Finite Group Databasebeta

← #10901294

Subgroups — #10901294

Order 768.

1102 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3460C211yesmaximal, characteristic
2#82533C212yesmaximal, complemented
3#82382C211yesmaximal, characteristic, complemented
4#71905C212yesmaximal, complemented
5#71881C212yesmaximal, complemented
6#71857C214yesmaximal, complemented
7#71646C212yesmaximal, complemented
8#71617C211yesmaximal, characteristic, complemented
9#29680—31nomaximal, Sylow
10C3×#544C2211yescharacteristic
11C3×#542—22no—
12C3×#540—22no—
13C3×#538—22no—
14C3×#537C2211yescharacteristic
15C3×#535C2212yes—
16C3×#532—22no—
17C3×#529—22no—
18C3×#525—22no—
19C3×C4×#41C2212yes—
20C2×C3×C4×D4C2211yescharacteristic
21C2×#951C2214yes—
22C2×#936C2212yes—
23C2×#869C2211yescharacteristic
24C2×#837C2214yes—
25C2×#833C2212yes—
26#5212—22no—
27#5031—22no—
28#5031—22no—
29#5019—22no—
30#5019—22no—
31#5004C2214yes—
32#4973C2212yes—
33#4953—22no—
34C4×C4⋊Dic3C2211yescharacteristic
35#4924C2212yes—
36#4532C2214yes—
37#4530C2212yes—
38#3460D311yescharacteristic
39#3039—32no—
40#2547—31no—
41#2028—32no—
42#2007—32no—
43#2004—34no—
44#1806—32no—
45#1800—31no—
46C2×C3×#41—22no—
47C2×C3×#41C2312yes—
48C22×C3×D4C2311yescharacteristic
49C23×C3×C4D412yes—
50C22×Dic6C2312yes—
51C3×C4⋊Q8—22no—
52C3×#127—22no—
53C3×#126—24no—
54C3×#125—22no—
55C3×#124—24no—
56C3×#124—22no—
57C3×#123—24no—
58C3×#123—24no—
59C3×C22⋊Q8—22no—
60C3×C22⋊Q8—22no—
61C3×C22⋊Q8—22no—
62C3×#121—22no—
63C3×#121—24no—
64C3×#121—22no—
65C3×#121—22no—
66C3×#121—22no—
67C3×#121—22no—
68C3×#120—24no—
69C3×C4×Q8D412yes—
70C3×C4×Q8D412yes—
71C3×C4×D4—24no—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C3×C4×D4D412yes—
77C3×C4×D4D412yes—
78C3×#117—22no—
79C3×#117—22no—
80C3×#117C2312yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4D412yes—
83C2×C3×C42—22no—
84C2×C3×C42C2311yescharacteristic
85C2×#217—24no—
86#951—44no—
87#951—44no—
88C2×#216—24no—
89#936—44no—
90C2×#214—24no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94C2×C4×Dic3C2312yes—
95C2×#208C2314yes—
96#869—22no—
97#869—22no—
98#837—44no—
99#833—42no—
100C3⋊#95—21no—
101C3⋊#95—24no—
102#810—24no—
103#810—24no—
104C4⋊(C3⋊C8)—22no—
105C4⋊(C3⋊C8)—22no—
106C2×#137—34no—
107#544C2×D311yescharacteristic
108#542—22no—
109#540—22no—
110#538—22no—
111#537C2×D311yescharacteristic
112#535C2×D312yes—
113#532—22no—
114C2×C4⋊Q8—31no—
115#529—22no—
116C2×#124—32no—
117#525—22no—
118C4×#41C2×D312yes—
119C2×C4×D4C2×D311yescharacteristic
120#462—62no—
121#462—62no—
122#461—62no—
123#461—62no—
124#423—32no—
125C2×#104—34no—
126#419—34no—
127C2×#100—32no—
128#397—62no—
129#391—32no—
130#380—62no—
131C4×C4⋊C4—31no—
132#343—32no—
133#339—34no—
134C24×C3C2×D412yes—
135C3×#41—22no—
136C3×#41—22no—
137C3×#41—22no—
138C3×#41—22no—
139C3×#41—84no—
140C2×C3×Q8—22no—
141C2×C3×Q8C2×D412yes—
142C2×C3×D4—22no—
143C2×C3×D4—24no—
144C2×C3×D4—24no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—21no—
148C2×C3×D4—22no—
149C2×C3×D4—21no—
150C2×C3×D4C2×D412yes—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—44no—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2411yescharacteristic
161C22×Dic3—22no—
162C22×Dic3—22no—
163C2×Dic6—24no—
164C2×Dic6—24no—
165C2×Dic6—22no—
166C2×Dic6—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4C2×D412yes—
174C3×C4⋊C4C2×D412yes—
175C3×C22⋊C4—24no—
176C3×C22⋊C4—24no—
177C3×C22⋊C4—24no—
178C3×C22⋊C4—22no—
179C3×C22⋊C4—24no—
180C3×C22⋊C4—22no—
181C3×C22⋊C4—22no—
182C3×C42—22no—
183C3×C42—22no—
184C3×C42—22no—
185C3×C42C2×D412yes—
186C3×C42C2×D412yes—
187#217—48no—
188#216—48no—
189#214—48no—
190C4⋊Dic3—42no—
191C4⋊Dic3—42no—
192C4⋊Dic3—41no—
193C4⋊Dic3—41no—
194C4×Dic3—24no—
195#208—24no—
196#208—24no—
197C2×C3⋊C8—24no—
198C2×#41—22no—
199C2×#41C22×D312yes—
200C22×Q8—32no—
201C22×D4C22×D311yescharacteristic
202C23×C4#6712yes—
203#137—124no—
204#137—124no—
205C2×Q16—64no—
206C2×SD16—64no—
207C2×M16—34no—
208C4⋊Q8—22no—
209C4⋊Q8—62no—
210C4⋊Q8—62no—
211#127—22no—
212#126—24no—
213#125—22no—
214#124—24no—
215#124—22no—
216#124—124no—
217#123—24no—
218#123—24no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222#121—22no—
223#121—24no—
224#121—22no—
225#121—22no—
226#121—22no—
227#121—22no—
228#120—24no—
229C4×Q8#6712yes—
230C4×Q8#6712yes—
231C4×D4—24no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4#6712yes—
237C4×D4#6712yes—
238#117—22no—
239#117—22no—
240#117C22×D312yes—
241C2×C4⋊C4—61no—
242C2×C4⋊C4—61no—
243C2×C4⋊C4—61no—
244C2×C4⋊C4C22×D311yescharacteristic
245C2×C22⋊C4—64no—
246C2×C22⋊C4#6712yes—
247C2×C42—22no—
248C2×C42—32no—
249C2×C42C22×D311yescharacteristic
250C4⋊C8—62no—
251C4⋊C8—62no—
252#104—124no—
253#103—64no—
254#103—64no—
255#100—122no—
256#95—61no—
257#95—64no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3C22×D411yescharacteristic
265C3×Q8—22no—
266C3×Q8—22no—
267C3×Q8—82no—
268C3×D4—22no—
269C3×D4—22no—
270C3×D4—42no—
271C3×D4—42no—
272C3×D4—42no—
273C3×D4—42no—
274C3×D4—84no—
275C3×D4—82no—
276C3×D4—84no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—44no—
286C2×C3×C4—44no—
287C2×C3×C4—42no—
288C2×C3×C4—82no—
289C2×C3×C4—82no—
290C2×C3×C4#12012yes—
291C2×C3×C4#12011yescharacteristic
292C2×C3×C4#12011yescharacteristic
293C2×C3×C4C22×D412yes—
294C2×C3×C4H(5,𝔽2)11yescharacteristic
295C2×C3×C4H(5,𝔽2)12yes—
296C2×C3×C42−1+411yescharacteristic
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303Dic6—44no—
304Dic6—44no—
305C3⋊C8—44no—
306C24C2×#6712yes—
307#41—22no—
308#41—22no—
309#41—22no—
310#41—22no—
311#41—84no—
312C2×Q8—22no—
313C2×Q8—64no—
314C2×Q8—62no—
315C2×Q8—62no—
316C2×Q8—64no—
317C2×Q8C2×#6712yes—
318C2×D4—22no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—22no—
322C2×D4—21no—
323C2×D4—22no—
324C2×D4—21no—
325C2×D4—22no—
326C2×D4C2×#6712yes—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—44no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4C2×#6711yescharacteristic
336C22×C4C2×#6712yes—
337C22×C4C2×#6711yescharacteristic
338C22×C4C23×D311yescharacteristic
339Q16—128no—
340SD16—128no—
341M16—64no—
342M16—64no—
343C2×C8—64no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—122no—
351C4⋊C4—122no—
352C4⋊C4—121no—
353C4⋊C4—121no—
354C4⋊C4C2×#6712yes—
355C4⋊C4C2×#6712yes—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—22no—
360C22⋊C4—24no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—128no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—64no—
368C42C2×#6712yes—
369C42C2×#6712yes—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—44no—
373C22×C3—42no—
374C22×C3—42no—
375C22×C3—42no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—82no—
380C22×C3C2×#12011yescharacteristic
381C22×C3#53511yescharacteristic
382C22×C3#53711yescharacteristic
383C3×C4—21no—
384C3×C4—22no—
385C3×C4—42no—
386C3×C4—42no—
387C3×C4—42no—
388C3×C4—42no—
389C3×C4#45512yes—
390C3×C4#53512yes—
391C3×C4#53712yes—
392Dic3—82no—
393Dic3—82no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—42no—
399C23—42no—
400C23C22×#6711yescharacteristic
401Q8—22no—
402Q8—22no—
403Q8—82no—
404Q8—124no—
405Q8—124no—
406D4—22no—
407D4—22no—
408D4—42no—
409D4—42no—
410D4—42no—
411D4—42no—
412D4—82no—
413D4—84no—
414D4—84no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4#95312yes—
435C2×C4#95311yescharacteristic
436C2×C4#95311yescharacteristic
437C2×C4#100911yescharacteristic
438C2×C4#100912yes—
439C2×C4#101011yescharacteristic
440C2×C4C22×#6712yes—
441C8—124no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—82no—
445C2×C3—82no—
446C2×C3#261511yescharacteristic
447C2×C3C2×#45511yescharacteristic
448C2×C3#302211yescharacteristic
449C22—21no—
450C22—21no—
451C22—44no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—82no—
459C22#583011yescharacteristic
460C22#583311yescharacteristic
461C22C2×#95311yescharacteristic
462C4—21no—
463C4—22no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—242no—
469C4—242no—
470C4#524212yes—
471C4#583012yes—
472C4#583312yes—
473C3#2968011yescharacteristic, complemented, Sylow
474C2—21no—
475C2—21no—
476C2—82no—
477C2—82no—
478C2#8409811yescharacteristic
479C2C2×#524211yescharacteristic
480C2#8412311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3460C2
socleC22×C3C2×#120
hypercenter#3460D3
hypocenterC3#29680
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3460

Complements

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

NH ≅ G/Ntype
2. #82533C2semidirect
3. #82382C2semidirect
4. #71905C2semidirect
5. #71881C2semidirect
6. #71857C2semidirect
7. #71646C2semidirect
8. #71617C2semidirect
473. C3#29680semidirect

Maximal subgroups

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

subgroupindex
C3×#34602
#825332
#823822
#719052
#718812
#718572
#716462
#716172
#296803