Finite Group Databasebeta

← #10694150

Subgroups — #10694150

Order 768.

584 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#947C211yesmaximal, characteristic
2C2×C3×Q8⋊C8C211yesmaximal, characteristic, complemented
3C2×#4535C211yesmaximal, characteristic, complemented
4#70441C211yesmaximal, characteristic, complemented
5C2×#4483C211yesmaximal, characteristic
6C4×Dic3⋊C8C211yesmaximal, characteristic, complemented
7C2×#4459C211yesmaximal, characteristic
8#9676—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#947C411yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14#5229C411yescharacteristic
15C2×#815—21no—
16C2×Dic3⋊C8—21no—
17#4990—21no—
18#4990—21no—
19#4988C411yescharacteristic
20#4977C411yescharacteristic
21C4×(C2×D3)⋊C4—21no—
22C4×(C2×D3)⋊C4—21no—
23C42×Dic3C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26C3×Q8⋊C8—22no—
27#4550—21no—
28#4550—21no—
29#4535—22no—
30#4483—22no—
31#4459—22no—
32C4×#124—31no—
33#1767—31no—
34C4×C4⋊C8—31no—
35C2×Q8⋊C8—31no—
36C2×Q8⋊C8D311yescharacteristic, complemented
37C2×D4⋊C8—31no—
38C2×D4⋊C8—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C22×D12C2×C411yescharacteristic
41C22×C4×D3—21no—
42C22×C3×C8—21no—
43C22×C3×C8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4C2×C411yescharacteristic
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52#947—22no—
53#947—22no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×C4⋊Dic3C2×C411yescharacteristic
60C2×C4×Dic3C2×C411yescharacteristic
61C2×C4×Dic3C2×C411yescharacteristic
62C2×C4×Dic3D411yescharacteristic
63C2×C4×Dic3D411yescharacteristic
64C2×C4×Dic3D411yescharacteristic
65C2×C4×Dic3D411yescharacteristic
66C22×C3⋊C8—21no—
67C22×C3⋊C8—21no—
68C4×D12—21no—
69C4×D12—21no—
70C4×D12—21no—
71C4×D12—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8D412yes—
74C3×C4×C8—21no—
75C3×C4×C8—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80#815—42no—
81Dic3⋊C8—24no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85C4×C3⋊C8—21no—
86C2×#124—31no—
87C2×C4×Q8C2×D311yescharacteristic
88C2×C4×D4—31no—
89C2×C4⋊C8—31no—
90C2×C4⋊C8—61no—
91C2×C4⋊C8C2×D311yescharacteristic
92C2×#97—61no—
93C2×C4×C8—31no—
94C2×C4×C8C2×D311yescharacteristic
95#391—31no—
96#390—31no—
97#388—61no—
98#388—61no—
99#382—31no—
100C4×C22⋊C4—61no—
101C4×C22⋊C4—61no—
102C43—31no—
103#336—61no—
104#336—61no—
105Q8⋊C8—22no—
106Q8⋊C8—62no—
107D4⋊C8—62no—
108D4⋊C8—62no—
109C23×D3—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8M1612yes—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4C22×C411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×Dic3—21no—
122C22×Dic3C22⋊C411yescharacteristic
123C22×Dic3C22⋊C411yescharacteristic
124C22×Dic3C22⋊C411yescharacteristic
125C22×Dic3C22⋊C411yescharacteristic
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12M1612yes—
132C2×C4×D3—41no—
133C2×C4×D3—41no—
134C2×C4×D3—41no—
135C2×C4×D3—41no—
136C2×C4×D3—41no—
137C2×C4×D3—41no—
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—41no—
141C2×C3×C8—41no—
142C2×C3×C8—41no—
143C2×C3×C8—41no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C42—41no—
149C3×C42—41no—
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42#4111yescharacteristic
153C3×C42#4111yescharacteristic
154(C2×D3)⋊C4—24no—
155(C2×D3)⋊C4—24no—
156(C2×D3)⋊C4—42no—
157(C2×D3)⋊C4—41no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—42no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4×Dic3—21no—
163C4×Dic3—21no—
164C4×Dic3—21no—
165C4×Dic3—21no—
166C4×Dic3—21no—
167C4×Dic3—21no—
168C4×Dic3—21no—
169C4×Dic3—21no—
170C4×Dic3—22no—
171C4×Dic3—41no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—41no—
175C2×C3⋊C8—41no—
176C2×C3⋊C8—41no—
177C2×C3⋊C8—41no—
178C22×Q8C4×D311yescharacteristic
179C22×D4—31no—
180C23×C4—61no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—61no—
184C22×C8—61no—
185#124—62no—
186#124—62no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×D4—61no—
192C4×D4—61no—
193C4×D4—61no—
194C4×D4—61no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4C4×D311yescharacteristic
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—61no—
202C2×C22⋊C4—61no—
203C2×C42—21no—
204C2×C42—31no—
205C2×C42—31no—
206C2×C42—31no—
207C2×C42—31no—
208C2×C42—31no—
209C2×C42—31no—
210C2×C42C22×D311yescharacteristic
211C4⋊C8—21no—
212C4⋊C8—32no—
213C4⋊C8—64no—
214C4⋊C8—61no—
215C4⋊C8#6712yes—
216#97—122no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—61no—
220C4×C8—61no—
221#95—61no—
222#95—61no—
223#95—61no—
224#95—61no—
225C23×C3C2×C22⋊C411yescharacteristic
226C22×D3—22no—
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3—41no—
232C22×D3—41no—
233C3×Q8—22no—
234C3×Q8—22no—
235C3×Q8—41no—
236C3×Q8—41no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—22no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4C4×D411yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12411yescharacteristic
250C2×C3×C4#12711yescharacteristic
251C2×C3×C4C2×M1611yescharacteristic
252C2×C3×C4C2×M1611yescharacteristic
253C2×C3×C4#13111yescharacteristic
254C2×C3×C4#13111yescharacteristic
255C2×C3×C4#13511yescharacteristic
256C2×C3×C4#13511yescharacteristic
257C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
258C2×C3×C4#13711yescharacteristic
259C2×Dic3—21no—
260C2×Dic3—21no—
261C2×Dic3—21no—
262C2×Dic3—21no—
263C2×Dic3—21no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3#10412yes—
274D12—22no—
275D12—22no—
276D12—41no—
277D12—41no—
278C4×D3—81no—
279C4×D3—81no—
280C4×D3—81no—
281C4×D3—81no—
282C3×C8—42no—
283C3×C8—42no—
284C3⋊C8—42no—
285C3⋊C8—42no—
286C24—61no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8ΓL(1,𝔽25)12yes—
293C2×D4—32no—
294C2×D4—61no—
295C2×D4—61no—
296C2×D4—61no—
297C2×D4—61no—
298C2×D4—61no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—31no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—31no—
305C22×C4—61no—
306C22×C4—121no—
307C22×C4—121no—
308C22×C4—121no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4—121no—
312C22×C4C2×C4×D311yescharacteristic
313C22×C4D3×D411yescharacteristic
314C22×C4D3×D411yescharacteristic
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—62no—
322C2×C8—62no—
323C2×C8—121no—
324C2×C8—121no—
325C2×C8—121no—
326C2×C8—121no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—61no—
332C4⋊C4—61no—
333C22⋊C4—64no—
334C22⋊C4—64no—
335C22⋊C4—122no—
336C22⋊C4—122no—
337C22⋊C4—121no—
338C22⋊C4—121no—
339C42—41no—
340C42—41no—
341C42—62no—
342C42—61no—
343C42—61no—
344C42—61no—
345C42—61no—
346C42—61no—
347C42—61no—
348C42—61no—
349C42—61no—
350C42—121no—
351C42#23511yescharacteristic
352C42D3×D411yescharacteristic
353C42#23711yescharacteristic
354C42C2×#6711yescharacteristic
355C22×C3#39011yescharacteristic
356C22×C3#40711yescharacteristic
357C22×C3#40811yescharacteristic
358C22×C3#41611yescharacteristic
359C22×C3#41711yescharacteristic
360C22×C3C2×#10411yescharacteristic
361C22×C3#42111yescharacteristic
362C2×D3—42no—
363C2×D3—42no—
364C2×D3—42no—
365C2×D3—81no—
366C2×D3—81no—
367C2×D3—81no—
368C2×D3—81no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—42no—
372C3×C4—81no—
373C3×C4#43511yescharacteristic
374C3×C4#43511yescharacteristic
375C3×C4#43511yescharacteristic
376C3×C4#43511yescharacteristic
377C3×C4#46311yescharacteristic
378C3×C4#46311yescharacteristic
379C3×C4#46411yescharacteristic
380C3×C4#46411yescharacteristic
381Dic3—22no—
382Dic3—41no—
383Dic3—41no—
384Dic3—41no—
385Dic3—81no—
386C23—62no—
387C23—121no—
388C23—121no—
389C23—121no—
390C23—121no—
391C23—121no—
392C23—121no—
393C23D3×C22⋊C411yescharacteristic
394Q8—22no—
395Q8—22no—
396Q8—41no—
397Q8—41no—
398D4—62no—
399D4—62no—
400D4—121no—
401D4—121no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—32no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—61no—
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—61no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—61no—
420C2×C4—61no—
421C2×C4—61no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—241no—
426C2×C4—241no—
427C2×C4—241no—
428C2×C4—241no—
429C2×C4#67⋊C411yescharacteristic
430C2×C4#88411yescharacteristic
431C2×C4#88511yescharacteristic
432C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
433C2×C4#90111yescharacteristic
434C2×C4D3×M1611yescharacteristic
435C2×C4#90711yescharacteristic
436C2×C4#91411yescharacteristic
437C2×C4#91611yescharacteristic
438C2×C4#91811yescharacteristic
439C2×C4#91911yescharacteristic
440C2×C4C4×#6711yescharacteristic
441C2×C4#93911yescharacteristic
442C2×C4#94011yescharacteristic
443C8—42no—
444C8—42no—
445C8—122no—
446C8—122no—
447C2×C3#148811yescharacteristic
448C2×C3#148911yescharacteristic
449C2×C3#149411yescharacteristic
450C2×C3#149511yescharacteristic
451C2×C3#195511yescharacteristic
452C2×C3#198011yescharacteristic
453C2×C3#198311yescharacteristic
454D3—82no—
455C22—122no—
456C22—122no—
457C22—122no—
458C22—241no—
459C22—241no—
460C22—241no—
461C22—241no—
462C22#472611yescharacteristic
463C22#472711yescharacteristic
464C22#480311yescharacteristic
465C22#480411yescharacteristic
466C22D3×#10411yescharacteristic
467C22#482111yescharacteristic
468C22#495711yescharacteristic
469C4—21no—
470C4—21no—
471C4—42no—
472C4—62no—
473C4—81no—
474C4—121no—
475C4—121no—
476C4—121no—
477C4—241no—
478C4#472811yescharacteristic
479C4#473011yescharacteristic
480C4#479811yescharacteristic
481C4#481911yescharacteristic
482C4#511111yescharacteristic
483C4#512711yescharacteristic
484C4#516511yescharacteristic
485C4#518511yescharacteristic
486C3#967611yescharacteristic, complemented, Sylow
487C2—242no—
488C2#6823711yescharacteristic
489C2#6823811yescharacteristic
490C2#6824111yescharacteristic
491C2#6824211yescharacteristic
492C2#6991711yescharacteristic
493C2#7023011yescharacteristic
494C2#7035811yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×Q8⋊C8C2
socleC23×C3C2×C22⋊C4
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenterC2×Q8⋊C8D3
hypocenterC3#9676
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ 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. C2×#4535C2semidirect
4. #70441C2semidirect
6. C4×Dic3⋊C8C2semidirect
36. C2×Q8⋊C8D3semidirect
486. C3#9676semidirect

Maximal subgroups

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

subgroupindex
C4×#9472
C2×C3×Q8⋊C82
C2×#45352
#704412
C2×#44832
C4×Dic3⋊C82
C2×#44592
#96763