C4C2×SD1611yescharacteristic
223C22×C4#13511yescharacteristic
224C22×C4#13511yescharacteristic
225C2×C8—22no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—84no—
244C2×C8—81no—
245C2×C8—81no—
246C2×C8—81no—
247C4⋊C4—21no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—81no—
258C4⋊C4—81no—
259C4⋊C4—81no—
260C4⋊C4#10212yes—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—81no—
266C22⋊C4—81no—
267C22⋊C4—81no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—41no—
275C42—41no—
276C42—41no—
277C42—41no—
278C42—81no—
279C42—81no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—22no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23#38611yescharacteristic
294C23#46511yescharacteristic
295C23#46611yescharacteristic
296D4—42no—
297D4—81no—
298D4—82no—
299D4—82no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
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—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—161no—
345C2×C4—161no—
346C2×C4—161no—
347C2×C4—161no—
348C2×C4C2×#10211yescharacteristic
349C2×C4#41611yescharacteristic
350C2×C4#44311yescharacteristic
351C2×C4#44411yescharacteristic
352C2×C4#45811yescharacteristic
353C2×C4#45811yescharacteristic
354C2×C4#46511yescharacteristic
355C2×C4#46611yescharacteristic
356C8—42no—
357C8—81no—
358C8—81no—
359C8—81no—
360C8—81no—
361C8—81no—
362C8—161no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22#187611yescharacteristic
381C22#188111yescharacteristic
382C22#189011yescharacteristic
383C22#189311yescharacteristic
384C22#190411yescharacteristic
385C22#190611yescharacteristic
386C22#190911yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—41no—
396C4—41no—
397C4—42no—
398C4—81no—
399C4—81no—
400C4—162no—
401C4—161no—
402C4—161no—
403C2—41no—
404C2—41no—
405C2—82no—
406C2#903811yescharacteristic
407C2#920811yescharacteristic
408C2#927911yescharacteristic
409C2#930511yescharacteristic
410C2#936911yescharacteristic
411C2#947011yescharacteristic
412C2#952511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ 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
3. #8716C2semidirect
4. #8709C2semidirect
6. #7044C2semidirect
7. #7024C2semidirect
11. #2084C4semidirect
12. #2083C4semidirect
13. #2071C4semidirect
14. #2071C4semidirect

Maximal subgroups

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

subgroupindex
#276502
#223292
#87162
#87092
#85752
#70442
#70242