Finite Group Databasebeta

← #10772439

Subgroups — #10772439

Order 768.

918 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86281C211yesmaximal, characteristic
2C3×#3209C211yesmaximal, characteristic, complemented
3#83212C211yesmaximal, characteristic
4C2×#5093C211yesmaximal, characteristic, complemented
5#71826C211yesmaximal, characteristic, complemented
6#71063C214yesmaximal, complemented
7#71062C214yesmaximal, complemented
8#70779C211yesmaximal, characteristic, complemented
9#70754C211yesmaximal, characteristic
10#18633—31nomaximal, Sylow
11C3×#533C2211yescharacteristic
12C2×C3×C4⋊Q8C2211yescharacteristic
13#5728—21no—
14D3×C4⋊Q8—21no—
15#5689—22no—
16#5635—22no—
17#5632—22no—
18#5579C2212yes—
19Q8×D12—21no—
20#5567—21no—
21Q8×Dic6C2212yes—
22#5523—21no—
23#5519—21no—
24C4×#235C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C3×#494—21no—
27C3×#487—21no—
28C3×#462C2214yes—
29C3×#451—22no—
30C3×#431C2211yescharacteristic
31#5094C2212yes—
32#5093—22no—
33#5093C2214yes—
34#5091—21no—
35#5087C2212yes—
36#5084—21no—
37C2×#810—22no—
38#4966—21no—
39#4964—21no—
40#4924C2211yescharacteristic
41C2×C4×C3⋊C8C2211yescharacteristic
42#4535C2214yes—
43#4484C2212yes—
44#4483C2212yes—
45#4481C2214yes—
46#3503—31no—
47#3209D311yescharacteristic, complemented
48C2×#487—31no—
49#3079—31no—
50#1645—34no—
51#1644—34no—
52#1548—31no—
53#1493—31no—
54#1476—31no—
55C22×C3×Q8—21no—
56C2×D3×Q8—22no—
57C2×#235D411yescharacteristic
58C22×Dic6D412yes—
59C2×C3×Q16—24no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8D411yescharacteristic
64C3×C4⋊Q8D411yescharacteristic
65C3×C4⋊Q8C2312yes—
66C3×#125—21no—
67C3×C22⋊Q8—21no—
68C3×C22⋊Q8D412yes—
69C3×C4×Q8—21no—
70C3×C4×Q8C2312yes—
71C3×#117—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C42C2311yescharacteristic
75#946—22no—
76Q8×Dic3—22no—
77#944—22no—
78C4×#67—21no—
79C4×#67—21no—
80#926D411yescharacteristic
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—21no—
83C2×#208—21no—
84C22×C3⋊C8—21no—
85#897—22no—
86#897—24no—
87#896—24no—
88#892—22no—
89D3×C4⋊C4—22no—
90D3×C4⋊C4—22no—
91#890—22no—
92#889—24no—
93#888—22no—
94#888—24no—
95#888—22no—
96#887—22no—
97#67⋊C4—24no—
98#878—24no—
99C4×D12—22no—
100C4×D12D411yescharacteristic
101C4×D12C2311yescharacteristic
102#872—21no—
103C42×D3—21no—
104#870—22no—
105#869—21no—
106#869—21no—
107#869—22no—
108#869D412yes—
109#869C2312yes—
110C4×Dic6—22no—
111C4×Dic6D411yescharacteristic
112C4×Dic6C2311yescharacteristic
113C3×C4⋊C8C2312yes—
114C3×#103—24no—
115C3×C22⋊C8—21no—
116C3×C4×C8—21no—
117#810—22no—
118#810—28no—
119#809—22no—
120#808—22no—
121#807—22no—
122C4⋊(C3⋊C8)—21no—
123C4⋊(C3⋊C8)C2312yes—
124C4×C3⋊C8—21no—
125C4×C3⋊C8C2312yes—
126Q82—32no—
127#557—61no—
128#554—32no—
129#554—62no—
130D4×Q8—61no—
131D4×Q8—62no—
132#544—61no—
133#544—62no—
134#533—61no—
135#533C2×D311yescharacteristic
136C2×C4⋊Q8—31no—
137C2×C4⋊Q8—61no—
138C2×C4⋊Q8C2×D311yescharacteristic
139#520—61no—
140C4×#41—31no—
141#494—21no—
142#487—21no—
143#487—34no—
144#487—62no—
145#477—61no—
146#476—61no—
147#475—32no—
148#474—32no—
149#462C2×D314yes—
150#451—22no—
151#431C2×D311yescharacteristic
152#428—61no—
153#423—31no—
154#417—61no—
155C2×#103—62no—
156C2×C4×C8—31no—
157#333—34no—
158Q8⋊C8—34no—
159Q8⋊C8—32no—
160D4⋊C8—32no—
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×Q8—41no—
164C2×C3×Q8—82no—
165C2×C3×Q8C2×D412yes—
166C22×C3×C4—41no—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C2×#67—21no—
171C22×Dic3—22no—
172D3×Q8—84no—
173#235—22no—
174#235Q1614yescomplemented
175C2×D12C2×D411yescharacteristic
176C2×C4×D3—21no—
177C2×C4×D3—21no—
178C2×C4×D3—21no—
179C2×C4×D3—24no—
180C2×Dic6—22no—
181C2×Dic6—24no—
182C2×Dic6—44no—
183C2×Dic6C2×D411yescharacteristic
184C2×Dic6C2×D414yes—
185C3×Q16—84no—
186C2×C3×C8—22no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—41no—
195C3×C4⋊C4—41no—
196C3×C4⋊C4C2×D412yes—
197C3×C22⋊C4—22no—
198C3×C42—22no—
199C3×C42C2×D411yescharacteristic
200C3×C42C2×D411yescharacteristic
201C3×C42C2×D411yescharacteristic
202C3×C42C2411yescharacteristic
203#217—21no—
204(C2×D3)⋊C4—24no—
205(C2×D3)⋊C4—21no—
206(C2×D3)⋊C4—21no—
207C4⋊Dic3—24no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3—22no—
211C4⋊Dic3—22no—
212C4⋊Dic3C2×D411yescharacteristic
213C4⋊Dic3C2×D411yescharacteristic
214Dic3⋊C4—24no—
215Dic3⋊C4—24no—
216Dic3⋊C4—24no—
217Dic3⋊C4—21no—
218Dic3⋊C4—21no—
219C4×Dic3—24no—
220C4×Dic3—21no—
221C4×Dic3—21no—
222C4×Dic3—21no—
223#208—42no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—42no—
227C2×#41—31no—
228C22×Q8—21no—
229C22×Q8—32no—
230C22×Q8—62no—
231C2×Q16—24no—
232C2×M16—61no—
233C22×C8—61no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—32no—
238C4⋊Q8—32no—
239C4⋊Q8—62no—
240C4⋊Q8—64no—
241C4⋊Q8—62no—
242C4⋊Q8—62no—
243C4⋊Q8—62no—
244C4⋊Q8—61no—
245C4⋊Q8—61no—
246C4⋊Q8#6711yescharacteristic
247C4⋊Q8#6711yescharacteristic
248C4⋊Q8C22×D312yes—
249#125—21no—
250#125—62no—
251#125—62no—
252#125—64no—
253C22⋊Q8—21no—
254C22⋊Q8—64no—
255C22⋊Q8—64no—
256C22⋊Q8—62no—
257C22⋊Q8—64no—
258C22⋊Q8—62no—
259C22⋊Q8#6712yes—
260C4×Q8—21no—
261C4×Q8—31no—
262C4×Q8—31no—
263C4×Q8—62no—
264C4×Q8—62no—
265C4×Q8—62no—
266C4×Q8C22×D312yes—
267C4×D4—31no—
268C4×D4—31no—
269C4×D4—62no—
270C4×D4—64no—
271C4×D4—61no—
272C4×D4—61no—
273#117—21no—
274#117—31no—
275#117—62no—
276#117—61no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—62no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—62no—
283C2×C42—61no—
284C2×C42C22×D311yescharacteristic
285#107—62no—
286#106—62no—
287C4⋊C8—32no—
288C4⋊C8—61no—
289C4⋊C8C22×D312yes—
290#103—24no—
291#103—68no—
292#103—62no—
293#102—62no—
294C22⋊C8—21no—
295C4×C8—21no—
296C4×C8—32no—
297C4×C8—61no—
298C23×C3#12011yescharacteristic
299C22×D3—21no—
300C3×Q8—24no—
301C3×Q8—42no—
302C3×Q8—44no—
303C3×Q8—82no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—42no—
309C2×C3×C4—82no—
310C2×C3×C4#12011yescharacteristic
311C2×C3×C4#12011yescharacteristic
312C2×C3×C4#12011yescharacteristic
313C2×C3×C4C2×Q1612yes—
314C2×C3×C4#13712yes—
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C22×D411yescharacteristic
318#67—24no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—24no—
323C2×Dic3—82no—
324D12—21no—
325D12C2×Q1612yes—
326C4×D3—24no—
327C4×D3—42no—
328C4×D3—42no—
329C4×D3—84no—
330Dic6—21no—
331Dic6—44no—
332Dic6—44no—
333Dic6—84no—
334Dic6C2×Q1612yes—
335C3×C8—42no—
336C3⋊C8—42no—
337C3⋊C8—42no—
338#41—34no—
339#41—62no—
340C2×Q8—21no—
341C2×Q8—22no—
342C2×Q8—31no—
343C2×Q8—34no—
344C2×Q8—41no—
345C2×Q8—64no—
346C2×Q8—62no—
347C2×Q8—82no—
348C2×Q8—124no—
349C2×Q8—244no—
350C2×Q8C2×#6712yes—
351C2×D4—31no—
352C2×D4—61no—
353C22×C4—41no—
354C22×C4—64no—
355C22×C4—61no—
356C22×C4—62no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4D3×D411yescharacteristic
360C22×C4D3×D411yescharacteristic
361C22×C4C2×#6711yescharacteristic
362Q16—84no—
363M16—122no—
364C2×C8—22no—
365C2×C8—62no—
366C2×C8—62no—
367C2×C8—122no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—31no—
376C4⋊C4—31no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—64no—
380C4⋊C4—64no—
381C4⋊C4—64no—
382C4⋊C4—64no—
383C4⋊C4—62no—
384C4⋊C4—62no—
385C4⋊C4—61no—
386C4⋊C4—62no—
387C4⋊C4—61no—
388C4⋊C4—62no—
389C4⋊C4C2×#6712yes—
390C22⋊C4—22no—
391C22⋊C4—64no—
392C22⋊C4—61no—
393C22⋊C4—61no—
394C22⋊C4—61no—
395C42—22no—
396C42—64no—
397C42—61no—
398C42—61no—
399C42—61no—
400C42D3×D411yescharacteristic
401C42D3×D411yescharacteristic
402C42C2×#6711yescharacteristic
403C42C23×D311yescharacteristic
404C22×C3—21no—
405C22×C3—21no—
406C22×C3#45112yes—
407C22×C3C2×#12011yescharacteristic
408C2×D3—22no—
409C2×D3—41no—
410C2×D3—41no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4—82no—
416C3×C4#45112yes—
417C3×C4#45512yes—
418C3×C4C22×Q1611yescharacteristic
419C3×C4C2×#13711yescharacteristic
420C3×C4#57712yes—
421Dic3—22no—
422Dic3—41no—
423Dic3—44no—
424Dic3—41no—
425C23—61no—
426C23#93711yescharacteristic
427Q8—24no—
428Q8—32no—
429Q8—42no—
430Q8—44no—
431Q8—61no—
432Q8—82no—
433Q8—124no—
434Q8—124no—
435Q8—244no—
436D4—32no—
437D4—64no—
438D4—61no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—42no—
444C2×C4—61no—
445C2×C4—64no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—64no—
449C2×C4—82no—
450C2×C4—122no—
451C2×C4—122no—
452C2×C4—244no—
453C2×C4—242no—
454C2×C4D3×Q1612yes—
455C2×C4#91812yes—
456C2×C4#93711yescharacteristic
457C2×C4#93711yescharacteristic
458C2×C4#93711yescharacteristic
459C2×C4C2×D3×D411yescharacteristic
460C2×C4C2×D3×D411yescharacteristic
461C2×C4C22×#6711yescharacteristic
462C8—42no—
463C8—122no—
464C8—122no—
465C2×C3—22no—
466C2×C3C2×#45111yescharacteristic
467C2×C3#301011yescharacteristic
468C2×C3C2×#45511yescharacteristic
469D3—42no—
470C22—21no—
471C22—21no—
472C22—62no—
473C22—121no—
474C22—121no—
475C22#518612yes—
476C22C2×#93711yescharacteristic
477C4—21no—
478C4—21no—
479C4—42no—
480C4—42no—
481C4—62no—
482C4—82no—
483C4—121no—
484C4—124no—
485C4—121no—
486C4#518612yes—
487C4#520212yes—
488C4C2×D3×Q1611yescharacteristic
489C4C2×#91811yescharacteristic
490C4#583612yes—
491C3#1863311yescharacteristic, complemented, Sylow
492C2—22no—
493C2—122no—
494C2#8339011yescharacteristic
495C2C2×#518611yescharacteristic
496C2C2×#520211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3209C2
socleC22×C3C2×#120
prime radicalC2×#235D4
hypercenter#3209D3
hypocenterC3#18633
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
2. C3×#3209C2semidirect
4. C2×#5093C2semidirect
5. #71826C2semidirect
6. #71063C2semidirect
7. #71062C2semidirect
8. #70779C2semidirect
47. #3209D3semidirect
174. #235Q16semidirect
491. C3#18633semidirect

Maximal subgroups

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

subgroupindex
#862812
C3×#32092
#832122
C2×#50932
#718262
#710632
#710622
#707792
#707542
#186333