Finite Group Databasebeta

← #117204

Subgroups — #117204

Order 512.

730 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21964C211yesmaximal, characteristic
2#21834C211yesmaximal, characteristic
3#21324C211yesmaximal, characteristic
4C2×#1966C211yesmaximal, characteristic
5C2×#1299C211yesmaximal, characteristic
6#8028C211yesmaximal, characteristic
7#7896C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C22×#101—22no—
12C2×C4×M16C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2285C2211yescharacteristic
15#2285C2211yescharacteristic
16#1993—22no—
17#1990—21no—
18#1990—21no—
19#1990—21no—
20#1990—21no—
21#1987—22no—
22#1966—22no—
23#1966C414yescomplemented
24#1948—21no—
25#1948—21no—
26#1947—21no—
27#1947—21no—
28#1824—22no—
29#1809—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×#101—28no—
69C2×#101—44no—
70C2×#101—44no—
71C2×C22⋊C8—21no—
72C4×M16—21no—
73C4×M16—21no—
74C4×M16—21no—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16C2×C412yes—
80C2×#97—21no—
81C2×#97D412yes—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85#384D411yescharacteristic
86#384D411yescharacteristic
87#384D411yescharacteristic
88#384D411yescharacteristic
89#382—21no—
90#382—21no—
91#380—21no—
92#380—21no—
93#380—21no—
94#380—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99#340—22no—
100#340—22no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—42no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8—42no—
111C22×Q8—41no—
112C22×Q8—41no—
113C22×Q8—41no—
114C22×Q8—41no—
115C23×C4—22no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4#4111yescharacteristic
119C2×M16—24no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—42no—
125C2×M16—41no—
126C2×M16—41no—
127C2×M16—41no—
128C2×M16—41no—
129C2×M16—41no—
130C2×M16—42no—
131C2×M16—42no—
132C2×M16C22⋊C414yes—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—41no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—41no—
146C4⋊Q8—41no—
147C4⋊Q8—41no—
148C4⋊Q8—41no—
149C4⋊Q8—41no—
150C4⋊Q8—41no—
151C4⋊Q8—41no—
152C4⋊Q8—41no—
153#117—44no—
154#117—82no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
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—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4C4⋊C414yes—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C42—41no—
182C2×C42—41no—
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×Q811yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195#107—82no—
196#106—82no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203#101—416no—
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—44no—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—44no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8—82no—
233C2×Q8—82no—
234C2×Q8—82no—
235C2×Q8—81no—
236C2×Q8—81no—
237C2×Q8—81no—
238C2×Q8—81no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C4⋊C411yescharacteristic
258C22×C4#11711yescharacteristic
259C22×C4#12011yescharacteristic
260C22×C4#12011yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4#12111yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4C22⋊Q811yescharacteristic
266C22×C4C22⋊Q811yescharacteristic
267C22×C4#12311yescharacteristic
268C22×C4#12311yescharacteristic
269C22×C4#12311yescharacteristic
270C22×C4#12411yescharacteristic
271C22×C4#12611yescharacteristic
272C22×C4#12611yescharacteristic
273C22×C4#12711yescharacteristic
274M16—24no—
275M16—44no—
276M16—44no—
277M16—42no—
278M16—42no—
279M16—42no—
280M16—44no—
281M16—82no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—81no—
317C4⋊C4—81no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C42—82no—
321C42—82no—
322C42C4×D411yescharacteristic
323C42C4×D411yescharacteristic
324C42C4×D411yescharacteristic
325C42C4×D411yescharacteristic
326C42#12111yescharacteristic
327C42#12111yescharacteristic
328C42#12111yescharacteristic
329C42#12111yescharacteristic
330C42C22⋊Q811yescharacteristic
331C42C22⋊Q811yescharacteristic
332C42C22⋊Q811yescharacteristic
333C42C22⋊Q811yescharacteristic
334C42#12311yescharacteristic
335C42#12311yescharacteristic
336C42#12411yescharacteristic
337C42#12411yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23#38011yescharacteristic
345C23#39211yescharacteristic
346C23#39711yescharacteristic
347Q8—84no—
348Q8—84no—
349Q8—82no—
350Q8—82no—
351Q8—82no—
352Q8—82no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—48no—
376C2×C4—44no—
377C2×C4—84no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4#38511yescharacteristic
393C2×C4#38511yescharacteristic
394C2×C4#38611yescharacteristic
395C2×C4#38811yescharacteristic
396C2×C4#38811yescharacteristic
397C2×C4#39011yescharacteristic
398C2×C4#39311yescharacteristic
399C2×C4#39411yescharacteristic
400C2×C4#39411yescharacteristic
401C2×C4#39611yescharacteristic
402C2×C4#39611yescharacteristic
403C2×C4#39911yescharacteristic
404C2×C4#45311yescharacteristic
405C2×C4#45311yescharacteristic
406C2×C4#45412yes—
407C2×C4#45411yescharacteristic
408C2×C4#45411yescharacteristic
409C2×C4#45612yes—
410C2×C4#47112yes—
411C2×C4#47312yes—
412C8—44no—
413C8—44no—
414C8—44no—
415C8—82no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22#143911yescharacteristic
427C22#200311yescharacteristic
428C22#200511yescharacteristic
429C22#200611yescharacteristic
430C22#200611yescharacteristic
431C22#201911yescharacteristic
432C22#202011yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—84no—
440C4—161no—
441C4—161no—
442C4—161no—
443C4—161no—
444C4#185712yes—
445C4#188711yescharacteristic
446C4#188811yescharacteristic
447C4#193012yes—
448C4#202711yescharacteristic
449C4#202711yescharacteristic
450C4#203812yes—
451C4#206412yes—
452C2—41no—
453C2—41no—
454C2#1010811yescharacteristic
455C2#1010911yescharacteristic
456C2#1011611yescharacteristic
457C2#1011711yescharacteristic
458C2#1105411yescharacteristic
459C2#1113611yescharacteristic
460C2#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
23. #1966C4semidirect

Maximal subgroups

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

subgroupindex
#219642
#218342
#213242
C2×#19662
C2×#12992
#80282
#78962