Finite Group Databasebeta

← C2×#71820 #10948017

Subgroups — C2×#71820 #10948017

Order 768.

1306 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×#5237C212yesmaximal, complemented
3#71820C218yesmaximal, direct product factor
4C2×#4548C212yesmaximal, complemented
5C2×#4542C212yesmaximal, complemented
6C2×#2132—31nomaximal, Sylow
7C22×C3×#41C411yescharacteristic
8C22×C3×#41C2211yescharacteristic
9C3×#532—24no—
10C3×#532C414yes—
11C3×#532C2214yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×#127C411yescharacteristic
14C2×C3×#124C411yescharacteristic
15C2×C3×#124C2211yescharacteristic
16C2×C3×#121—22no—
17C2×C3×C4×D4—22no—
18C22×C3×C42C411yescharacteristic
19C22×#208—22no—
20#5237—24no—
21#5237C2218yescomplemented
22C2×#836C2214yes—
23C2×#833—22no—
24#4548C2218yescomplemented
25#4542C2218yescomplemented
26C2×#532D311yescharacteristic
27C2×#413—32no—
28#2132—38no—
29C2×#356—32no—
30C2×#355—32no—
31C2×C3×#41—22no—
32C2×C3×#41—41no—
33C2×C3×#41—41no—
34C2×C3×#41—42no—
35C2×C3×#41—42no—
36C2×C3×#41—41no—
37C2×C3×#41—41no—
38C2×C3×#41C2×C414yes—
39C2×C3×#41C2×C412yes—
40C2×C3×#41D414yes—
41C2×C3×#41C2312yes—
42C22×C3×Q8C2×C411yescharacteristic
43C22×C3×Q8C2311yescharacteristic
44C22×C3×D4—21no—
45C22×C3×D4—22no—
46C22×C3×D4C2×C411yescharacteristic
47C22×C3×D4C2×C411yescharacteristic
48C22×C3×D4D412yes—
49C23×C3×C4—21no—
50C23×C3×C4—21no—
51C23×C3×C4C2×C411yescharacteristic
52C23×C3×C4D412yes—
53C3×C4⋊Q8—22no—
54C3×C4⋊Q8C2×C412yes—
55C3×C4⋊Q8C2312yes—
56C3×#127—22no—
57C3×#127C2×C414yes—
58C3×#124—22no—
59C3×#124—22no—
60C3×#124C2×C414yes—
61C3×#124C2×C412yes—
62C3×#124C2312yes—
63C3×#121—28no—
64C3×#121—28no—
65C3×C4×D4—28no—
66C3×C4×D4—28no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C42—21no—
71C2×C3×C42—22no—
72C2×C3×C42C2×C411yescharacteristic
73C2×C3×C42C2×C411yescharacteristic
74C2×C3×C42C2×C414yes—
75C2×#208—24no—
76C2×#208—28no—
77C2×#208—42no—
78C2×#208—44no—
79C22×C3⋊C8—42no—
80#836—24no—
81#836C2318yes—
82#833—28no—
83C22×#41Dic311yescharacteristic
84C22×#41C2×D311yescharacteristic
85C22×M16—62no—
86#532—24no—
87#532Dic314yes—
88#532C2×D314yes—
89C2×C4⋊Q8C2×D311yescharacteristic
90C2×#127Dic311yescharacteristic
91C2×#124Dic311yescharacteristic
92C2×#124C2×D311yescharacteristic
93C2×#121—22no—
94C2×C4×D4—22no—
95C22×C42Dic311yescharacteristic
96#413—38no—
97#413—64no—
98C2×#101—34no—
99C2×#100—62no—
100#356—38no—
101#355—38no—
102C24×C3—21no—
103C24×C3C22⋊C412yes—
104C24×C3C22⋊C411yescharacteristic
105C3×#41—42no—
106C3×#41—42no—
107C3×#41—44no—
108C3×#41—44no—
109C3×#41—42no—
110C3×#41—42no—
111C3×#41—44no—
112C3×#41—44no—
113C3×#41—44no—
114C3×#41—44no—
115C2×C3×Q8—21no—
116C2×C3×Q8—41no—
117C2×C3×Q8—41no—
118C2×C3×Q8—41no—
119C2×C3×Q8—41no—
120C2×C3×Q8C22×C412yes—
121C2×C3×Q8C22×C411yescharacteristic
122C2×C3×Q8C2×D412yes—
123C2×C3×Q8C2411yescharacteristic
124C2×C3×D4—28no—
125C2×C3×D4—24no—
126C2×C3×D4—21no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—42no—
130C2×C3×D4—42no—
131C2×C3×D4—41no—
132C2×C3×D4—41no—
133C2×C3×D4—48no—
134C2×C3×D4—42no—
135C2×C3×D4—42no—
136C2×C3×D4C22⋊C414yes—
137C2×C3×D4C22⋊C412yes—
138C2×C3×D4C22×C412yes—
139C2×C3×D4C22×C412yes—
140C2×C3×D4C2×D414yes—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—24no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—24no—
150C22×C3×C4—21no—
151C22×C3×C4—24no—
152C22×C3×C4—44no—
153C22×C3×C4—42no—
154C22×C3×C4—41no—
155C22×C3×C4—41no—
156C22×C3×C4C22⋊C412yes—
157C22×C3×C4C22⋊C414yes—
158C22×C3×C4C22⋊C411yescharacteristic
159C22×C3×C4C22×C411yescharacteristic
160C22×C3×C4C22×C412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D414yes—
163C3×C4⋊C4—24no—
164C3×C4⋊C4—24no—
165C3×C22⋊C4—28no—
166C3×C22⋊C4—24no—
167C3×C22⋊C4—24no—
168C3×C42—24no—
169C3×C42—21no—
170C3×C42—21no—
171C3×C42C22×C412yes—
172C3×C42C22×C412yes—
173#208—28no—
174#208—44no—
175#208—48no—
176C2×C3⋊C8—44no—
177C2×C3⋊C8—48no—
178C2×#41—22no—
179C2×#41—41no—
180C2×#41—42no—
181C2×#41—41no—
182C2×#41—42no—
183C2×#41—41no—
184C2×#41—41no—
185C2×#41C2×Dic314yes—
186C2×#41C2×Dic312yes—
187C2×#41#6714yes—
188C2×#41C22×D312yes—
189C22×Q8C2×Dic311yescharacteristic
190C22×Q8C22×D311yescharacteristic
191C22×D4—22no—
192C22×D4—21no—
193C22×D4C2×Dic311yescharacteristic
194C22×D4C2×Dic311yescharacteristic
195C22×D4#6712yes—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4C2×Dic311yescharacteristic
199C23×C4#6712yes—
200C2×M16—64no—
201C2×M16—68no—
202C2×M16—122no—
203C2×M16—124no—
204C22×C8—122no—
205C4⋊Q8—22no—
206C4⋊Q8C2×Dic312yes—
207C4⋊Q8C22×D312yes—
208#127—22no—
209#127C2×Dic314yes—
210#124—22no—
211#124—22no—
212#124C2×Dic314yes—
213#124C2×Dic312yes—
214#124C22×D312yes—
215#121—28no—
216#121—28no—
217C4×D4—28no—
218C4×D4—28no—
219C2×C4⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42C2×Dic311yescharacteristic
225C2×C42C2×Dic311yescharacteristic
226C2×C42C2×Dic314yes—
227#101—38no—
228#101—64no—
229#100—68no—
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—22no—
233C23×C3—22no—
234C23×C3—22no—
235C23×C3—42no—
236C23×C3—44no—
237C23×C3—44no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3C2×C22⋊C411yescharacteristic
241C23×C3C2×C22⋊C414yes—
242C23×C3C2×C22⋊C412yes—
243C3×Q8—41no—
244C3×Q8—42no—
245C3×Q8—41no—
246C3×Q8—42no—
247C3×Q8—42no—
248C3×D4—42no—
249C3×D4—44no—
250C3×D4—42no—
251C3×D4—44no—
252C3×D4—42no—
253C3×D4—416no—
254C3×D4—42no—
255C3×D4—48no—
256C2×C3×C4—24no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—48no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—44no—
277C2×C3×C4—44no—
278C2×C3×C4#9912yes—
279C2×C3×C4#9911yescharacteristic
280C2×C3×C4#9911yescharacteristic
281C2×C3×C4C2×C22⋊C411yescharacteristic
282C2×C3×C4C2×C22⋊C412yes—
283C2×C3×C4C2×C22⋊C412yes—
284C2×C3×C4C23×C411yescharacteristic
285C2×C3×C4C22×D412yes—
286C3⋊C8—48no—
287C24—21no—
288C24#21712yes—
289C24#21711yescharacteristic
290#41—42no—
291#41—42no—
292#41—44no—
293#41—44no—
294#41—42no—
295#41—42no—
296#41—44no—
297#41—44no—
298#41—44no—
299#41—44no—
300C2×Q8—21no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8C22×Dic312yes—
306C2×Q8C22×Dic311yescharacteristic
307C2×Q8C2×#6712yes—
308C2×Q8C23×D311yescharacteristic
309C2×D4—28no—
310C2×D4—24no—
311C2×D4—21no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—48no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4#21714yes—
322C2×D4#21712yes—
323C2×D4C22×Dic312yes—
324C2×D4C22×Dic312yes—
325C2×D4C2×#6714yes—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—21no—
336C22×C4—24no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4#21712yes—
342C22×C4#21714yes—
343C22×C4#21711yescharacteristic
344C22×C4C22×Dic311yescharacteristic
345C22×C4C22×Dic312yes—
346C22×C4C2×#6712yes—
347C22×C4C2×#6714yes—
348M16—68no—
349M16—124no—
350M16—128no—
351C2×C8—124no—
352C2×C8—128no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C22⋊C4—28no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C42—24no—
359C42—21no—
360C42—21no—
361C42C22×Dic312yes—
362C42C22×Dic312yes—
363C22×C3—22no—
364C22×C3—22no—
365C22×C3—24no—
366C22×C3—42no—
367C22×C3—44no—
368C22×C3—42no—
369C22×C3—44no—
370C22×C3—42no—
371C22×C3—44no—
372C22×C3—42no—
373C22×C3—82no—
374C22×C3—82no—
375C22×C3—82no—
376C22×C3C2×#9911yescharacteristic
377C22×C3C2×#9911yescharacteristic
378C22×C3C22×C22⋊C411yescharacteristic
379C3×C4—24no—
380C3×C4—22no—
381C3×C4—21no—
382C3×C4—22no—
383C3×C4—21no—
384C3×C4—42no—
385C3×C4—42no—
386C3×C4C2×#9912yes—
387C3×C4C2×#9912yes—
388C23—21no—
389C23—21no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—42no—
394C23—44no—
395C23—44no—
396C23—41no—
397C23—41no—
398C23C2×#21711yescharacteristic
399C23C2×#21714yes—
400C23C2×#21712yes—
401Q8—41no—
402Q8—42no—
403Q8—41no—
404Q8—42no—
405Q8—42no—
406D4—42no—
407D4—44no—
408D4—42no—
409D4—44no—
410D4—42no—
411D4—42no—
412D4—416no—
413D4—48no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—48no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4#83411yescharacteristic
437C2×C4#83412yes—
438C2×C4#83411yescharacteristic
439C2×C4C2×#21711yescharacteristic
440C2×C4C2×#21712yes—
441C2×C4C2×#21712yes—
442C2×C4C23×Dic311yescharacteristic
443C2×C4C22×#6712yes—
444C8—128no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—42no—
448C2×C3—44no—
449C2×C3—82no—
450C2×C3#213212yescomplemented
451C2×C3C22×#9911yescharacteristic
452C22—22no—
453C22—22no—
454C22—24no—
455C22—42no—
456C22—44no—
457C22—42no—
458C22—44no—
459C22—42no—
460C22—44no—
461C22—42no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22C2×#83411yescharacteristic
466C22C2×#83411yescharacteristic
467C22C22×#21711yescharacteristic
468C4—24no—
469C4—22no—
470C4—21no—
471C4—22no—
472C4—21no—
473C4—42no—
474C4—42no—
475C4C2×#83412yes—
476C4C2×#83412yes—
477C3C2×#213211yescharacteristic, complemented, Sylow
478C2—21no—
479C2—21no—
480C2—42no—
481C2—44no—
482C2—82no—
483C2#7182012yesdirect product factor
484C2C22×#83411yescharacteristic
subgroupG/H
centerC2×C4#834
commutatorC2×C3×C4C23×C4
FrattiniC2×Q8C23×D3
Fitting, prime radicalC2×C3×#532C2
socleC22×C3C2×#99
cosocleC2×C3×Q8C24
hypercenterC2×#532D3
hypocenterC3C2×#2132
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
2. C2×#5237C2semidirect
3. #71820C2direct (N × H)
4. C2×#4548C2semidirect
5. C2×#4542C2semidirect
21. #5237C22semidirect
24. #4548C22semidirect
25. #4542C22semidirect
450. C2×C3#2132semidirect
477. C3C2×#2132semidirect
483. C2#71820direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×#52372
#718202
C2×#45482
C2×#45422
C2×#21323