Finite Group Databasebeta

← #106901

Subgroups — #106901

Order 512.

807 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34192C211yesmaximal, characteristic
2#17449C211yesmaximal, characteristic
3#17449C211yesmaximal, characteristic
4#8116C211yesmaximal, characteristic
5#8106C211yesmaximal, characteristic
6#8070C211yesmaximal, characteristic
7#7976C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13#2356—22no—
14#2356—21no—
15#2351—21no—
16#2345—22no—
17#2345C412yes—
18Q8×C22⋊C4C412yes—
19#2320—22no—
20#2320—21no—
21#2283—21no—
22#2283—21no—
23#2283C2211yescharacteristic
24#2283C2211yescharacteristic
25#2276—21no—
26#1491—22no—
27#1491C414yescomplemented
28#1487—21no—
29#1487—21no—
30#1487—21no—
31#1487—21no—
32#1479—22no—
33#1479C414yescomplemented
34#1306—21no—
35#1306—21no—
36#1305—22no—
37#1300—22no—
38#1288—22no—
39C23×Q8C2×C411yescharacteristic
40C2×C4×Q8—22no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8C2×C412yes—
48C2×C4×Q8C2×C412yes—
49C2×C4×Q8D411yescharacteristic
50C2×C4×Q8D411yescharacteristic
51C2×C4×Q8D411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8D412yes—
54C2×C4×Q8Q812yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432C2×C412yes—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431C2×C412yes—
74#431C2×C412yes—
75#423—21no—
76#423—21no—
77#423—21no—
78#423—21no—
79#407—21no—
80#407—21no—
81#407—21no—
82#407—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16C2×C412yes—
87#391—22no—
88#391—24no—
89#389—24no—
90#389—22no—
91#387—24no—
92#387—24no—
93#387—24no—
94#383—22no—
95#383—22no—
96#383—21no—
97#383D412yes—
98#382—22no—
99#382—22no—
100#382—21no—
101#382—21no—
102#381—21no—
103#381D411yescharacteristic
104#381D411yescharacteristic
105#380—22no—
106#380—22no—
107#380C2×C412yes—
108#380D412yes—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4C2×C412yes—
119#376—22no—
120#376—22no—
121#376—21no—
122#376D412yes—
123Q8⋊C8—24no—
124Q8⋊C8—24no—
125Q8⋊C8—24no—
126Q8⋊C8—24no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—44no—
131C22×Q8C4212yes—
132C22×Q8C22⋊C411yescharacteristic
133C22×Q8C22⋊C411yescharacteristic
134C22×Q8C4⋊C412yes—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4C22×C411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C2×M16—22no—
141C2×M16—41no—
142C2×M16—41no—
143C2×M16—41no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—21no—
156C4×Q8—21no—
157C4×Q8—21no—
158C4×Q8—21no—
159C4×Q8—21no—
160C4×Q8—48no—
161C4×Q8C22⋊C412yes—
162C4×Q8C22⋊C412yes—
163C4×Q8C22⋊C412yes—
164C4×Q8C22⋊C412yes—
165C4×Q8C4⋊C412yes—
166C4×Q8C4⋊C412yes—
167C4×Q8C4⋊C412yes—
168C4×Q8C4⋊C412yes—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4C4212yes—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C4⋊C412yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C22⋊C412yes—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—24no—
200C2×C42—22no—
201C2×C42—24no—
202C2×C42—22no—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×Q811yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42#4111yescharacteristic
213C2×C42#4111yescharacteristic
214C2×C42#4111yescharacteristic
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—41no—
219C4⋊C8—41no—
220C4⋊C8—41no—
221C4⋊C8—41no—
222C4⋊C8—41no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231#97—22no—
232#97—22no—
233#97—41no—
234#97—41no—
235C4×C8—22no—
236C4×C8—22no—
237C4×C8—22no—
238C4×C8—41no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—22no—
244#95—21no—
245#95—21no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—21no—
250#95—22no—
251#95—21no—
252#95C22⋊C414yes—
253C24C2×C22⋊C411yescharacteristic
254C2×Q8—24no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×Q8#9512yes—
268C2×Q8#9512yes—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—81no—
284C22×C4—82no—
285C22×C4C2×C4211yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C4⋊C411yescharacteristic
289C22×C4#11711yescharacteristic
290C22×C4#11711yescharacteristic
291C22×C4C4×D411yescharacteristic
292C22×C4C4×D411yescharacteristic
293C22×C4C4×D411yescharacteristic
294C22×C4C4×D411yescharacteristic
295C22×C4#12011yescharacteristic
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4#12111yescharacteristic
301C22×C4C22⋊Q811yescharacteristic
302C22×C4C22⋊Q811yescharacteristic
303M16—82no—
304M16—82no—
305M16—82no—
306M16—82no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—48no—
324C4⋊C4—44no—
325C4⋊C4—82no—
326C4⋊C4—84no—
327C4⋊C4#9512yes—
328C4⋊C4#9512yes—
329C22⋊C4—44no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—48no—
341C42—44no—
342C42—82no—
343C42—84no—
344C42C2×C22⋊C411yescharacteristic
345C42C2×C22⋊C411yescharacteristic
346C42C2×C4⋊C411yescharacteristic
347C42C2×C4⋊C411yescharacteristic
348C42C4×D411yescharacteristic
349C42C4×D411yescharacteristic
350C42C4×D411yescharacteristic
351C42C4×D411yescharacteristic
352C42#12111yescharacteristic
353C42#12111yescharacteristic
354C42C22⋊Q811yescharacteristic
355C42C22⋊Q811yescharacteristic
356C42#12311yescharacteristic
357C42#12311yescharacteristic
358C42#12311yescharacteristic
359C42#12311yescharacteristic
360C23—22no—
361C23—41no—
362C23—41no—
363C23C4×C22⋊C411yescharacteristic
364C23#37911yescharacteristic
365C23#38011yescharacteristic
366Q8—24no—
367Q8—22no—
368Q8—22no—
369Q8—22no—
370Q8—44no—
371Q8—44no—
372Q8—42no—
373Q8#34414yes—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4C2×#9511yescharacteristic
414C2×C4C4×C22⋊C411yescharacteristic
415C2×C4#38011yescharacteristic
416C2×C4#38111yescharacteristic
417C2×C4#38511yescharacteristic
418C2×C4#38511yescharacteristic
419C2×C4#38511yescharacteristic
420C2×C4#38511yescharacteristic
421C2×C4#38611yescharacteristic
422C2×C4#38611yescharacteristic
423C2×C4#38611yescharacteristic
424C2×C4#38611yescharacteristic
425C8—82no—
426C8—82no—
427C8—82no—
428C8—82no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—44no—
438C22C22⋊#9511yescharacteristic
439C22#179612yes—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—24no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—42no—
453C4—42no—
454C4—42no—
455C4—82no—
456C4—84no—
457C4—82no—
458C4C2×#34412yes—
459C2—22no—
460C2—41no—
461C2—41no—
462C2#1073811yescharacteristic
463C2#1077711yescharacteristic
464C2#1078311yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ 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
27. #1491C4semidirect
33. #1479C4semidirect

Maximal subgroups

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

subgroupindex
#341922
#174492
#81162
#81062
#80702
#79762