Finite Group Databasebeta

← #10690660

Subgroups — #10690660

Order 768.

514 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#909C211yesmaximal, characteristic
2C8×#235C211yesmaximal, characteristic
3#70786C211yesmaximal, characteristic, complemented
4#70755C211yesmaximal, characteristic
5C2×#4456C211yesmaximal, characteristic, complemented
6C3×#1278C211yesmaximal, characteristic, complemented
7#67668C211yesmaximal, characteristic, complemented
8#9096—31nomaximal, Sylow
9C2×#909C411yescharacteristic
10C4×#235C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15#5133C411yescharacteristic
16#5121C411yescharacteristic
17#5120C411yescharacteristic
18C8×#67—22no—
19#4925C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21#4716—21no—
22#4710—21no—
23C4×Dic12—21no—
24C4×#204—21no—
25C8×D12—21no—
26#4685—21no—
27C4×D3×C8—21no—
28C8×Dic6—21no—
29#4469—22no—
30#4459—21no—
31#4456—21no—
32#4456—21no—
33#4456—21no—
34C8×#41—31no—
35C4×#137—31no—
36#1477—31no—
37C2×Q8⋊C8—31no—
38#1461—31no—
39#1283—31no—
40#1278D311yescharacteristic, complemented
41C2×#235C2×C411yescharacteristic
42C22×Dic6C2×C411yescharacteristic
43C2×C3×M16C2×C411yescharacteristic
44C2×C3×M16C2×C411yescharacteristic
45C22×C3×C8—21no—
46C2×C3×C42C2311yescharacteristic
47C4×#67—21no—
48C4×#67—21no—
49#926C2×C411yescharacteristic
50C2×C4⋊Dic3C2×C411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3—21no—
53C22×C3⋊C8—21no—
54#909—22no—
55#909C814yescomplemented
56C2×Dic12—21no—
57C2×#204—21no—
58C2×D3×C8—22no—
59C4×D12D411yescharacteristic
60C4×D12D411yescharacteristic
61#872—21no—
62C42×D3—21no—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6D411yescharacteristic
68C4×Dic6D411yescharacteristic
69C3×C4⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#97—21no—
72C3×C4×C8—21no—
73C3×C4×C8—21no—
74C3×C4×C8—21no—
75C3×C4×C8D411yescharacteristic
76C3×C4×C8D411yescharacteristic
77#830D412yes—
78#821—21no—
79(C2×D3)⋊C8—22no—
80#818—21no—
81#817—21no—
82#816—21no—
83#816—21no—
84#816—21no—
85Dic3⋊C8—22no—
86C8×Dic3—22no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4×C3⋊C8—21no—
91C2×#137—31no—
92C4×#41—31no—
93C2×C4×Q8—31no—
94C8×Q8—61no—
95#441—61no—
96#440—61no—
97C4×Q16—61no—
98C4×SD16—61no—
99C8×D4—61no—
100C8×D4—62no—
101#431—31no—
102#431—61no—
103#431C2×D311yescharacteristic
104#428—31no—
105C2×C4⋊C8—31no—
106#419—31no—
107#417—31no—
108C4×M16C2×D311yescharacteristic
109C2×C4×C8—61no—
110C2×C4×C8C2×D311yescharacteristic
111Q8⋊C8—61no—
112Q8⋊C8—61no—
113Q8⋊C8—61no—
114D4⋊C8—61no—
115#324—62no—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C2×#67—21no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122#235—21no—
123#235—21no—
124#235C2×C812yes—
125#235M1612yes—
126C2×D12C22⋊C411yescharacteristic
127C2×C4×D3—21no—
128C2×C4×D3—21no—
129C2×C4×D3—21no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6—21no—
135C2×Dic6C22⋊C411yescharacteristic
136C2×Dic6C2×C812yescomplemented
137C3×M16—21no—
138C3×M16—21no—
139C3×M16—21no—
140C3×M16C2×C812yescomplemented
141C2×C3×C8—21no—
142C2×C3×C8—21no—
143C2×C3×C8—21no—
144C2×C3×C8—21no—
145C2×C3×C8—42no—
146C2×C3×C8—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151#217—21no—
152(C2×D3)⋊C4—21no—
153(C2×D3)⋊C4—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3C22⋊C411yescharacteristic
157C4⋊Dic3C22⋊C411yescharacteristic
158Dic3⋊C4—21no—
159Dic3⋊C4—21no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162C4×Dic3—21no—
163C4×Dic3—21no—
164C4×Dic3—21no—
165C4×Dic3—41no—
166C4×Dic3—41no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8—81no—
170Dic12—24no—
171#204—24no—
172D3×C8—44no—
173C2×#41—31no—
174C22×Q8—31no—
175#137—34no—
176#137—62no—
177C2×Q16—61no—
178C2×SD16—61no—
179C2×M16C4×D311yescharacteristic
180C2×M16C4×D311yescharacteristic
181C22×C8—21no—
182C22×C8—62no—
183C22×C8—61no—
184C4×Q8—31no—
185C4×Q8—31no—
186C4×Q8—61no—
187C4×Q8—61no—
188C4×Q8—61no—
189C4×Q8—61no—
190C4×D4—31no—
191C4×D4—31no—
192C4×D4—61no—
193C4×D4—61no—
194#117—31no—
195#117—61no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—61no—
198C2×C42—61no—
199C2×C42—61no—
200C2×C42C22×D311yescharacteristic
201#107—61no—
202#106—61no—
203C4⋊C8—21no—
204C4⋊C8—62no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208#103—61no—
209#103—61no—
210#103—61no—
211#102—61no—
212C22⋊C8—21no—
213C22⋊C8—32no—
214C22⋊C8—62no—
215#97—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—62no—
220C4×C8—61no—
221C4×C8D1211yescharacteristic
222C4×C8D1211yescharacteristic
223C23×C3C4×D411yescharacteristic
224C22×D3—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4C2×C22⋊C411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12311yescharacteristic
234C2×C3×C4C22×C811yescharacteristic
235C2×C3×C4C2×M1611yescharacteristic
236C2×C3×C4#13111yescharacteristic
237C2×C3×C4#13111yescharacteristic
238#67—22no—
239#67—22no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—22no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—81no—
248D12—21no—
249D12C22⋊C812yes—
250C4×D3—22no—
251C4×D3—22no—
252C4×D3—42no—
253C4×D3—41no—
254C4×D3—41no—
255Dic6—21no—
256Dic6—22no—
257Dic6—22no—
258Dic6—41no—
259Dic6—41no—
260Dic6C22⋊C812yes—
261C3×C8—22no—
262C3×C8—22no—
263C3×C8—41no—
264C3×C8—41no—
265C3×C8—81no—
266C3⋊C8—42no—
267C3⋊C8—81no—
268#41—32no—
269#41—32no—
270#41—61no—
271#41—61no—
272C2×Q8—31no—
273C2×Q8—32no—
274C2×Q8—61no—
275C2×Q8—61no—
276C2×Q8—61no—
277C2×Q8—61no—
278C2×Q8—61no—
279C2×D4—31no—
280C2×D4—61no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4C2×C4×D311yescharacteristic
287C22×C4D3×D411yescharacteristic
288C22×C4#23911yescharacteristic
289Q16—64no—
290SD16—64no—
291M16—21no—
292M16—21no—
293M16—21no—
294M16D3×C812yescomplemented
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—62no—
302C2×C8—124no—
303C2×C8—121no—
304C2×C8—241no—
305C4⋊C4—31no—
306C4⋊C4—31no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—121no—
312C4⋊C4—121no—
313C22⋊C4—61no—
314C22⋊C4—61no—
315C22⋊C4—61no—
316C42—61no—
317C42—61no—
318C42—61no—
319C42—121no—
320C42—121no—
321C42C2×D1211yescharacteristic
322C42#23511yescharacteristic
323C42D3×D411yescharacteristic
324C42D3×D411yescharacteristic
325C22×C3—21no—
326C22×C3—21no—
327C22×C3#38611yescharacteristic
328C22×C3C8×D411yescharacteristic
329C22×C3#43511yescharacteristic
330C2×D3—22no—
331C2×D3—41no—
332C2×D3—41no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4C2×C22⋊C811yescharacteristic
337C3×C4#40811yescharacteristic
338C3×C4C8×D411yescharacteristic
339C3×C4#43511yescharacteristic
340C3×C4#47111yescharacteristic
341C3×C4#47311yescharacteristic
342Dic3—22no—
343Dic3—42no—
344Dic3—41no—
345Dic3—41no—
346Dic3—81no—
347C23—61no—
348C23#89311yescharacteristic
349Q8—32no—
350Q8—61no—
351Q8—62no—
352Q8—62no—
353Q8—121no—
354Q8—121no—
355D4—32no—
356D4—62no—
357D4—62no—
358D4—61no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—61no—
363C2×C4—62no—
364C2×C4—62no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—62no—
368C2×C4—122no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—241no—
375C2×C4C4×D1211yescharacteristic
376C2×C4D3×C22⋊C411yescharacteristic
377C2×C4#88211yescharacteristic
378C2×C4#88411yescharacteristic
379C2×C4#89411yescharacteristic
380C2×C4#89511yescharacteristic
381C2×C4C2×D3×C811yescharacteristic
382C2×C4#90111yescharacteristic
383C2×C4D3×M1611yescharacteristic
384C2×C4#90711yescharacteristic
385C8—22no—
386C8—22no—
387C8—41no—
388C8—41no—
389C8—81no—
390C8—122no—
391C8—241no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3#186911yescharacteristic
395C2×C3#190611yescharacteristic
396C2×C3#190811yescharacteristic
397D3—42no—
398C22—21no—
399C22—21no—
400C22—62no—
401C22—121no—
402C22—121no—
403C22#466511yescharacteristic
404C22#483411yescharacteristic
405C22#483611yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—62no—
410C4—122no—
411C4—121no—
412C4—121no—
413C4—241no—
414C4C8×D1211yescharacteristic
415C4#470611yescharacteristic
416C4D3×C22⋊C811yescharacteristic
417C4#472711yescharacteristic
418C4#489611yescharacteristic
419C4#489811yescharacteristic
420C3#909611yescharacteristic, complemented, Sylow
421C2—21no—
422C2—21no—
423C2—122no—
424C2#6866811yescharacteristic
425C2#6877811yescharacteristic
426C2#6894911yescharacteristic
subgroupG/H
centerC2×C4#895
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×#1278C2
socleC22×C3#386
prime radicalC2×#235C2×C4
cosocleC2×C3×C42C23
hypercenter#1278D3
hypocenterC3#9096
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1278

Complements

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

NH ≅ G/Ntype
3. #70786C2semidirect
5. C2×#4456C2semidirect
6. C3×#1278C2semidirect
7. #67668C2semidirect
40. #1278D3semidirect
55. #909C8semidirect
136. C2×Dic6C2×C8semidirect
140. C3×M16C2×C8semidirect
294. M16D3×C8semidirect
420. C3#9096semidirect

Maximal subgroups

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

subgroupindex
C4×#9092
C8×#2352
#707862
#707552
C2×#44562
C3×#12782
#676682
#90963