Finite Group Databasebeta

← #122170

Subgroups — #122170

Order 512.

719 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34966C211yesmaximal, characteristic
2#10852C211yesmaximal, characteristic
3#8097C211yesmaximal, characteristic
4#8093C211yesmaximal, characteristic
5#8048C211yesmaximal, characteristic
6#7950C211yesmaximal, characteristic
7#7941C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#423C2211yescharacteristic
11#2606C412yescomplemented
12#2605C412yescomplemented
13#2570—21no—
14#2550—21no—
15#2442—21no—
16#2440—21no—
17#2438—22no—
18#2402—21no—
19#2399—21no—
20C4×C4⋊Q8—21no—
21C4×#125—21no—
22C4×C22⋊Q8—22no—
23#2283C2211yescharacteristic
24C4×#117C2211yescharacteristic
25C4×#117C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×#339—22no—
28#1303—21no—
29#1303—21no—
30#1303—21no—
31#1303—21no—
32#1297—21no—
33#1297—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×C4×Q8—22no—
42C2×#117—21no—
43C2×#117—21no—
44C22×C4⋊C4—21no—
45C22×C42D412yes—
46C22×C42C2311yescharacteristic
47#432—21no—
48#432—21no—
49#432—21no—
50#432Q812yescomplemented
51#431—21no—
52#431—21no—
53#431—21no—
54#431—21no—
55#423—21no—
56#423—21no—
57#423—21no—
58#423—21no—
59#423Q812yescomplemented
60#423Q812yescomplemented
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63#407—21no—
64#407—21no—
65C4×M16—21no—
66C4×M16—21no—
67#400—22no—
68#400—22no—
69#398—22no—
70#398—22no—
71#398—22no—
72#398—22no—
73#395—22no—
74#395—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#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#385—22no—
88#385—22no—
89#384—21no—
90#383C2×C412yes—
91#382—21no—
92#382—21no—
93#381D412yes—
94#380—22no—
95#380—21no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4D411yescharacteristic
105C4×C4⋊C4D411yescharacteristic
106C4×C4⋊C4D411yescharacteristic
107C4×C4⋊C4D411yescharacteristic
108C4×C4⋊C4D411yescharacteristic
109C4×C4⋊C4D411yescharacteristic
110C4×C4⋊C4Q812yescomplemented
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—21no—
115C4×C22⋊C4D411yescharacteristic
116C4×C22⋊C4D411yescharacteristic
117#376—21no—
118#376—21no—
119#376C2×C411yescharacteristic
120#376C2×C411yescharacteristic
121C2×#95—21no—
122C43—21no—
123C43—21no—
124#339—44no—
125C22×Q8—22no—
126C23×C4—22no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—41no—
133C2×M16—41no—
134C2×M16—42no—
135C2×M16—42no—
136C22×C8—41no—
137C4⋊Q8—44no—
138#125—44no—
139C22⋊Q8—48no—
140C4×Q8—82no—
141C4×Q8—82no—
142#117—22no—
143#117—22no—
144#117—44no—
145#117—41no—
146#117—41no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—44no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—41no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—41no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42—41no—
183C2×C42C22⋊C412yes—
184C2×C42C22⋊C412yes—
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×Q811yescharacteristic
190C2×C42C2×Q811yescharacteristic
191C2×C42C2×Q811yescharacteristic
192C2×C42C2×Q811yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C22⋊C8—22no—
206C22⋊C8—41no—
207C22⋊C8—41no—
208C22⋊C8—41no—
209#97—22no—
210#97—41no—
211C4×C8—41no—
212C4×C8—41no—
213#95—22no—
214#95—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—21no—
225#95—21no—
226#95—22no—
227#95—22no—
228#95—41no—
229#95C22⋊C411yescharacteristic
230#95C22⋊C411yescharacteristic
231#95C22⋊C411yescharacteristic
232#95C22⋊C411yescharacteristic
233#95C22⋊C412yes—
234#95C22⋊C412yes—
235C24#12011yescharacteristic
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—82no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—82no—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4#12011yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4#12011yescharacteristic
263C22×C4C22⋊Q811yescharacteristic
264C22×C4C22⋊Q811yescharacteristic
265C22×C4C22⋊Q811yescharacteristic
266C22×C4C22⋊Q811yescharacteristic
267C22×C4C22⋊Q811yescharacteristic
268C22×C4C22⋊Q811yescharacteristic
269C22×C4#12411yescharacteristic
270C22×C4#12411yescharacteristic
271C22×C4#12411yescharacteristic
272C22×C4C4⋊Q811yescharacteristic
273C22×C4C4⋊Q811yescharacteristic
274C22×C4C4⋊Q811yescharacteristic
275M16—44no—
276M16—82no—
277M16—81no—
278M16—81no—
279M16—82no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—82no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C22⋊C4—24no—
303C22⋊C4—42no—
304C22⋊C4—44no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C42—42no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—41no—
317C42—41no—
318C42—41no—
319C42—41no—
320C42—42no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—82no—
326C42—82no—
327C42C4×Q811yescharacteristic
328C42C4×Q811yescharacteristic
329C42C4×Q811yescharacteristic
330C42C4×Q811yescharacteristic
331C42C22⋊Q811yescharacteristic
332C42C22⋊Q811yescharacteristic
333C42C22⋊Q811yescharacteristic
334C42C22⋊Q811yescharacteristic
335C42C22⋊Q811yescharacteristic
336C42C22⋊Q811yescharacteristic
337C42#12411yescharacteristic
338C42#12411yescharacteristic
339C42#12411yescharacteristic
340C42C4⋊Q811yescharacteristic
341C42C4⋊Q811yescharacteristic
342C42C4⋊Q811yescharacteristic
343C23—21no—
344C23—21no—
345C23—22no—
346C23—41no—
347C23#37911yescharacteristic
348C23#39311yescharacteristic
349C23#39511yescharacteristic
350Q8—88no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—44no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—88no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4#39111yescharacteristic
402C2×C4#39111yescharacteristic
403C2×C4#39111yescharacteristic
404C2×C4#39111yescharacteristic
405C2×C4#39111yescharacteristic
406C2×C4#39111yescharacteristic
407C2×C4#39311yescharacteristic
408C2×C4#39311yescharacteristic
409C2×C4#39311yescharacteristic
410C2×C4#39511yescharacteristic
411C2×C4#39511yescharacteristic
412C2×C4#39511yescharacteristic
413C2×C4#45311yescharacteristic
414C2×C4#45411yescharacteristic
415C2×C4#45511yescharacteristic
416C2×C4#45611yescharacteristic
417C8—82no—
418C8—82no—
419C8—82no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—42no—
428C22—42no—
429C22#144611yescharacteristic
430C22#179211yescharacteristic
431C22#179311yescharacteristic
432C22#203112yes—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—42no—
443C4—44no—
444C4—42no—
445C4—81no—
446C4—81no—
447C4—82no—
448C4—82no—
449C4—82no—
450C4—84no—
451C4#202711yescharacteristic
452C4#202811yescharacteristic
453C4#203112yes—
454C4#206011yescharacteristic
455C4#206111yescharacteristic
456C2—21no—
457C2—21no—
458C2—22no—
459C2—41no—
460C2#1138711yescharacteristic
461C2#1138711yescharacteristic
462C2#1139211yescharacteristic
subgroupG/H
centerC2×C4#393
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1446
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #2606C4semidirect
12. #2605C4semidirect
50. #432Q8semidirect
59. #423Q8semidirect
60. #423Q8semidirect
110. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#349662
#108522
#80972
#80932
#80482
#79502
#79412