Finite Group Databasebeta

← C2×#34336 #7625618

Subgroups — C2×#34336 #7625618

Order 512.

7198 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2#34336C2132yesmaximal, direct product factor
3C2×#2370C213yesmaximal
4C2×#2370C213yesmaximal
5C2×#2369C213yesmaximal
6C2×#2355C213yesmaximal
7C2×#2351C211yesmaximal, characteristic
8C2×#2351C211yesmaximal, characteristic
9C2×Q8×C4⋊C4C211yesmaximal, characteristic
10C2×#2331C213yesmaximal
11C2×#2330C213yesmaximal
12C2×C4×#125C213yesmaximal
13C2×#2295C213yesmaximal
14C2×#2276C211yesmaximal, characteristic
15C2×#2276C211yesmaximal, characteristic
16C2×C42×Q8C211yesmaximal, characteristic
17C2×#557C4132yescomplemented
18C2×#557C22116yes—
19C22×C4⋊Q8C2213yes—
20C22×#125C2213yes—
21C22×#125C2213yes—
22C22×C4×Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2211yescharacteristic
28C22×C4×Q8C2213yes—
29C22×C4×Q8C2211yescharacteristic
30#2370C22148yes—
31#2370C22148yes—
32#2369C22148yes—
33#2355C22148yes—
34#2351C22116yes—
35#2351C22116yes—
36Q8×C4⋊C4C22116yes—
37#2331C22148yes—
38#2330C22148yes—
39C4×#125C22148yes—
40#2295C22148yes—
41C2×#391C2213yes—
42C2×#391C2216yes—
43C2×#391C2213yes—
44C2×#389C2213yes—
45C2×#389C2213yes—
46C2×#389C2213yes—
47C2×#389C2216yes—
48C2×#389C2213yes—
49C2×#389C2213yes—
50C2×#389C2213yes—
51C2×#389C2213yes—
52C2×#389C2213yes—
53C2×#387C2216yes—
54C2×#387C2216yes—
55C2×#387C2216yes—
56C2×#387C2213yes—
57C2×#387C2213yes—
58C2×#387C2211yescharacteristic
59C2×#387C2213yes—
60C2×#384C2213yes—
61C2×#382C2213yes—
62C2×#382C2213yes—
63#2276C22116yes—
64#2276C22116yes—
65C42×Q8C22116yes—
66C2×C4×C4⋊C4C2213yes—
67C2×C4×C4⋊C4C2213yes—
68C2×C4×C4⋊C4C2213yes—
69C2×C4×C4⋊C4C2213yes—
70C2×C4×C4⋊C4C2213yes—
71C2×C4×C4⋊C4C2213yes—
72C2×C4×C4⋊C4C2213yes—
73C2×C4×C4⋊C4C2213yes—
74C2×C4×C4⋊C4C2213yes—
75C2×C4×C4⋊C4C2216yes—
76C2×C4×C4⋊C4C2213yes—
77C2×C4×C4⋊C4C2213yes—
78C2×C4×C4⋊C4C2216yes—
79C2×C4×C4⋊C4C2213yes—
80C2×#376C2213yes—
81C2×#376C2213yes—
82C2×C43C2213yes—
83C23×Q8C2313yes—
84C23×Q8C2311yescharacteristic
85#557C2×C41256yescomplemented
86C2×C4⋊Q8C2×C4148yes—
87C2×C4⋊Q8C23124yes—
88C2×#125C2×C4148yes—
89C2×#125C2×C4148yes—
90C2×#125C23124yes—
91C2×#125C23124yes—
92C2×C4×Q8—28no—
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8C2×C4116yes—
96C2×C4×Q8C2×C4116yes—
97C2×C4×Q8C2×C4148yes—
98C2×C4×Q8C2×C4116yes—
99C2×C4×Q8Q8116yescomplemented
100C2×C4×Q8C2318yes—
101C2×C4×Q8C2318yes—
102C2×C4×Q8C2318yes—
103C2×C4×Q8C2318yes—
104C2×C4×Q8C2318yes—
105C2×C4×Q8C2318yes—
106C2×C4×Q8C23124yes—
107C2×C4×Q8C2318yes—
108C22×C4⋊C4C2313yes—
109C22×C4⋊C4C2313yes—
110C22×C4⋊C4C2313yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2313yes—
113C22×C4⋊C4C2313yes—
114C22×C4⋊C4C2316yes—
115C22×C4⋊C4C2313yes—
116C22×C4⋊C4C2313yes—
117C22×C4⋊C4C2313yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2313yes—
120C22×C4⋊C4C2313yes—
121C22×C4⋊C4C2313yes—
122C22×C4⋊C4C2313yes—
123C22×C4⋊C4C2313yes—
124C22×C4⋊C4C2316yes—
125C22×C4⋊C4C2313yes—
126C22×C42C2313yes—
127C22×C42C2313yes—
128C22×C42C2313yes—
129C22×C42C2311yescharacteristic
130C22×C42C2313yes—
131C22×C42C2313yes—
132C22×C42C2313yes—
133C22×C42C2316yes—
134C22×C42C2316yes—
135C22×C42C2313yes—
136C22×C42C2313yes—
137C22×C42C2313yes—
138C22×C42C2313yes—
139#391C23124yes—
140#391C23148yes—
141#391C23124yes—
142#389C23124yes—
143#389C23124yes—
144#389C23124yes—
145#389C23124yes—
146#389C23124yes—
147#389C23148yes—
148#389C23124yes—
149#389C23124yes—
150#389C23124yes—
151#387C23124yes—
152#387C23148yes—
153#387C23148yes—
154#387C23148yes—
155#387C23124yes—
156#387C23124yes—
157#387C2318yes—
158#384C23124yes—
159#382C23124yes—
160#382C23124yes—
161C4×C4⋊C4C23124yes—
162C4×C4⋊C4C23124yes—
163C4×C4⋊C4C23148yes—
164C4×C4⋊C4C23148yes—
165C4×C4⋊C4C23124yes—
166C4×C4⋊C4C23124yes—
167C4×C4⋊C4C23124yes—
168C4×C4⋊C4C23124yes—
169C4×C4⋊C4C23124yes—
170C4×C4⋊C4C23124yes—
171C4×C4⋊C4C23124yes—
172C4×C4⋊C4C23124yes—
173C4×C4⋊C4C23124yes—
174C4×C4⋊C4C23124yes—
175#376C23124yes—
176#376C23124yes—
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2313yes—
180C2×#95C2313yes—
181C2×#95C2313yes—
182C2×#95C2313yes—
183C2×#95C2313yes—
184C2×#95C2313yes—
185C2×#95C2313yes—
186C2×#95C2311yescharacteristic
187C2×#95C2316yes—
188C2×#95C2313yes—
189C2×#95C2311yescharacteristic
190C2×#95C2313yes—
191C2×#95C2316yes—
192C2×#95C2316yes—
193C2×#95C2313yes—
194C43C23124yes—
195C22×Q8—46no—
196C22×Q8—46no—
197C22×Q8C22×C4124yes—
198C22×Q8C22×C418yes—
199C22×Q8C2×Q818yes—
200C22×Q8#4118yes—
201C22×Q8C24112yes—
202C22×Q8C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2413yes—
205C23×C4C2413yes—
206C23×C4C2411yescharacteristic
207C23×C4C2413yes—
208C23×C4C2413yes—
209C23×C4C2411yescharacteristic
210C23×C4C2413yes—
211C23×C4C2413yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2413yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2413yes—
218C4⋊Q8C22×C41192yes—
219#125C22×C41192yes—
220#125C22×C41192yes—
221C4×Q8—232no—
222C4×Q8—232no—
223C4×Q8—232no—
224C4×Q8C22×C4164yes—
225C4×Q8C22×C4164yes—
226C4×Q8C22×C41192yes—
227C4×Q8C22×C4164yes—
228C4×Q8C2×Q8164yescomplemented
229C2×C4⋊C4—212no—
230C2×C4⋊C4—212no—
231C2×C4⋊C4—212no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—212no—
234C2×C4⋊C4—46no—
235C2×C4⋊C4—46no—
236C2×C4⋊C4—46no—
237C2×C4⋊C4—46no—
238C2×C4⋊C4—412no—
239C2×C4⋊C4C22×C418yes—
240C2×C4⋊C4C22×C4124yes—
241C2×C4⋊C4C22×C4124yes—
242C2×C4⋊C4C22×C4124yes—
243C2×C4⋊C4C22×C4124yes—
244C2×C4⋊C4C22×C4124yes—
245C2×C4⋊C4C22×C4148yes—
246C2×C4⋊C4C2×Q8124yes—
247C2×C4⋊C4C24112yes—
248C2×C4⋊C4C24112yes—
249C2×C4⋊C4C24112yes—
250C2×C4⋊C4C24112yes—
251C2×C4⋊C4C2414yes—
252C2×C4⋊C4C24112yes—
253C2×C4⋊C4C24124yes—
254C2×C4⋊C4C24112yes—
255C2×C4⋊C4C24112yes—
256C2×C4⋊C4C24112yes—
257C2×C4⋊C4C2414yes—
258C2×C4⋊C4C24112yes—
259C2×C4⋊C4C24112yes—
260C2×C4⋊C4C24112yes—
261C2×C4⋊C4C24112yes—
262C2×C4⋊C4C24112yes—
263C2×C4⋊C4C24124yes—
264C2×C4⋊C4C24112yes—
265C2×C42—212no—
266C2×C42—212no—
267C2×C42—212no—
268C2×C42—24no—
269C2×C42—46no—
270C2×C42—46no—
271C2×C42—412no—
272C2×C42—412no—
273C2×C42—46no—
274C2×C42C22×C4124yes—
275C2×C42C22×C4124yes—
276C2×C42C22×C4124yes—
277C2×C42C2×Q8124yes—
278C2×C42C24112yes—
279C2×C42C24112yes—
280C2×C42C24112yes—
281C2×C42C2414yes—
282C2×C42C24112yes—
283C2×C42C24112yes—
284C2×C42C24112yes—
285C2×C42C24124yes—
286C2×C42C24124yes—
287C2×C42C24112yes—
288C2×C42C24112yes—
289C2×C42C24112yes—
290C2×C42C24112yes—
291#95C2414yes—
292#95C2414yes—
293#95C24112yes—
294#95C24112yes—
295#95C24112yes—
296#95C24112yes—
297#95C24112yes—
298#95C24112yes—
299#95C24112yes—
300#95C2414yes—
301#95C24124yes—
302#95C24112yes—
303#95C2414yes—
304#95C24112yes—
305#95C24124yes—
306#95C24124yes—
307#95C24112yes—
308C24C2511yescharacteristic
309C2×Q8—448no—
310C2×Q8—412no—
311C2×Q8—412no—
312C2×Q8C4×Q8164yes—
313C2×Q8C23×C4148yes—
314C2×Q8C23×C4116yes—
315C2×Q8C22×Q8116yes—
316C2×Q8C2×#41116yes—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—26no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—26no—
325C22×C4—26no—
326C22×C4—22no—
327C22×C4—26no—
328C22×C4—26no—
329C22×C4—26no—
330C22×C4—26no—
331C22×C4—22no—
332C22×C4—26no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—26no—
338C22×C4—26no—
339C22×C4—26no—
340C22×C4—22no—
341C22×C4—26no—
342C22×C4—26no—
343C22×C4—26no—
344C22×C4—26no—
345C22×C4—26no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—26no—
349C22×C4—22no—
350C22×C4C23×C414yes—
351C22×C4C23×C414yes—
352C22×C4C23×C414yes—
353C22×C4C23×C4112yes—
354C22×C4C23×C4112yes—
355C22×C4C23×C4112yes—
356C22×C4C23×C4112yes—
357C22×C4C22×Q814yes—
358C22×C4C22×Q8112yes—
359C22×C4C22×Q8112yes—
360C22×C4C2×#41112yes—
361C22×C42−1+414yes—
362C22×C4C2512yes—
363C22×C4C2516yes—
364C22×C4C2512yes—
365C22×C4C2516yes—
366C22×C4C2516yes—
367C22×C4C2512yes—
368C22×C4C2512yes—
369C22×C4C2516yes—
370C22×C4C2512yes—
371C22×C4C2516yes—
372C22×C4C2516yes—
373C22×C4C2512yes—
374C22×C4C2516yes—
375C22×C4C2512yes—
376C22×C4C2516yes—
377C4⋊C4—224no—
378C4⋊C4—224no—
379C4⋊C4—224no—
380C4⋊C4—224no—
381C4⋊C4—28no—
382C4⋊C4—412no—
383C4⋊C4—412no—
384C4⋊C4—412no—
385C4⋊C4—412no—
386C4⋊C4—424no—
387C4⋊C4C23×C4116yes—
388C4⋊C4C23×C4148yes—
389C4⋊C4C23×C4148yes—
390C4⋊C4C23×C4148yes—
391C4⋊C4C23×C4148yes—
392C4⋊C4C23×C4148yes—
393C4⋊C4C23×C4196yes—
394C4⋊C4C22×Q8148yes—
395C42—224no—
396C42—224no—
397C42—224no—
398C42—28no—
399C42—412no—
400C42—412no—
401C42—424no—
402C42—424no—
403C42—412no—
404C42C23×C4148yes—
405C42C23×C4148yes—
406C42C23×C4148yes—
407C42C22×Q8148yes—
408C23C24×C412yes—
409C23C23×Q812yes—
410C23C22×#4112yes—
411C23C2×2−1+412yes—
412C23#58512yes—
413C23#58512yes—
414C23#58512yes—
415C23C2611yescharacteristic
416Q8—448no—
417Q8C2×C4×Q8164yes—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—26no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—26no—
426C2×C4—26no—
427C2×C4—22no—
428C2×C4—26no—
429C2×C4—26no—
430C2×C4—26no—
431C2×C4—26no—
432C2×C4—22no—
433C2×C4—26no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—224no—
437C2×C4—224no—
438C2×C4—28no—
439C2×C4—28no—
440C2×C4—224no—
441C2×C4—28no—
442C2×C4—28no—
443C2×C4—224no—
444C2×C4—28no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—26no—
448C2×C4—26no—
449C2×C4—26no—
450C2×C4—22no—
451C2×C4—26no—
452C2×C4—26no—
453C2×C4—26no—
454C2×C4—26no—
455C2×C4—26no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—26no—
459C2×C4—22no—
460C2×C4—44no—
461C2×C4—412no—
462C2×C4—44no—
463C2×C4—412no—
464C2×C4—412no—
465C2×C4C2×C4×Q8148yes—
466C2×C4C24×C414yes—
467C2×C4C24×C414yes—
468C2×C4C24×C414yes—
469C2×C4C24×C4112yes—
470C2×C4C24×C4112yes—
471C2×C4C24×C4112yes—
472C2×C4C24×C4112yes—
473C2×C4C23×Q814yes—
474C2×C4C23×Q8112yes—
475C2×C4C23×Q8112yes—
476C2×C4C22×#41112yes—
477C2×C4C2×2−1+414yes—
478C22C22×C4×Q814yes—
479C22#342814yes—
480C22C4×2−1+414yes—
481C22#350214yes—
482C22#350414yes—
483C22#351614yes—
484C22#352414yes—
485C22C25×C411yescharacteristic
486C22C24×Q811yescharacteristic
487C22C23×#4111yescharacteristic
488C22C22×2−1+411yescharacteristic
489C22C2×#58511yescharacteristic
490C22C2×#58511yescharacteristic
491C22C2×#58511yescharacteristic
492C4—24no—
493C4—24no—
494C4—212no—
495C4—212no—
496C4—212no—
497C4—24no—
498C4—24no—
499C4—212no—
500C4—24no—
501C4—42no—
502C4—46no—
503C4—42no—
504C4—46no—
505C4—46no—
506C4C22×C4×Q8124yes—
507C2#3433618yesdirect product factor
508C2C23×C4×Q811yescharacteristic
509C2C2×#342811yescharacteristic
510C2C2×C4×2−1+411yescharacteristic
511C2C2×#350211yescharacteristic
512C2C2×#351611yescharacteristic
513C2C2×#352411yescharacteristic
514C2C2×#350411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C25
commutatorC22C25×C4
Frattini, cosocleC23C26
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34336C2direct (N × H)
17. C2×#557C4semidirect
85. #557C2×C4semidirect
99. C2×C4×Q8Q8semidirect
228. C4×Q8C2×Q8semidirect
507. C2#34336direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5572
#343362
C2×#23702
C2×#23692
C2×#23552
C2×#23512
C2×Q8×C4⋊C42
C2×#23312
C2×#23302
C2×C4×#1252
C2×#22952
C2×#22762
C2×C42×Q82