Finite Group Databasebeta

← #122149

Subgroups — #122149

Order 512.

669 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34966C211yesmaximal, characteristic
2#8098C211yesmaximal, characteristic
3#8071C211yesmaximal, characteristic
4#8052C211yesmaximal, characteristic
5#8021C211yesmaximal, characteristic
6#7951C211yesmaximal, characteristic
7#7944C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11#2606—21no—
12#2605—21no—
13#2570—21no—
14#2550—21no—
15#2442—21no—
16#2440—21no—
17#2438—22no—
18#2402C412yescomplemented
19#2399C412yescomplemented
20C4×C4⋊Q8—21no—
21C4×#125—21no—
22C4×C22⋊Q8—22no—
23#2283C2211yescharacteristic
24#2283C2211yescharacteristic
25C4×#117C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1303—21no—
28#1303—21no—
29#1300—21no—
30#1300—21no—
31#1297—21no—
32#1297—21no—
33#1294—22no—
34#1285—21no—
35#1285—21no—
36#1284—22no—
37C2×C4⋊Q8—21no—
38C2×#125—21no—
39C2×C22⋊Q8—22no—
40C2×C4×Q8—22no—
41C2×#117—21no—
42C22×C4⋊C4—21no—
43C22×C42—21no—
44C22×C42C2311yescharacteristic
45#432—21no—
46#432—21no—
47#432—21no—
48#432—21no—
49#432—21no—
50#432—21no—
51#432Q812yescomplemented
52#432Q812yescomplemented
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#423—21no—
58#423Q812yescomplemented
59C2×C4⋊C8—21no—
60#407—21no—
61#407—21no—
62C2×C22⋊C8—21no—
63C4×M16—21no—
64C4×M16—21no—
65C2×#97—21no—
66#400—22no—
67#400—22no—
68#398—22no—
69#398—22no—
70#398—22no—
71#398—22no—
72#395—22no—
73#395—22no—
74#391—22no—
75#391—22no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—22no—
80#389—21no—
81#389—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#385—22no—
87#385—22no—
88#384—21no—
89#383—21no—
90#383—21no—
91#382—21no—
92#382—21no—
93#382Q812yescomplemented
94#381—21no—
95#381—21no—
96#380—22no—
97#380—21no—
98#380—21no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4C2×C411yescharacteristic
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C4⋊C4D411yescharacteristic
110C4×C4⋊C4D411yescharacteristic
111C4×C4⋊C4D411yescharacteristic
112C4×C4⋊C4D411yescharacteristic
113C4×C22⋊C4—21no—
114C4×C22⋊C4—21no—
115C4×C22⋊C4D412yes—
116C4×C22⋊C4D411yescharacteristic
117C4×C22⋊C4D411yescharacteristic
118#376—21no—
119#376C2×C412yes—
120#376D411yescharacteristic
121#376D411yescharacteristic
122C2×#95D412yes—
123C43—21no—
124C22×Q8—22no—
125C23×C4—22no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C2×M16—22no—
130C2×M16—41no—
131C22×C8—41no—
132C4⋊Q8—44no—
133#125—44no—
134C22⋊Q8—48no—
135C4×Q8—84no—
136#117—44no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—44no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—41no—
159C2×C42—21no—
160C2×C42—22no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—41no—
169C2×C42—82no—
170C2×C42C22⋊C412yes—
171C2×C42C22⋊C412yes—
172C2×C42C22×C411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×Q811yescharacteristic
177C2×C42C2×Q811yescharacteristic
178C2×C42C2×Q811yescharacteristic
179C2×C42C2×Q811yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196#97—22no—
197#97—22no—
198#97—41no—
199#97—41no—
200C4×C8—41no—
201C4×C8—41no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—21no—
212#95—21no—
213#95—22no—
214#95—21no—
215#95—21no—
216#95—22no—
217#95—22no—
218#95—41no—
219#95—41no—
220#95C22⋊C412yes—
221#95C22⋊C412yes—
222#95C22⋊C411yescharacteristic
223#95C22⋊C411yescharacteristic
224#95C22⋊C411yescharacteristic
225#95C22⋊C411yescharacteristic
226C24#12011yescharacteristic
227C2×Q8—44no—
228C2×Q8—44no—
229C2×Q8—82no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—82no—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12011yescharacteristic
252C22×C4C22⋊Q811yescharacteristic
253C22×C4C22⋊Q811yescharacteristic
254C22×C4C22⋊Q811yescharacteristic
255C22×C4C22⋊Q811yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4C22⋊Q811yescharacteristic
258C22×C4#12411yescharacteristic
259C22×C4#12411yescharacteristic
260C22×C4#12411yescharacteristic
261C22×C4C4⋊Q811yescharacteristic
262C22×C4C4⋊Q811yescharacteristic
263C22×C4C4⋊Q811yescharacteristic
264M16—81no—
265M16—81no—
266M16—82no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—82no—
271C4⋊C4—24no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C22⋊C4—24no—
289C22⋊C4—42no—
290C22⋊C4—44no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C42—42no—
294C42—42no—
295C42—42no—
296C42—44no—
297C42—42no—
298C42—42no—
299C42—42no—
300C42—42no—
301C42—42no—
302C42—81no—
303C42—81no—
304C42—82no—
305C42C4×Q811yescharacteristic
306C42C4×Q811yescharacteristic
307C42C4×Q811yescharacteristic
308C42C4×Q811yescharacteristic
309C42C22⋊Q811yescharacteristic
310C42C22⋊Q811yescharacteristic
311C42C22⋊Q811yescharacteristic
312C42C22⋊Q811yescharacteristic
313C42C22⋊Q811yescharacteristic
314C42C22⋊Q811yescharacteristic
315C42#12411yescharacteristic
316C42#12411yescharacteristic
317C42#12411yescharacteristic
318C42C4⋊Q811yescharacteristic
319C42C4⋊Q811yescharacteristic
320C42C4⋊Q811yescharacteristic
321C23—21no—
322C23—21no—
323C23—22no—
324C23—41no—
325C23#37911yescharacteristic
326C23#39311yescharacteristic
327C23#39511yescharacteristic
328Q8—88no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—88no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4#39111yescharacteristic
378C2×C4#39111yescharacteristic
379C2×C4#39111yescharacteristic
380C2×C4#39111yescharacteristic
381C2×C4#39111yescharacteristic
382C2×C4#39111yescharacteristic
383C2×C4#39311yescharacteristic
384C2×C4#39311yescharacteristic
385C2×C4#39311yescharacteristic
386C2×C4#39511yescharacteristic
387C2×C4#39511yescharacteristic
388C2×C4#39511yescharacteristic
389C8—82no—
390C8—82no—
391C8—82no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—42no—
400C22—42no—
401C22#144611yescharacteristic
402C22#179511yescharacteristic
403C22#179611yescharacteristic
404C22#203112yes—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—42no—
417C4—44no—
418C4—42no—
419C4—81no—
420C4—81no—
421C4—82no—
422C4—82no—
423C4—82no—
424C4—162no—
425C4#203112yes—
426C2—21no—
427C2—21no—
428C2—22no—
429C2—41no—
430C2#1138711yescharacteristic
431C2#1138711yescharacteristic
432C2#1138811yescharacteristic
subgroupG/H
center, socleC22#1446
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #2402C4semidirect
19. #2399C4semidirect
51. #432Q8semidirect
52. #432Q8semidirect
58. #423Q8semidirect
93. #382Q8semidirect

Maximal subgroups

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

subgroupindex
#349662
#80982
#80712
#80522
#80212
#79512
#79442