Finite Group Databasebeta

← #10697517

Subgroups — #10697517

Order 768.

644 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82500C211yesmaximal, characteristic
2C2×#5031C211yesmaximal, characteristic, complemented
3C2×#4846C211yesmaximal, characteristic
4C2×C3×Q8⋊C8C211yesmaximal, characteristic, complemented
5#69777C211yesmaximal, characteristic, complemented
6C2×#4459C211yesmaximal, characteristic
7#67702C211yesmaximal, characteristic, complemented
8#10233—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#946—21no—
11C22×#216—21no—
12C2×C4×D12C2211yescharacteristic
13C2×#869C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4×C8C2211yescharacteristic
16C2×#821—21no—
17C2×#818—21no—
18#5031—22no—
19#5031—22no—
20#4994—22no—
21#4987—21no—
22C2×#810—21no—
23#4939—21no—
24#4935C2211yescharacteristic
25#4928—21no—
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27#4846—22no—
28#4846—22no—
29C3×Q8⋊C8—22no—
30#4551—22no—
31#4459—22no—
32C2×#474—31no—
33C2×#462—31no—
34#2538—31no—
35#1990—31no—
36C2×Q8⋊C8D311yescharacteristic, complemented
37C2×D4⋊C8—31no—
38#1296—31no—
39C22×C3×Q8D411yescharacteristic
40C22×D12D411yescharacteristic
41C22×C4×D3—21no—
42C22×Dic6—21no—
43C22×C3×C8—21no—
44C22×C3×C8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53#946—42no—
54#946—42no—
55C2×#216—24no—
56C2×#216—42no—
57C2×#216—42no—
58C2×(C2×D3)⋊C4—22no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—22no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3D411yescharacteristic
66C2×C4⋊Dic3D412yes—
67C2×Dic3⋊C4—22no—
68C22×C3⋊C8—21no—
69C4×D12—21no—
70C4×D12—21no—
71C4×D12—21no—
72C4×D12—21no—
73#869—21no—
74#869—21no—
75#869—21no—
76#869D412yes—
77C3×C4⋊C8—21no—
78C3×C4⋊C8D412yes—
79C3×C4×C8—21no—
80C3×C4×C8—21no—
81C3⋊#95—22no—
82#821—24no—
83#818—24no—
84#810—24no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)D412yes—
87C22×Q16—61no—
88C2×C4⋊Q8—31no—
89C2×C22⋊Q8—61no—
90C2×C4×Q8C2×D311yescharacteristic
91C2×C4×D4—31no—
92#474—62no—
93#474—62no—
94#462—62no—
95#462—62no—
96C2×#107—61no—
97C2×C4⋊C8—31no—
98C2×C4⋊C8C2×D311yescharacteristic
99C2×#103—61no—
100C2×#102—61no—
101C2×C4×C8C2×D311yescharacteristic
102#396—62no—
103#390—61no—
104#389—61no—
105#384—31no—
106#380—61no—
107#340—62no—
108Q8⋊C8—22no—
109D4⋊C8—62no—
110C23×D3—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8D812yes—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×Dic3—21no—
123C22×Dic3—22no—
124C22×Dic3—41no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12Q1612yescomplemented
131C2×C4×D3—41no—
132C2×C4×D3—41no—
133C2×C4×D3—41no—
134C2×C4×D3—41no—
135C2×C4×D3—41no—
136C2×C4×D3—41no—
137C2×Dic6—22no—
138C2×Dic6—41no—
139C2×Dic6—41no—
140C2×Dic6—41no—
141C2×Dic6—41no—
142C2×Dic6—41no—
143C2×C3×C8—22no—
144C2×C3×C8—22no—
145C2×C3×C8—41no—
146C2×C3×C8—41no—
147C2×C3×C8—41no—
148C2×C3×C8—41no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C42—41no—
154C3×C42—41no—
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42#4111yescharacteristic
159#216—48no—
160(C2×D3)⋊C4—44no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163(C2×D3)⋊C4—82no—
164C4⋊Dic3—21no—
165C4⋊Dic3—22no—
166C4⋊Dic3—22no—
167C4⋊Dic3—21no—
168C4⋊Dic3—41no—
169C4⋊Dic3—41no—
170C4⋊Dic3—41no—
171C4⋊Dic3—41no—
172C4⋊Dic3—41no—
173C4⋊Dic3—41no—
174C4⋊Dic3—42no—
175Dic3⋊C4—44no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—41no—
179C22×Q8—61no—
180C22×Q8D1211yescharacteristic
181C22×D4—31no—
182C23×C4—61no—
183C2×Q16—64no—
184C2×Q16—122no—
185C2×Q16—122no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—61no—
189C4⋊Q8—32no—
190C4⋊Q8—61no—
191C4⋊Q8—61no—
192C4⋊Q8—61no—
193C22⋊Q8—122no—
194C22⋊Q8—122no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×D4—61no—
200C4×D4—61no—
201C4×D4—61no—
202C4×D4—61no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—32no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4—62no—
209C2×C4⋊C4—61no—
210C2×C4⋊C4—61no—
211C2×C4⋊C4D1211yescharacteristic
212C2×C22⋊C4—62no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—62no—
215C2×C42—21no—
216C2×C42C22×D311yescharacteristic
217#107—64no—
218C4⋊C8—21no—
219C4⋊C8—32no—
220C4⋊C8—61no—
221C4⋊C8#6712yes—
222#103—64no—
223#102—64no—
224C4×C8—21no—
225C4×C8—21no—
226#95—62no—
227C23×C3#12011yescharacteristic
228C22×D3—22no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3—41no—
232C22×D3—41no—
233C22×D3—41no—
234C22×D3—41no—
235C3×Q8—22no—
236C3×Q8—22no—
237C3×Q8—41no—
238C3×Q8—41no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—22no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12711yescharacteristic
253C2×C3×C4C2×D811yescharacteristic
254C2×C3×C4C2×Q1611yescharacteristic
255C2×C3×C4#13511yescharacteristic
256C2×C3×C4#13511yescharacteristic
257C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
258C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
259C2×C3×C4#13711yescharacteristic
260C2×C3×C4#13711yescharacteristic
261C2×Dic3—42no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—42no—
265C2×Dic3—41no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—81no—
269C2×Dic3—81no—
270D12—22no—
271D12—22no—
272D12—41no—
273D12—41no—
274C4×D3—81no—
275C4×D3—81no—
276C4×D3—81no—
277C4×D3—81no—
278Dic6—42no—
279Dic6—42no—
280Dic6—81no—
281Dic6—81no—
282C3×C8—42no—
283C3×C8—42no—
284C3⋊C8—42no—
285C24—61no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—62no—
292C2×Q8—121no—
293C2×Q8—121no—
294C2×Q8—121no—
295C2×Q8—121no—
296C2×Q8—121no—
297C2×Q8D2412yes—
298C2×D4—32no—
299C2×D4—61no—
300C2×D4—61no—
301C2×D4—61no—
302C2×D4—61no—
303C2×D4—61no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—61no—
307C22×C4—62no—
308C22×C4—121no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4—121no—
312C22×C4—121no—
313C22×C4—121no—
314C22×C4—121no—
315C22×C4C2×D1211yescharacteristic
316C22×C4D3×D411yescharacteristic
317C22×C4D3×D411yescharacteristic
318Q16—128no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—62no—
326C2×C8—121no—
327C2×C8—121no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—61no—
333C4⋊C4—62no—
334C4⋊C4—62no—
335C4⋊C4—61no—
336C4⋊C4—122no—
337C4⋊C4—124no—
338C4⋊C4—121no—
339C4⋊C4—121no—
340C4⋊C4—121no—
341C4⋊C4—121no—
342C4⋊C4—121no—
343C4⋊C4—121no—
344C22⋊C4—124no—
345C22⋊C4—121no—
346C22⋊C4—121no—
347C22⋊C4—242no—
348C42—41no—
349C42—41no—
350C42#23511yescharacteristic
351C42D3×D411yescharacteristic
352C42D3×D411yescharacteristic
353C42C2×#6711yescharacteristic
354C22×C3#39211yescharacteristic
355C22×C3#44711yescharacteristic
356C22×C3#44911yescharacteristic
357C22×C3#45111yescharacteristic
358C22×C3#45211yescharacteristic
359C22×C3#45511yescharacteristic
360C22×C3#45611yescharacteristic
361C2×D3—42no—
362C2×D3—42no—
363C2×D3—42no—
364C2×D3—81no—
365C2×D3—81no—
366C2×D3—81no—
367C2×D3—81no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—42no—
371C3×C4—81no—
372C3×C4#44711yescharacteristic
373C3×C4#44911yescharacteristic
374C3×C4#45111yescharacteristic
375C3×C4#45211yescharacteristic
376C3×C4#46611yescharacteristic
377C3×C4#46711yescharacteristic
378C3×C4#46911yescharacteristic
379C3×C4#47011yescharacteristic
380Dic3—82no—
381Dic3—82no—
382Dic3—81no—
383C23—62no—
384C23—121no—
385C23—121no—
386C23—121no—
387C23—121no—
388C23—121no—
389C23—121no—
390C23#88211yescharacteristic
391Q8—22no—
392Q8—22no—
393Q8—41no—
394Q8—41no—
395Q8—122no—
396Q8—122no—
397Q8—241no—
398Q8—241no—
399D4—62no—
400D4—62no—
401D4—121no—
402D4—121no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—121no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4—122no—
417C2×C4—122no—
418C2×C4—241no—
419C2×C4—241no—
420C2×C4—241no—
421C2×C4—241no—
422C2×C4—241no—
423C2×C4—241no—
424C2×C4#88211yescharacteristic
425C2×C4#88411yescharacteristic
426C2×C4#88411yescharacteristic
427C2×C4C2×D2411yescharacteristic
428C2×C4#90411yescharacteristic
429C2×C4#90811yescharacteristic
430C2×C4#90811yescharacteristic
431C2×C4#91511yescharacteristic
432C2×C4#91611yescharacteristic
433C2×C4D3×Q1611yescharacteristic
434C2×C4#91811yescharacteristic
435C2×C4#93011yescharacteristic
436C2×C4#93711yescharacteristic
437C2×C4#94011yescharacteristic
438C8—42no—
439C8—42no—
440C8—122no—
441C2×C3#164111yescharacteristic
442C2×C3#164411yescharacteristic
443C2×C3#164911yescharacteristic
444C2×C3#165211yescharacteristic
445C2×C3#199911yescharacteristic
446C2×C3#200111yescharacteristic
447C2×C3#200711yescharacteristic
448D3—82no—
449C22—122no—
450C22—122no—
451C22—122no—
452C22—241no—
453C22—241no—
454C22—241no—
455C22—241no—
456C22#473211yescharacteristic
457C22#473411yescharacteristic
458C22#480811yescharacteristic
459C22#480911yescharacteristic
460C22#482511yescharacteristic
461C22#482611yescharacteristic
462C22#496011yescharacteristic
463C4—21no—
464C4—21no—
465C4—42no—
466C4—81no—
467C4—241no—
468C4—242no—
469C4—242no—
470C4#473211yescharacteristic
471C4#473411yescharacteristic
472C4#481311yescharacteristic
473C4#481611yescharacteristic
474C4#511511yescharacteristic
475C4#513511yescharacteristic
476C4#517311yescharacteristic
477C4#518611yescharacteristic
478C3#1023311yescharacteristic, complemented, Sylow
479C2—242no—
480C2#6826311yescharacteristic
481C2#6827011yescharacteristic
482C2#6827311yescharacteristic
483C2#6827411yescharacteristic
484C2#6996111yescharacteristic
485C2#7027311yescharacteristic
486C2#7039311yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×Q8⋊C8C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenterC2×Q8⋊C8D3
hypocenterC3#10233
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#5031C2semidirect
4. C2×C3×Q8⋊C8C2semidirect
5. #69777C2semidirect
7. #67702C2semidirect
36. C2×Q8⋊C8D3semidirect
130. C2×D12Q16semidirect
478. C3#10233semidirect

Maximal subgroups

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

subgroupindex
#825002
C2×#50312
C2×#48462
C2×C3×Q8⋊C82
#697772
C2×#44592
#677022
#102333