Finite Group Databasebeta

← #117205

Subgroups — #117205

Order 512.

746 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21964C211yesmaximal, characteristic
2#21804C211yesmaximal, characteristic
3C2×#1997C211yesmaximal, characteristic
4#21202C211yesmaximal, characteristic
5C2×#1299C211yesmaximal, characteristic
6#8028C211yesmaximal, characteristic
7#7896C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C2×#419—22no—
12C2×C4×M16C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2285C2211yescharacteristic
15#2285C2211yescharacteristic
16#1997—22no—
17#1997C414yescomplemented
18#1981—22no—
19#1960—22no—
20#1956—21no—
21#1956—21no—
22#1956—21no—
23#1956—21no—
24#1948—21no—
25#1948—21no—
26#1947—21no—
27#1947—21no—
28#1809—22no—
29#1787—22no—
30C2×#343—22no—
31#1299—22no—
32#1295—22no—
33#1279—22no—
34C23×Q8D412yes—
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⋊C4—21no—
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—
68#419—48no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—24no—
73C2×C22⋊C8—21no—
74C4×M16—21no—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16C2×C412yes—
82C2×#97—21no—
83C2×#97D412yes—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87#384D411yescharacteristic
88#384D411yescharacteristic
89#384D411yescharacteristic
90#384D411yescharacteristic
91#382—21no—
92#382—21no—
93#380—21no—
94#380—21no—
95#380—21no—
96#380—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101#343—28no—
102#340—22no—
103#340—22no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—44no—
110C22×Q8—44no—
111C22×Q8C4⋊C414yes—
112C23×C4—22no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4#4111yescharacteristic
116C2×M16—24no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—42no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—41no—
127C2×M16—42no—
128C2×M16—42no—
129C2×M16C22⋊C414yes—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—41no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—41no—
143C4⋊Q8—41no—
144C4⋊Q8—41no—
145C4⋊Q8—41no—
146C4⋊Q8—41no—
147C4⋊Q8—41no—
148C4⋊Q8—41no—
149C4⋊Q8—41no—
150#117—44no—
151#117—82no—
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—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—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42C22×C411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×Q811yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192#107—82no—
193#106—82no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200#103—48no—
201#103—48no—
202#103—84no—
203#103—84no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206#97—22no—
207#97—22no—
208#97—22no—
209#97—41no—
210C4×C8—22no—
211C4×C8—41no—
212C4×C8—41no—
213C4×C8—41no—
214C4×C8—41no—
215C4×C8—41no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220C24#12111yescharacteristic
221C2×Q8—24no—
222C2×Q8—24no—
223C2×Q8—48no—
224C2×Q8—44no—
225C2×Q8—44no—
226C2×Q8—48no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—44no—
230C2×Q8—44no—
231C2×Q8—84no—
232C2×Q8—84no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C4⋊C411yescharacteristic
252C22×C4#11711yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12011yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4#12311yescharacteristic
262C22×C4#12311yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12411yescharacteristic
265C22×C4#12611yescharacteristic
266C22×C4#12611yescharacteristic
267C22×C4#12711yescharacteristic
268M16—24no—
269M16—44no—
270M16—44no—
271M16—42no—
272M16—42no—
273M16—42no—
274M16—44no—
275M16—82no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—81no—
311C4⋊C4—81no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C42—82no—
315C42—82no—
316C42C4×D411yescharacteristic
317C42C4×D411yescharacteristic
318C42C4×D411yescharacteristic
319C42C4×D411yescharacteristic
320C42#12111yescharacteristic
321C42#12111yescharacteristic
322C42#12111yescharacteristic
323C42#12111yescharacteristic
324C42C22⋊Q811yescharacteristic
325C42C22⋊Q811yescharacteristic
326C42C22⋊Q811yescharacteristic
327C42C22⋊Q811yescharacteristic
328C42#12311yescharacteristic
329C42#12311yescharacteristic
330C42#12411yescharacteristic
331C42#12411yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23#38011yescharacteristic
339C23#39211yescharacteristic
340C23#39711yescharacteristic
341Q8—48no—
342Q8—48no—
343Q8—84no—
344Q8—84no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—44no—
368C2×C4—84no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—84no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4#38511yescharacteristic
385C2×C4#38511yescharacteristic
386C2×C4#38611yescharacteristic
387C2×C4#38811yescharacteristic
388C2×C4#38811yescharacteristic
389C2×C4#39011yescharacteristic
390C2×C4#39311yescharacteristic
391C2×C4#39411yescharacteristic
392C2×C4#39411yescharacteristic
393C2×C4#39611yescharacteristic
394C2×C4#39611yescharacteristic
395C2×C4#39911yescharacteristic
396C2×C4#45312yes—
397C2×C4#45512yes—
398C2×C4#45511yescharacteristic
399C2×C4#45511yescharacteristic
400C2×C4#45611yescharacteristic
401C2×C4#45611yescharacteristic
402C2×C4#47112yes—
403C2×C4#47312yes—
404C8—44no—
405C8—44no—
406C8—44no—
407C8—82no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22#143911yescharacteristic
419C22#200211yescharacteristic
420C22#200411yescharacteristic
421C22#200711yescharacteristic
422C22#200711yescharacteristic
423C22#201911yescharacteristic
424C22#202011yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—84no—
432C4—161no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4#185612yes—
437C4#188511yescharacteristic
438C4#188611yescharacteristic
439C4#193012yes—
440C4#202811yescharacteristic
441C4#202811yescharacteristic
442C4#203712yes—
443C4#206412yes—
444C2—41no—
445C2—41no—
446C2#1010611yescharacteristic
447C2#1010711yescharacteristic
448C2#1011411yescharacteristic
449C2#1011511yescharacteristic
450C2#1105511yescharacteristic
451C2#1113511yescharacteristic
452C2#1114511yescharacteristic
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
17. #1997C4semidirect

Maximal subgroups

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

subgroupindex
#219642
#218042
C2×#19972
#212022
C2×#12992
#80282
#78962