Finite Group Databasebeta

← #117183

Subgroups — #117183

Order 512.

790 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21961C211yesmaximal, characteristic
2#21833C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic
4#21202C211yesmaximal, characteristic
5#10851C211yesmaximal, characteristic
6#8014C211yesmaximal, characteristic
7C2×#1283C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C22×C4⋊C8C2211yescharacteristic
11C2×#419—22no—
12C2×C4×M16C2211yescharacteristic
13#2285C2211yescharacteristic
14#2285C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1990—22no—
17#1988—22no—
18#1988—22no—
19#1988C414yescomplemented
20#1960—22no—
21#1956—22no—
22#1947—22no—
23#1947—22no—
24#1823—22no—
25#1808—22no—
26C2×#340—22no—
27C2×#339—22no—
28#1294—22no—
29#1283—22no—
30C23×Q8—21no—
31C22×M16D412yes—
32C23×C8D412yes—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8D412yes—
44C2×C4⋊Q8Q812yes—
45C2×#117—21no—
46C2×#117—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C42—21no—
51C22×C42C2311yescharacteristic
52#431—21no—
53#431—21no—
54#431—21no—
55#431—21no—
56C2×#107—24no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2×C412yes—
63C2×C4⋊C8D412yes—
64#419—44no—
65#419—44no—
66C2×#103—28no—
67C2×C22⋊C8D412yes—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16—21no—
72C2×#97—21no—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75#384D412yes—
76#384D412yes—
77#382—21no—
78#382—21no—
79#380—21no—
80#380—21no—
81#380—21no—
82#380—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4C2×C412yes—
91C2×#95—21no—
92#340—24no—
93#340—28no—
94#340—28no—
95#339—44no—
96C22×Q8—22no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—41no—
100C22×Q8—41no—
101C22×Q8—41no—
102C22×Q8—42no—
103C22×Q8—42no—
104C22×Q8—42no—
105C22×Q8—42no—
106C23×C4—41no—
107C23×C4—41no—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16D814yes—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—44no—
122C22×C8C22⋊C414yes—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—41no—
133C4⋊Q8—41no—
134C4⋊Q8—41no—
135C4⋊Q8—41no—
136C4⋊Q8—41no—
137C4⋊Q8—41no—
138C4⋊Q8C4⋊C414yes—
139#117—42no—
140#117—42no—
141#117—42no—
142#117—42no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
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—22no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C42—22no—
168C2×C42—42no—
169C2×C42—42no—
170C2×C42—41no—
171C2×C42—41no—
172C2×C42—41no—
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×Q811yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185#107—84no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—41no—
191C4⋊C8—41no—
192#103—416no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195#97—41no—
196#97—41no—
197C4×C8—41no—
198C4×C8—41no—
199C4×C8—41no—
200C4×C8—41no—
201#95—22no—
202#95—41no—
203#95—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207C24#12711yescharacteristic
208C2×Q8—24no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—44no—
213C2×Q8—44no—
214C2×Q8—44no—
215C2×Q8—44no—
216C2×Q8—44no—
217C2×Q8—44no—
218C2×Q8—82no—
219C2×Q8—82no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×Q8—82no—
223C2×Q8—82no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—82no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—81no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4#12011yescharacteristic
245C22×C4#12011yescharacteristic
246C22×C4#12011yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4C22⋊Q811yescharacteristic
253C22×C4C22⋊Q811yescharacteristic
254C22×C4C22⋊Q811yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12311yescharacteristic
258C22×C4#12411yescharacteristic
259C22×C4C2×D812yes—
260C22×C4#13512yes—
261C22×C4AGL(1,ℤ/8)12yes—
262C22×C4#13712yes—
263M16—24no—
264M16—24no—
265M16—42no—
266M16—42no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—44no—
272C2×C8—44no—
273C2×C8—44no—
274C2×C8—42no—
275C2×C8—82no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—82no—
295C22⋊C4—81no—
296C22⋊C4—81no—
297C22⋊C4—81no—
298C22⋊C4—81no—
299C42—44no—
300C42—82no—
301C42—82no—
302C42—82no—
303C42C2×C4⋊C411yescharacteristic
304C42#11711yescharacteristic
305C42C4×D411yescharacteristic
306C42C4×D411yescharacteristic
307C42#12111yescharacteristic
308C42#12111yescharacteristic
309C42#12111yescharacteristic
310C42C22⋊Q811yescharacteristic
311C42C22⋊Q811yescharacteristic
312C42C22⋊Q811yescharacteristic
313C42#12311yescharacteristic
314C42#12311yescharacteristic
315C42#12411yescharacteristic
316C42#12411yescharacteristic
317C42#12611yescharacteristic
318C42#12611yescharacteristic
319C23—22no—
320C23—22no—
321C23#39011yescharacteristic
322C23#39211yescharacteristic
323C23#39911yescharacteristic
324C23#49311yescharacteristic
325C23#49511yescharacteristic
326C23#49611yescharacteristic
327C23#49711yescharacteristic
328Q8—48no—
329Q8—84no—
330Q8—84no—
331Q8—84no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—82no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4#38011yescharacteristic
364C2×C4#38511yescharacteristic
365C2×C4#38511yescharacteristic
366C2×C4#38611yescharacteristic
367C2×C4#38811yescharacteristic
368C2×C4#38811yescharacteristic
369C2×C4#39311yescharacteristic
370C2×C4#39411yescharacteristic
371C2×C4#39411yescharacteristic
372C2×C4#39611yescharacteristic
373C2×C4#39611yescharacteristic
374C2×C4#39711yescharacteristic
375C2×C4C4×D812yes—
376C2×C4#44012yes—
377C2×C4#44712yes—
378C2×C4#45212yes—
379C2×C4#45512yes—
380C2×C4#45612yes—
381C2×C4#45912yes—
382C2×C4#46412yes—
383C2×C4#46612yes—
384C2×C4#46812yes—
385C2×C4#47412yes—
386C2×C4#47812yes—
387C2×C4#48012yes—
388C2×C4#48212yes—
389C8—44no—
390C8—44no—
391C8—82no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22#143911yescharacteristic
401C22#197011yescharacteristic
402C22#197311yescharacteristic
403C22#200712yes—
404C22#200912yes—
405C22#201111yescharacteristic
406C22#201511yescharacteristic
407C22#209211yescharacteristic
408C22#209311yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—84no—
416C4—84no—
417C4—162no—
418C4—162no—
419C4#186212yes—
420C4#188511yescharacteristic
421C4#188611yescharacteristic
422C4#192512yes—
423C4#202812yes—
424C4#203312yes—
425C4#206712yes—
426C2—21no—
427C2—21no—
428C2—21no—
429C2—21no—
430C2#1020412yes—
431C2#1020712yes—
432C2#1105511yescharacteristic
433C2#1112311yescharacteristic
434C2#1113011yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #1988C4semidirect

Maximal subgroups

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

subgroupindex
#219612
#218332
C2×#19882
#212022
#108512
#80142
C2×#12832