Finite Group Databasebeta

← #122113

Subgroups — #122113

Order 512.

670 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic
2#10852C211yesmaximal, characteristic
3#8052C211yesmaximal, characteristic
4#7962C211yesmaximal, characteristic
5#7960C211yesmaximal, characteristic
6#7950C211yesmaximal, characteristic
7#7940C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#423C2211yescharacteristic
11#2857—21no—
12#2851—21no—
13#2627—22no—
14#2606—21no—
15#2606C412yescomplemented
16#2605—21no—
17#2605C412yescomplemented
18#2570—21no—
19#2570—21no—
20#2550—21no—
21#2550—21no—
22#2329—22no—
23#2283C2211yescharacteristic
24#2283C2211yescharacteristic
25C4×#117C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×#339—21no—
28C2×#339—21no—
29#1297—22no—
30#1286—21no—
31#1286—21no—
32#1286—21no—
33#1286—21no—
34#1285—21no—
35#1285—21no—
36#1284—22no—
37C22×M16—21no—
38C2×C4⋊Q8—21no—
39C2×#125—21no—
40C2×C22⋊Q8—22no—
41C2×#117—21no—
42C22×C4⋊C4—21no—
43C22×C42D411yescharacteristic
44C22×C42D411yescharacteristic
45C22×C42C2311yescharacteristic
46#432—21no—
47#432—21no—
48#432—21no—
49#432—21no—
50#431—21no—
51#431—21no—
52#431—21no—
53#431Q812yescomplemented
54#423—21no—
55#423—21no—
56#423—21no—
57#423—21no—
58#423—21no—
59#423Q812yescomplemented
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62#407—21no—
63#407—21no—
64C4×M16—21no—
65C4×M16Q812yescomplemented
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#391—22no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#389—22no—
82#389—22no—
83#389—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#385—22no—
91#385—22no—
92#384—21no—
93#383—21no—
94#383C2×C412yes—
95#382—21no—
96#382D411yescharacteristic
97#382D411yescharacteristic
98#382D411yescharacteristic
99#382D411yescharacteristic
100#381—21no—
101#381D412yes—
102#380—22no—
103#380—21no—
104#380—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4C2×C411yescharacteristic
111C4×C4⋊C4C2×C411yescharacteristic
112C4×C4⋊C4D411yescharacteristic
113C4×C4⋊C4D411yescharacteristic
114C4×C4⋊C4Q812yescomplemented
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4D412yes—
119#376—21no—
120#376—21no—
121#376C2×C412yes—
122C2×#95—21no—
123C43—21no—
124#339—42no—
125#339—42no—
126C22×Q8—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—41no—
135C2×M16—41no—
136C2×M16—41no—
137C2×M16—41no—
138C2×M16—41no—
139C2×M16—41no—
140C22×C8—41no—
141C4⋊Q8—81no—
142C4⋊Q8—81no—
143#125—81no—
144#125—81no—
145C22⋊Q8—48no—
146#117—22no—
147#117—22no—
148#117—41no—
149#117—41no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—44no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—41no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—41no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42—41no—
181C2×C42—41no—
182C2×C42—41no—
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×Q811yescharacteristic
192C2×C42C2×Q811yescharacteristic
193C2×C42C2×Q811yescharacteristic
194C2×C42C2×Q811yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207C22⋊C8—22no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—41no—
211#97—22no—
212#97—41no—
213C4×C8—22no—
214C4×C8—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—21no—
222#95—21no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95—41no—
235#95C22⋊C412yes—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239C24#12011yescharacteristic
240C2×Q8—44no—
241C2×Q8—82no—
242C2×Q8—82no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4C22⋊Q811yescharacteristic
271C22×C4C22⋊Q811yescharacteristic
272C22×C4C22⋊Q811yescharacteristic
273C22×C4C22⋊Q811yescharacteristic
274C22×C4C22⋊Q811yescharacteristic
275C22×C4C22⋊Q811yescharacteristic
276C22×C4#12411yescharacteristic
277C22×C4#12411yescharacteristic
278C22×C4#12411yescharacteristic
279C22×C4C4⋊Q811yescharacteristic
280C22×C4C4⋊Q811yescharacteristic
281C22×C4C4⋊Q811yescharacteristic
282M16—42no—
283M16—42no—
284M16—82no—
285M16—81no—
286M16—81no—
287M16—82no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—81no—
292C2×C8—81no—
293C4⋊C4—24no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C42—41no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—42no—
325C42—42no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—82no—
331C42—82no—
332C42—82no—
333C42—82no—
334C42C4×Q811yescharacteristic
335C42C4×Q811yescharacteristic
336C42C4×Q811yescharacteristic
337C42C4×Q811yescharacteristic
338C42C22⋊Q811yescharacteristic
339C42C22⋊Q811yescharacteristic
340C42C22⋊Q811yescharacteristic
341C42C22⋊Q811yescharacteristic
342C42C22⋊Q811yescharacteristic
343C42C22⋊Q811yescharacteristic
344C42#12411yescharacteristic
345C42#12411yescharacteristic
346C42#12411yescharacteristic
347C42C4⋊Q811yescharacteristic
348C42C4⋊Q811yescharacteristic
349C42C4⋊Q811yescharacteristic
350C23—21no—
351C23—21no—
352C23—22no—
353C23—41no—
354C23#37911yescharacteristic
355C23#39311yescharacteristic
356C23#39511yescharacteristic
357Q8—164no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—84no—
406C2×C4—82no—
407C2×C4—84no—
408C2×C4—82no—
409C2×C4#39111yescharacteristic
410C2×C4#39111yescharacteristic
411C2×C4#39111yescharacteristic
412C2×C4#39111yescharacteristic
413C2×C4#39111yescharacteristic
414C2×C4#39111yescharacteristic
415C2×C4#39311yescharacteristic
416C2×C4#39311yescharacteristic
417C2×C4#39311yescharacteristic
418C2×C4#39511yescharacteristic
419C2×C4#39511yescharacteristic
420C2×C4#39511yescharacteristic
421C2×C4#45311yescharacteristic
422C2×C4#45411yescharacteristic
423C2×C4#45511yescharacteristic
424C2×C4#45611yescharacteristic
425C8—82no—
426C8—82no—
427C8—82no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—42no—
436C22—42no—
437C22#144611yescharacteristic
438C22#179211yescharacteristic
439C22#179311yescharacteristic
440C22#203112yes—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—42no—
452C4—42no—
453C4—82no—
454C4—82no—
455C4—81no—
456C4—81no—
457C4—82no—
458C4—82no—
459C4—84no—
460C4#202711yescharacteristic
461C4#202811yescharacteristic
462C4#206011yescharacteristic
463C4#206111yescharacteristic
464C2—21no—
465C2—21no—
466C2—22no—
467C2—41no—
468C2#1138711yescharacteristic
469C2#1138911yescharacteristic
470C2#1139211yescharacteristic
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
15. #2606C4semidirect
17. #2605C4semidirect
53. #431Q8semidirect
59. #423Q8semidirect
65. C4×M16Q8semidirect
114. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#351042
#108522
#80522
#79622
#79602
#79502
#79402