Finite Group Databasebeta

← #117206

Subgroups — #117206

Order 512.

686 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21964C211yesmaximal, characteristic
2#21833C211yesmaximal, characteristic
3C2×#1997C211yesmaximal, characteristic
4#21324C211yesmaximal, characteristic
5C2×#1299C211yesmaximal, characteristic
6#8028C211yesmaximal, characteristic
7#7896C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2285C2211yescharacteristic
14#2285C2211yescharacteristic
15#1997—22no—
16#1997C414yescomplemented
17#1993—22no—
18#1990—21no—
19#1990—21no—
20#1990—21no—
21#1990—21no—
22#1988—22no—
23#1948—21no—
24#1948—21no—
25#1947—21no—
26#1947—21no—
27#1823—22no—
28#1809—22no—
29C2×#343—22no—
30#1736—22no—
31#1299—22no—
32#1295—22no—
33#1279—22no—
34C23×Q8—21no—
35C22×M16—21no—
36C22×M16—21no—
37C22×M16D412yes—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
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⋊Q8—21no—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8Q812yes—
50C2×#117—21no—
51C2×#117—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D412yes—
54C22×C42C2311yescharacteristic
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59C2×#107—22no—
60C2×#106—22no—
61#423—21no—
62#423—21no—
63#423C2×C412yes—
64#423D412yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8D412yes—
68C2×C22⋊C8—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16—21no—
72C4×M16—21no—
73C4×M16—21no—
74C4×M16—21no—
75C4×M16—21no—
76C4×M16C2×C412yes—
77C2×#97—21no—
78C2×#97D412yes—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82#384D411yescharacteristic
83#384D411yescharacteristic
84#384D411yescharacteristic
85#384D411yescharacteristic
86#382—21no—
87#382—21no—
88#380—21no—
89#380—21no—
90#380—21no—
91#380—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96#343—28no—
97#340—22no—
98#340—22no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—41no—
106C22×Q8—41no—
107C22×Q8—41no—
108C22×Q8—41no—
109C22×Q8—82no—
110C22×Q8—82no—
111C23×C4—22no—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4#4111yescharacteristic
115C2×M16—24no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—42no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—42no—
127C2×M16—42no—
128C2×M16C22⋊C414yes—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—41no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—41no—
142C4⋊Q8—41no—
143C4⋊Q8—41no—
144C4⋊Q8—41no—
145C4⋊Q8—41no—
146C4⋊Q8—41no—
147C4⋊Q8—41no—
148C4⋊Q8—41no—
149#117—44no—
150#117—82no—
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—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4C4⋊C414yes—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C42—41no—
178C2×C42—41no—
179C2×C42C22×C411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×Q811yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191#107—82no—
192#106—82no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201#97—22no—
202#97—22no—
203#97—22no—
204#97—41no—
205C4×C8—22no—
206C4×C8—41no—
207C4×C8—41no—
208C4×C8—41no—
209C4×C8—41no—
210C4×C8—41no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215C24#12111yescharacteristic
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—84no—
221C2×Q8—84no—
222C2×Q8—82no—
223C2×Q8—82no—
224C2×Q8—84no—
225C2×Q8—84no—
226C2×Q8—82no—
227C2×Q8—82no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C4⋊C411yescharacteristic
247C22×C4#11711yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4C22⋊Q811yescharacteristic
255C22×C4C22⋊Q811yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12311yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12411yescharacteristic
260C22×C4#12611yescharacteristic
261C22×C4#12611yescharacteristic
262C22×C4#12711yescharacteristic
263M16—24no—
264M16—44no—
265M16—44no—
266M16—42no—
267M16—42no—
268M16—42no—
269M16—44no—
270M16—82no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C42—82no—
310C42—82no—
311C42C4×D411yescharacteristic
312C42C4×D411yescharacteristic
313C42C4×D411yescharacteristic
314C42C4×D411yescharacteristic
315C42#12111yescharacteristic
316C42#12111yescharacteristic
317C42#12111yescharacteristic
318C42#12111yescharacteristic
319C42C22⋊Q811yescharacteristic
320C42C22⋊Q811yescharacteristic
321C42C22⋊Q811yescharacteristic
322C42C22⋊Q811yescharacteristic
323C42#12311yescharacteristic
324C42#12311yescharacteristic
325C42#12411yescharacteristic
326C42#12411yescharacteristic
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23#38011yescharacteristic
334C23#39211yescharacteristic
335C23#39711yescharacteristic
336Q8—84no—
337Q8—84no—
338Q8—84no—
339Q8—84no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—44no—
363C2×C4—84no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—84no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4#38511yescharacteristic
380C2×C4#38511yescharacteristic
381C2×C4#38611yescharacteristic
382C2×C4#38811yescharacteristic
383C2×C4#38811yescharacteristic
384C2×C4#39011yescharacteristic
385C2×C4#39311yescharacteristic
386C2×C4#39411yescharacteristic
387C2×C4#39411yescharacteristic
388C2×C4#39611yescharacteristic
389C2×C4#39611yescharacteristic
390C2×C4#39911yescharacteristic
391C2×C4#45412yes—
392C2×C4#45511yescharacteristic
393C2×C4#45511yescharacteristic
394C2×C4#45612yes—
395C2×C4#45611yescharacteristic
396C2×C4#45611yescharacteristic
397C2×C4#47112yes—
398C2×C4#47212yes—
399C8—44no—
400C8—44no—
401C8—44no—
402C8—82no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22#143911yescharacteristic
414C22#200311yescharacteristic
415C22#200511yescharacteristic
416C22#200711yescharacteristic
417C22#200711yescharacteristic
418C22#201811yescharacteristic
419C22#201911yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—84no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4#185712yes—
432C4#188511yescharacteristic
433C4#188611yescharacteristic
434C4#193112yes—
435C4#202811yescharacteristic
436C4#202811yescharacteristic
437C4#203812yes—
438C4#206312yes—
439C2—41no—
440C2—41no—
441C2#1011011yescharacteristic
442C2#1011111yescharacteristic
443C2#1011811yescharacteristic
444C2#1011911yescharacteristic
445C2#1105511yescharacteristic
446C2#1113611yescharacteristic
447C2#1114411yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
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
16. #1997C4semidirect

Maximal subgroups

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

subgroupindex
#219642
#218332
C2×#19972
#213242
C2×#12992
#80282
#78962