Finite Group Databasebeta

← #190361

Subgroups — #190361

Order 512.

956 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34197C211yesmaximal, characteristic
2C4×#410C211yesmaximal, characteristic
3#17366C211yesmaximal, characteristic
4#8779C214yesmaximal, complemented
5#8778C214yesmaximal, complemented
6#8052C212yesmaximal, complemented
7#7941C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9#2920C2212yes—
10C2×#410—21no—
11#2357C411yescharacteristic, complemented
12#2357C411yescharacteristic, complemented
13#2357C2212yes—
14#2357C2212yes—
15#2351C411yescharacteristic, complemented
16#2351C411yescharacteristic, complemented
17#2346—21no—
18#2346—22no—
19#2341—21no—
20#2319—22no—
21#2319—21no—
22#2283C412yescomplemented
23#2283C412yescomplemented
24#2283C2211yescharacteristic
25#2283C2211yescharacteristic
26#2276—21no—
27C4×#117C2212yes—
28C4×#99—22no—
29C4×#99C2214yes—
30#1740—21no—
31#1465C2214yes—
32#1464—22no—
33#1461—21no—
34#1460—21no—
35#1297C2218yes—
36#1284C2214yes—
37#1284C2214yes—
38C22×#41D411yescharacteristic
39C4×#41—22no—
40C4×#41—22no—
41C4×#41—22no—
42C4×#41—22no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8C2×C411yescharacteristic
46C2×C4×Q8D411yescharacteristic
47C2×C4×D4C2×C411yescharacteristic
48C2×C4×D4D412yes—
49C2×C4×D4D411yescharacteristic
50C2×C4×D4C2312yes—
51C2×#117—22no—
52C2×#117D411yescharacteristic
53C2×#117D412yes—
54C22×C42C2×C412yes—
55C22×C42C2311yescharacteristic
56#432—22no—
57#432—22no—
58#431—22no—
59#431—22no—
60#423—22no—
61#410—28no—
62C2×#99—24no—
63#407C2314yes—
64C4×M16—22no—
65#390—21no—
66#390—21no—
67#390—22no—
68#390—22no—
69#389—21no—
70#389—21no—
71#389—22no—
72#389—22no—
73#388—22no—
74#388—22no—
75#388—22no—
76#388—22no—
77#388—22no—
78#388—22no—
79#383C2×C411yescharacteristic
80#383C2×C411yescharacteristic
81#383C2×C412yes—
82#382—22no—
83#382—21no—
84#382C2×C411yescharacteristic
85#382C2×C411yescharacteristic
86#382C2×C412yes—
87#382C2×C412yes—
88#381—22no—
89#381—21no—
90#381C2×C412yes—
91#381C2×C412yes—
92#381C2311yescharacteristic
93#381C2311yescharacteristic
94#380—22no—
95#380—22no—
96#380—21no—
97#380—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4C2×C411yescharacteristic
107C4×C4⋊C4C2×C411yescharacteristic
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C412yes—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4C2×C412yes—
114C4×C22⋊C4C2×C412yes—
115C4×C22⋊C4C2314yes—
116C4×C22⋊C4C2312yes—
117#376—22no—
118#376—22no—
119#376—21no—
120#376—21no—
121#376—22no—
122C43—22no—
123#342—24no—
124#324—24no—
125#323—24no—
126C2×#41—24no—
127C2×#41—22no—
128C2×#41—22no—
129C2×#41—42no—
130C22×Q8C22⋊C411yescharacteristic
131C22×D4C22⋊C411yescharacteristic
132C22×D4C2×D412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22×C412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2411yescharacteristic
139C2×M16—24no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—44no—
145C4×Q8—44no—
146C4×D4—24no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—22no—
154#117—22no—
155#117—22no—
156#117—24no—
157#117—24no—
158#117—28no—
159#117—44no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22⋊C411yescharacteristic
172C2×C4⋊C4C2×D412yes—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C2×D412yes—
183C2×C22⋊C4C2×D412yes—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42C22⋊C412yes—
199C2×C42C22⋊C412yes—
200C2×C42C22×C412yes—
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42#4112yes—
208C2×C42#4112yes—
209C4⋊C8—42no—
210C4⋊C8—42no—
211#99—216no—
212C22⋊C8—24no—
213C22⋊C8—24no—
214#97—42no—
215C4×C8—42no—
216#95—21no—
217#95—21no—
218#95—22no—
219#95—22no—
220#95—21no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—24no—
231#95—42no—
232#95C22×C411yescharacteristic
233#95C22×C411yescharacteristic
234#95C22×C412yes—
235#95C22×C412yes—
236#95C22×C411yescharacteristic
237#95C22×C411yescharacteristic
238C24C2×C22⋊C412yes—
239C24C22×D411yescharacteristic
240#41—48no—
241#41—48no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8#10414yes—
245C2×D4—24no—
246C2×D4—22no—
247C2×D4—24no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—42no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4#12312yes—
280C22×C4#12312yes—
281C22×C4#12312yes—
282C22×C4#12312yes—
283C22×C4C23×C411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C2×#4112yes—
286C22×C4C2×#4112yes—
287M16—84no—
288C2×C8—44no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—44no—
294C4⋊C4—48no—
295C4⋊C4—44no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—81no—
299C4⋊C4—81no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—28no—
306C22⋊C4—44no—
307C22⋊C4—81no—
308C22⋊C4—81no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—44no—
321C42—48no—
322C42—44no—
323C42—42no—
324C42—42no—
325C42—82no—
326C42—82no—
327C42—82no—
328C42—84no—
329C42—84no—
330C42#11711yescharacteristic
331C42#11711yescharacteristic
332C42#11711yescharacteristic
333C42#11711yescharacteristic
334C42#11712yes—
335C42#11712yes—
336C42#12312yes—
337C42#12312yes—
338C42#12312yes—
339C42#12312yes—
340C23—21no—
341C23—21no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—41no—
346C23—41no—
347C23—42no—
348C23—42no—
349C23C22×C22⋊C411yescharacteristic
350C23C2×#12312yes—
351Q8—28no—
352D4—48no—
353D4—44no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—48no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—84no—
398C2×C4—84no—
399C2×C4#38111yescharacteristic
400C2×C4#38111yescharacteristic
401C2×C4#38111yescharacteristic
402C2×C4#38111yescharacteristic
403C2×C4#38112yes—
404C2×C4#38112yes—
405C2×C4C2×#10414yes—
406C2×C4C2×#10412yes—
407C2×C4#42112yes—
408C2×C4C2×#11711yescharacteristic
409C2×C4C2×#11711yescharacteristic
410C2×C4C2×#12312yes—
411C8—84no—
412C22—21no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—81no—
421C22—81no—
422C22—82no—
423C22—82no—
424C22C2×#38111yescharacteristic
425C22C22×#10411yescharacteristic
426C22C2×#42111yescharacteristic
427C4—21no—
428C4—21no—
429C4—24no—
430C4—21no—
431C4—21no—
432C4—22no—
433C4—22no—
434C4—21no—
435C4—21no—
436C4—42no—
437C4—42no—
438C4—82no—
439C4—81no—
440C4—81no—
441C4—82no—
442C4—81no—
443C4—81no—
444C4—88no—
445C4#182614yes—
446C2—21no—
447C2—21no—
448C2—42no—
449C2—82no—
450C2C2×#182611yescharacteristic
451C2#2070811yescharacteristic
452C2#2071011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ 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
4. #8779C2semidirect
5. #8778C2semidirect
6. #8052C2semidirect
7. #7941C2semidirect
11. #2357C4semidirect
12. #2357C4semidirect
15. #2351C4semidirect
16. #2351C4semidirect
22. #2283C4semidirect
23. #2283C4semidirect

Maximal subgroups

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

subgroupindex
#341972
C4×#4102
#173662
#87792
#87782
#80522
#79412