Finite Group Databasebeta

← #190437

Subgroups — #190437

Order 512.

900 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34179C211yesmaximal, characteristic
2C2×#1464C211yesmaximal, characteristic
3C2×#1458C211yesmaximal, characteristic
4#8736C214yesmaximal
5#8735C214yesmaximal, complemented
6#7928C212yesmaximal, complemented
7#7926C212yesmaximal, complemented
8C22×#117C2211yescharacteristic
9C2×#432C2212yes—
10C2×#431C2212yes—
11#2347—22no—
12#2347—22no—
13#2346—22no—
14#2346—22no—
15#2316C411yescharacteristic
16#2316C411yescharacteristic
17#2312C411yescharacteristic
18#2312C411yescharacteristic
19#2312C2212yes—
20#2283C412yes—
21#2283C2211yescharacteristic
22#2283C2211yescharacteristic
23#2280C412yes—
24#2274C2212yes—
25#1464—22no—
26#1464—22no—
27#1458—22no—
28#1458—22no—
29C2×#324C2212yes—
30C2×#324C2212yes—
31C2×#323C2212yes—
32C2×#323C2212yes—
33#1282C2218yes—
34#1282C2218yes—
35C24×C4C2×C411yescharacteristic
36C2×#117—22no—
37C2×#117—22no—
38C2×#117—22no—
39C2×#117—22no—
40C2×#117C2×C412yes—
41C2×#117C2×C412yes—
42C2×#117D412yes—
43C2×#117D412yes—
44C2×#117D412yes—
45C2×#117D412yes—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C22⋊C4C2×C411yescharacteristic
49C22×C22⋊C4C2311yescharacteristic
50C22×C42C2×C411yescharacteristic
51C22×C42C2311yescharacteristic
52#432—22no—
53#432—22no—
54#432—22no—
55#432—22no—
56#431—22no—
57#431—22no—
58#431—22no—
59#431—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C22⋊C8C2314yes—
63C2×C22⋊C8C2314yes—
64C2×#97—22no—
65C2×C4×C8—22no—
66#389—22no—
67#389—22no—
68#389—22no—
69#389—22no—
70#388—22no—
71#388—22no—
72#388—22no—
73#388—22no—
74#388—22no—
75#388—22no—
76#388—22no—
77#388—22no—
78#383—22no—
79#383—21no—
80#383—21no—
81#382—22no—
82#382—21no—
83#382—21no—
84#381C2×C412yes—
85#381C2×C412yes—
86#380—22no—
87#380—22no—
88#380C2×C411yescharacteristic
89#380C2×C411yescharacteristic
90#380C2×C411yescharacteristic
91#380C2×C411yescharacteristic
92#380C2×C412yes—
93#380C2×C412yes—
94#379C2×C412yes—
95#379C2×C412yes—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4C2×C412yes—
106C4×C22⋊C4C2×C412yes—
107C4×C22⋊C4C2312yes—
108C4×C22⋊C4C2312yes—
109#376—22no—
110#376—21no—
111#376—21no—
112#324—24no—
113#324—24no—
114#323—24no—
115#323—24no—
116C25C22×C411yescharacteristic
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—24no—
122C23×C4—42no—
123C23×C4C22⋊C412yes—
124C23×C4C22⋊C412yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8—24no—
132C22×C8—24no—
133#117—24no—
134#117—24no—
135#117—24no—
136#117—24no—
137#117—24no—
138#117—24no—
139#117—24no—
140#117—24no—
141#117—28no—
142#117—44no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22⋊C412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4C22⋊C412yes—
168C2×C22⋊C4C22⋊C412yes—
169C2×C22⋊C4C22×C412yes—
170C2×C22⋊C4C22×C412yes—
171C2×C22⋊C4C22×C412yes—
172C2×C22⋊C4C22×C412yes—
173C2×C22⋊C4C2×D412yes—
174C2×C22⋊C4C2×D412yes—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42C22⋊C412yes—
186C2×C42C22⋊C412yes—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42#4112yes—
192C2×C42#4112yes—
193C2×C42#4112yes—
194C2×C42#4112yes—
195C4⋊C8—42no—
196C4⋊C8—42no—
197C4⋊C8—42no—
198C4⋊C8—42no—
199C22⋊C8—24no—
200C22⋊C8—24no—
201C22⋊C8—24no—
202C22⋊C8—24no—
203#97—42no—
204#97—42no—
205C4×C8—42no—
206C4×C8—42no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221C24—21no—
222C24—21no—
223C24—22no—
224C24—22no—
225C24—81no—
226C24—81no—
227C24C2×C22⋊C412yes—
228C24C23×C411yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4#11711yescharacteristic
266C22×C4#11711yescharacteristic
267C22×C4#11711yescharacteristic
268C22×C4#11711yescharacteristic
269C22×C4#11712yes—
270C22×C4#11712yes—
271C22×C4#13114yes—
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C2×#4112yes—
275C22×C4C2×#4112yes—
276C2×C8—44no—
277C2×C8—44no—
278C2×C8—84no—
279C2×C8—84no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—81no—
287C4⋊C4—81no—
288C4⋊C4—81no—
289C4⋊C4—81no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—44no—
309C42—44no—
310C42—82no—
311C42—82no—
312C42—82no—
313C42—82no—
314C42#12312yes—
315C42#12312yes—
316C42#12312yes—
317C42#12312yes—
318C42#12312yes—
319C42#12312yes—
320C42#12312yes—
321C42#12312yes—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—41no—
326C23—41no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—84no—
336C23—84no—
337C23C22×C22⋊C411yescharacteristic
338C23C2×#11711yescharacteristic
339C23C2×#11711yescharacteristic
340C23C2×#13112yes—
341C23#56812yes—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4#38111yescharacteristic
387C2×C4#38111yescharacteristic
388C2×C4#38111yescharacteristic
389C2×C4#38111yescharacteristic
390C2×C4#38112yes—
391C2×C4#38112yes—
392C2×C4#40814yes—
393C2×C4C2×#12312yes—
394C2×C4C2×#12312yes—
395C8—84no—
396C8—84no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—82no—
409C22—82no—
410C22—84no—
411C22—84no—
412C22C2×#38111yescharacteristic
413C22C2×#40811yescharacteristic
414C22#287911yescharacteristic
415C22#292412yes—
416C22#292412yes—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—24no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—42no—
427C4—42no—
428C4—82no—
429C4—82no—
430C4—82no—
431C4—82no—
432C2—21no—
433C2—21no—
434C2—22no—
435C2—22no—
436C2—82no—
437C2#2068511yescharacteristic
438C2#2071011yescharacteristic
439C2#2071311yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8735C2semidirect
6. #7928C2semidirect
7. #7926C2semidirect

Maximal subgroups

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

subgroupindex
#341792
C2×#14642
C2×#14582
#87362
#87352
#79282
#79262