Finite Group Databasebeta

← C2×#9374 #200167

Subgroups — C2×#9374 #200167

Order 512.

972 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3237C211yesmaximal, characteristic
2C2×#2985C211yesmaximal, characteristic
3C2×#1552C211yesmaximal, characteristic
4C2×#1537C211yesmaximal, characteristic
5C2×#1481C211yesmaximal, characteristic
6#9374C218yesmaximal, direct product factor
7C2×#1290C211yesmaximal, characteristic
8C2×#1287C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#533C2211yescharacteristic
11#3237—22no—
12#3237—22no—
13#3237C414yes—
14#3237C2214yes—
15C2×#501C412yes—
16C2×#491C412yes—
17C2×#485—22no—
18C2×#477—21no—
19C2×#475—21no—
20#2985—22no—
21#2985—22no—
22#2985—22no—
23#2985C2214yes—
24C2×#446—22no—
25C2×#446—21no—
26C2×C8×Q8—21no—
27C2×#433—22no—
28C2×#432C2211yescharacteristic
29C2×#432C2211yescharacteristic
30C2×#432C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33#1552—22no—
34#1552C2214yes—
35C2×#332—21no—
36C2×#332—21no—
37#1537—22no—
38#1537C2214yes—
39C2×#330—22no—
40#1481—22no—
41#1481C2214yes—
42C2×Q8⋊C8—21no—
43C2×Q8⋊C8—21no—
44#1290C2214yes—
45#1287C2214yes—
46C23×C8—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533C2×C412yes—
58#533C2×C412yes—
59#533D412yes—
60#533D412yes—
61#533C2312yes—
62#533C2312yes—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×C4⋊Q8D411yescharacteristic
66C2×C4⋊Q8D411yescharacteristic
67C2×#125—21no—
68C2×#125C2×C412yes—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×#117—21no—
78C2×#117D412yes—
79C22×C42C2311yescharacteristic
80#501—24no—
81#501C2×C418yes—
82#491—24no—
83#491C2×C418yes—
84#485—28no—
85#485—28no—
86#477—24no—
87#477—24no—
88#475—24no—
89#475—24no—
90#446—28no—
91#446—24no—
92#446—28no—
93#446—24no—
94C8×Q8—24no—
95C8×Q8—24no—
96#433—28no—
97#433—28no—
98#432—21no—
99#432—21no—
100#432—21no—
101#432—21no—
102#432—21no—
103#432—21no—
104#432—21no—
105#432C2×C412yes—
106#432D412yes—
107#432C2312yes—
108#432C2312yes—
109#432C2312yes—
110#431—21no—
111#431—21no—
112#431D412yes—
113#431C2312yes—
114C2×#107—22no—
115C2×#106—22no—
116C2×C4⋊C8—22no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8C2312yes—
128C2×#103—22no—
129C2×#103—22no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×#97—22no—
135C2×#97—21no—
136C2×#97C2×C412yes—
137C2×#97D412yes—
138C2×C4×C8—22no—
139C2×C4×C8D411yescharacteristic
140C2×C4×C8D411yescharacteristic
141#332—24no—
142#332—24no—
143#330—28no—
144Q8⋊C8—24no—
145Q8⋊C8—24no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C23×C4C2×D411yescharacteristic
151C23×C4#4111yescharacteristic
152C23×C4#4111yescharacteristic
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—41no—
159C22×C8—41no—
160C22×C8—41no—
161C22×C8—82no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8C22⋊C412yes—
173C4⋊Q8C22⋊C412yes—
174C4⋊Q8C22×C412yes—
175C4⋊Q8C22×C412yes—
176C4⋊Q8C2×D412yes—
177C4⋊Q8C2×D412yes—
178#125—22no—
179#125—22no—
180#125—22no—
181#125—22no—
182#125—41no—
183#125—41no—
184#125C22⋊C414yes—
185#125C22×C414yes—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—41no—
199C22⋊Q8—41no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×Q8—41no—
218#117—22no—
219#117—22no—
220#117—22no—
221#117—22no—
222#117—41no—
223#117—41no—
224#117C22⋊C414yes—
225#117C2×D414yes—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4C22⋊C412yes—
237C2×C4⋊C4C22⋊C412yes—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—41no—
240C2×C42—22no—
241C2×C42—41no—
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42#4111yescharacteristic
252C2×C42#4111yescharacteristic
253C2×C42C2411yescharacteristic
254#107—28no—
255#106—28no—
256C4⋊C8—28no—
257C4⋊C8—28no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—22no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—41no—
268C4⋊C8—41no—
269C4⋊C8—41no—
270#103—28no—
271#103—28no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—22no—
276C22⋊C8—22no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—41no—
280#97—28no—
281#97—22no—
282#97—22no—
283#97—22no—
284#97—41no—
285#97C22×C414yes—
286#97C2×D414yes—
287C4×C8—28no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8C2×D412yes—
291C4×C8C2×D412yes—
292C24#12311yescharacteristic
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—44no—
300C2×Q8—44no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—82no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4C4×D411yescharacteristic
319C22×C4C4×D411yescharacteristic
320C22×C4#12011yescharacteristic
321C22×C4#12011yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12111yescharacteristic
326C22×C4#12311yescharacteristic
327C22×C4#12311yescharacteristic
328C22×C4#12311yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C2×C8—24no—
333C2×C8—22no—
334C2×C8—24no—
335C2×C8—22no—
336C2×C8—42no—
337C2×C8—44no—
338C2×C8—42no—
339C2×C8—44no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—82no—
343C2×C8—82no—
344C2×C8—82no—
345C2×C8—82no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4C2×C22⋊C414yes—
369C4⋊C4C2×C22⋊C414yes—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C42—24no—
375C42—24no—
376C42—42no—
377C42—42no—
378C42C2×C22⋊C411yescharacteristic
379C42C2×C22⋊C411yescharacteristic
380C42C4×D411yescharacteristic
381C42C4×D411yescharacteristic
382C42C4×D411yescharacteristic
383C42C4×D411yescharacteristic
384C42#12011yescharacteristic
385C42#12011yescharacteristic
386C42#12111yescharacteristic
387C42#12111yescharacteristic
388C42#12111yescharacteristic
389C42#12111yescharacteristic
390C42C23×C411yescharacteristic
391C42C22×D411yescharacteristic
392C42C22×D411yescharacteristic
393C42C22×D411yescharacteristic
394C23—22no—
395C23—22no—
396C23—22no—
397C23#38611yescharacteristic
398C23#38611yescharacteristic
399C23C2×#12311yescharacteristic
400Q8—44no—
401Q8—44no—
402Q8—44no—
403Q8—42no—
404Q8—42no—
405Q8—82no—
406Q8—82no—
407Q8—82no—
408Q8—82no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4#38611yescharacteristic
437C2×C4#38611yescharacteristic
438C2×C4#38611yescharacteristic
439C2×C4#38611yescharacteristic
440C2×C4#38611yescharacteristic
441C2×C4#38611yescharacteristic
442C2×C4#47112yes—
443C2×C4#47312yes—
444C2×C4C22×C22⋊C411yescharacteristic
445C2×C4C2×C4×D411yescharacteristic
446C2×C4C2×C4×D411yescharacteristic
447C2×C4C2×#12011yescharacteristic
448C2×C4C2×#12111yescharacteristic
449C2×C4C2×#12111yescharacteristic
450C8—42no—
451C8—44no—
452C8—42no—
453C8—44no—
454C8—82no—
455C8—82no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—44no—
463C22#188112yes—
464C22#190812yes—
465C22#191212yes—
466C22C2×#38611yescharacteristic
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—44no—
478C4—44no—
479C4—82no—
480C4—82no—
481C4#190812yes—
482C4C2×#47111yescharacteristic
483C4C2×#47311yescharacteristic
484C2—41no—
485C2—41no—
486C2#937414yesdirect product factor
487C2C2×#188111yescharacteristic
488C2C2×#190811yescharacteristic
489C2C2×#191211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#123
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9374C2direct (N × H)
486. C2#9374direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32372
C2×#29852
C2×#15522
C2×#15372
C2×#14812
#93742
C2×#12902
C2×#12872