Finite Group Databasebeta

← #10698560

Subgroups — #10698560

Order 768.

572 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82476C211yesmaximal, characteristic
2C2×C3×Q8⋊C8C211yesmaximal, characteristic, complemented
3#71626C211yesmaximal, characteristic, complemented
4#70454C211yesmaximal, characteristic, complemented
5C2×#4485C211yesmaximal, characteristic
6#69772C211yesmaximal, characteristic
7#67655C211yesmaximal, characteristic, complemented
8#10410—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#947—21no—
11C2×#874C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14C2×#835—21no—
15C2×#821—22no—
16C2×#817—21no—
17#4991—22no—
18#4988—21no—
19#4939—21no—
20#4939—21no—
21#4935C2211yescharacteristic
22C4×C4⋊Dic3C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C3×Q8⋊C8—22no—
25#4551—21no—
26#4529—21no—
27#4485—22no—
28#2548—31no—
29#1957—31no—
30#1948—31no—
31#1803—31no—
32C2×#331—31no—
33C2×Q8⋊C8D311yescharacteristic, complemented
34#1279—31no—
35C22×C3×Q8D411yescharacteristic
36C22×D12—21no—
37C22×D12—21no—
38C22×D12D412yes—
39C22×C3×C8—21no—
40C22×C3×C8—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C42—21no—
48C2×C3×C42C2311yescharacteristic
49#947—42no—
50#947—42no—
51C2×(C2×D3)⋊C4—22no—
52C2×(C2×D3)⋊C4—22no—
53C2×(C2×D3)⋊C4—22no—
54C2×(C2×D3)⋊C4—22no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3D411yescharacteristic
59C2×C4⋊Dic3D411yescharacteristic
60C2×C4×Dic3—21no—
61C22×C3⋊C8—21no—
62#874—21no—
63#874—21no—
64#874—21no—
65#874—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8D412yes—
68C3×C4×C8—21no—
69C3×C4×C8—21no—
70#835—42no—
71C3⋊#95—21no—
72#821—44no—
73#817—42no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C2×#127—31no—
77C2×#124—61no—
78C2×C4×Q8C2×D311yescharacteristic
79C2×#106—61no—
80C2×C4⋊C8—31no—
81C2×C4⋊C8C2×D311yescharacteristic
82C2×#103—61no—
83C2×#102—62no—
84C2×C4×C8C2×D311yescharacteristic
85#394—62no—
86#390—61no—
87#390—61no—
88#390—61no—
89#384—31no—
90C4×C4⋊C4—31no—
91#340—61no—
92#340—61no—
93#331—62no—
94Q8⋊C8—22no—
95C23×D3—22no—
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C2×C3×Q8—21no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8SD1612yes—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—41no—
110C2×D12—22no—
111C2×D12—22no—
112C2×D12—42no—
113C2×D12—42no—
114C2×D12—41no—
115C2×D12—41no—
116C2×D12—41no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—41no—
126C2×C3×C8—22no—
127C2×C3×C8—22no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C2×C3×C8—41no—
131C2×C3×C8—41no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C42—41no—
137C3×C42—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140C3×C42#4111yescharacteristic
141C3×C42#4111yescharacteristic
142(C2×D3)⋊C4—44no—
143(C2×D3)⋊C4—44no—
144(C2×D3)⋊C4—82no—
145(C2×D3)⋊C4—82no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—42no—
151C4⋊Dic3—41no—
152C4⋊Dic3—41no—
153C4⋊Dic3—41no—
154C4⋊Dic3—41no—
155C4×Dic3—41no—
156C4×Dic3—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C22×Q8D1211yescharacteristic
161C22×D4—32no—
162C22×D4—61no—
163C22×D4—61no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—61no—
167#127—61no—
168#127—61no—
169#127—61no—
170#127—61no—
171#124—122no—
172#124—122no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4D1211yescharacteristic
184C2×C22⋊C4—62no—
185C2×C22⋊C4—62no—
186C2×C22⋊C4—62no—
187C2×C22⋊C4—62no—
188C2×C42—21no—
189C2×C42—61no—
190C2×C42C22×D311yescharacteristic
191#106—122no—
192C4⋊C8—21no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195C4⋊C8#6712yes—
196#103—122no—
197#102—124no—
198C4×C8—21no—
199C4×C8—21no—
200#95—61no—
201C23×C3#12011yescharacteristic
202C22×D3—42no—
203C22×D3—42no—
204C22×D3—42no—
205C22×D3—42no—
206C22×D3—42no—
207C22×D3—42no—
208C22×D3—42no—
209C3×Q8—22no—
210C3×Q8—22no—
211C3×Q8—41no—
212C3×Q8—41no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—22no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12311yescharacteristic
225C2×C3×C4#12311yescharacteristic
226C2×C3×C4#12411yescharacteristic
227C2×C3×C4C2×SD1611yescharacteristic
228C2×C3×C4#13511yescharacteristic
229C2×C3×C4#13511yescharacteristic
230C2×C3×C4#13511yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
234C2×C3×C4#13711yescharacteristic
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—81no—
242C2×Dic3—81no—
243C2×Dic3—81no—
244D12—44no—
245D12—82no—
246D12—81no—
247D12—81no—
248D12—81no—
249D12—81no—
250D12—81no—
251D12—81no—
252D12—81no—
253D12—81no—
254C3×C8—42no—
255C3×C8—42no—
256C3⋊C8—81no—
257C24—62no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8#20412yes—
264C2×D4—62no—
265C2×D4—62no—
266C2×D4—122no—
267C2×D4—122no—
268C2×D4—121no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4—121no—
275C2×D4—121no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—121no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—121no—
285C22×C4C2×D1211yescharacteristic
286C22×C4D3×D411yescharacteristic
287C22×C4D3×D411yescharacteristic
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—121no—
295C2×C8—121no—
296C2×C8—121no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—122no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C4⋊C4—121no—
309C4⋊C4—121no—
310C22⋊C4—124no—
311C22⋊C4—124no—
312C22⋊C4—242no—
313C22⋊C4—242no—
314C42—41no—
315C42—41no—
316C42—121no—
317C42—121no—
318C42#23511yescharacteristic
319C42#23711yescharacteristic
320C42#23711yescharacteristic
321C42C2×#6711yescharacteristic
322C22×C3#39411yescharacteristic
323C22×C3#44911yescharacteristic
324C22×C3#44911yescharacteristic
325C22×C3#45011yescharacteristic
326C22×C3#45211yescharacteristic
327C22×C3#45311yescharacteristic
328C22×C3#45511yescharacteristic
329C2×D3—82no—
330C2×D3—82no—
331C2×D3—82no—
332C2×D3—82no—
333C2×D3—82no—
334C2×D3—82no—
335C2×D3—82no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—42no—
339C3×C4—81no—
340C3×C4#46311yescharacteristic
341C3×C4#46411yescharacteristic
342C3×C4#46511yescharacteristic
343C3×C4#46911yescharacteristic
344C3×C4#48111yescharacteristic
345C3×C4#48211yescharacteristic
346C3×C4#48811yescharacteristic
347C3×C4#48911yescharacteristic
348Dic3—81no—
349Dic3—81no—
350Dic3—161no—
351C23—122no—
352C23—122no—
353C23—122no—
354C23—122no—
355C23—122no—
356C23—122no—
357C23—122no—
358C23#88211yescharacteristic
359Q8—22no—
360Q8—22no—
361Q8—41no—
362Q8—41no—
363D4—124no—
364D4—242no—
365D4—241no—
366D4—241no—
367D4—241no—
368D4—241no—
369D4—241no—
370D4—241no—
371D4—241no—
372D4—241no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—241no—
388C2×C4—241no—
389C2×C4—241no—
390C2×C4#88311yescharacteristic
391C2×C4#88511yescharacteristic
392C2×C4#88611yescharacteristic
393C2×C4C2×#20411yescharacteristic
394C2×C4#90411yescharacteristic
395C2×C4#90811yescharacteristic
396C2×C4#90811yescharacteristic
397C2×C4#91411yescharacteristic
398C2×C4#91611yescharacteristic
399C2×C4#91811yescharacteristic
400C2×C4#91911yescharacteristic
401C2×C4#93011yescharacteristic
402C2×C4#93811yescharacteristic
403C2×C4#93911yescharacteristic
404C8—42no—
405C8—42no—
406C8—241no—
407C2×C3#162011yescharacteristic
408C2×C3#163111yescharacteristic
409C2×C3#164011yescharacteristic
410C2×C3#164211yescharacteristic
411C2×C3#203411yescharacteristic
412C2×C3#203711yescharacteristic
413C2×C3#204211yescharacteristic
414D3—162no—
415C22—242no—
416C22—242no—
417C22—242no—
418C22—242no—
419C22—242no—
420C22—242no—
421C22—242no—
422C22#473111yescharacteristic
423C22#473411yescharacteristic
424C22#480811yescharacteristic
425C22#481011yescharacteristic
426C22#482211yescharacteristic
427C22#482511yescharacteristic
428C22#496111yescharacteristic
429C4—21no—
430C4—21no—
431C4—42no—
432C4—81no—
433C4—241no—
434C4—241no—
435C4—481no—
436C4#473511yescharacteristic
437C4#473711yescharacteristic
438C4#479411yescharacteristic
439C4#480011yescharacteristic
440C4#511411yescharacteristic
441C4#513511yescharacteristic
442C4#516511yescharacteristic
443C4#518511yescharacteristic
444C3#1041011yescharacteristic, complemented, Sylow
445C2—482no—
446C2#6824811yescharacteristic
447C2#6825811yescharacteristic
448C2#6826111yescharacteristic
449C2#6826211yescharacteristic
450C2#6993511yescharacteristic
451C2#7024511yescharacteristic
452C2#7035911yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×Q8⋊C8C2
socleC23×C3#120
prime radicalC2×#874C22
hypercenterC2×Q8⋊C8D3
hypocenterC3#10410
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×C3×Q8⋊C8C2semidirect
3. #71626C2semidirect
4. #70454C2semidirect
7. #67655C2semidirect
33. C2×Q8⋊C8D3semidirect
444. C3#10410semidirect

Maximal subgroups

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

subgroupindex
#824762
C2×C3×Q8⋊C82
#716262
#704542
C2×#44852
#697722
#676552
#104103