Finite Group Databasebeta

← #10840282

Subgroups — #10840282

Order 768.

642 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
1C3×#3067C211yesmaximal, characteristic
2#83897C211yesmaximal, characteristic, complemented
3#72115C212yesmaximal, complemented
4#72114C212yesmaximal, complemented
5#72086C211yesmaximal, characteristic, complemented
6#72012C211yesmaximal, characteristic, complemented
7#72009C211yesmaximal, characteristic, complemented
8#70705C211yesmaximal, characteristic, complemented
9#70698C211yesmaximal, characteristic, complemented
10#70132C212yesmaximal, complemented
11#70131C212yesmaximal, complemented
12#24986—31nomaximal, Sylow
13C3×#575C2212yes—
14C2×C3×#135C2211yescharacteristic
15C2×C3×#131C2211yescharacteristic
16Dic3×#41C2212yes—
17C3×#473—21no—
18C3×#472—21no—
19C3×#471—22no—
20C2×C3×#108C2211yescharacteristic
21C3×#413C2212yes—
22#5240C2212yes—
23#5240C2212yes—
24#5236C2212yes—
25C2×#837C2211yescharacteristic
26#5233C2211yescharacteristic
27#5233C2212yes—
28#5184—21no—
29#5164—22no—
30#5152—21no—
31#5119C2212yes—
32#5106C2211yescharacteristic
33#5106C2211yescharacteristic
34#5101C2211yescharacteristic
35C2×#815C2211yescharacteristic
36#4559C2212yes—
37#4558C2211yescharacteristic
38#4557C2212yes—
39#4557C2212yes—
40#4555C2212yes—
41#4555C2211yescharacteristic
42#3067D311yescharacteristic
43#2943—31no—
44#2080—32no—
45#2053—32no—
46#1976—31no—
47#1946—31no—
48#1906—32no—
49#1833—31no—
50#1827—32no—
51#1812—31no—
52#1765—31no—
53C2×C3×#41C2312yes—
54C2×C3×#41C2311yescharacteristic
55C3×#137—22no—
56C3×#137—22no—
57C3×AGL(1,ℤ/8)—22no—
58C3×AGL(1,ℤ/8)—22no—
59C3×#135—21no—
60C3×#135—21no—
61C3×#135—21no—
62C3×#135—21no—
63C3×#135—42no—
64C2×C3×Q16D411yescharacteristic
65C2×C3×SD16D412yes—
66C2×C3×D8D411yescharacteristic
67C3×#131—21no—
68C3×#131—21no—
69C3×#131—21no—
70C3×#131—21no—
71C2×C3×M16—21no—
72C2×C3×M16C2311yescharacteristic
73C2×C3×M16C2312yes—
74C22×C3×C8—21no—
75C22×C3×C8C2311yescharacteristic
76Q8×Dic3—22no—
77D4×Dic3—22no—
78D4×Dic3—42no—
79#926—22no—
80#926C2311yescharacteristic
81#926C2312yes—
82#926C2312yes—
83C2×C4×Dic3—22no—
84C2×C4×Dic3C2311yescharacteristic
85C2×#208C2311yescharacteristic
86C22×C3⋊C8C2311yescharacteristic
87C3×#108—21no—
88C3×#108—21no—
89C3×#101—22no—
90C3×#100—22no—
91#837—22no—
92#837—42no—
93#835—22no—
94#835—21no—
95#834—42no—
96#834—42no—
97#832—22no—
98#832—21no—
99#818—21no—
100#818—21no—
101#817—21no—
102#817—21no—
103#815—21no—
104#815—21no—
105#815—21no—
106Dic3⋊C8—42no—
107C8×Dic3—21no—
108#575C2×D312yes—
109C2×#135C2×D311yescharacteristic
110C2×#131C2×D311yescharacteristic
111C4×#41—32no—
112#473—21no—
113#472—21no—
114#471—22no—
115#442—61no—
116#441—61no—
117#440—62no—
118C2×#108C2×D311yescharacteristic
119#427—31no—
120#427—31no—
121#424—32no—
122#421—32no—
123C2×#104—31no—
124#416—31no—
125#416—32no—
126#413C2×D312yes—
127#410—32no—
128#410—32no—
129#405—31no—
130C2×#97—31no—
131#344—32no—
132#341—31no—
133#339—31no—
134#339—32no—
135H(3,ℤ/4)—32no—
136H(3,ℤ/4)—32no—
137C3×#41—21no—
138C3×#41—21no—
139C3×#41—21no—
140C3×#41—22no—
141C3×#41—22no—
142C3×#41—22no—
143C3×#41—22no—
144C3×#41—21no—
145C2×C3×Q8C2×D412yes—
146C2×C3×Q8#4111yescharacteristic
147C2×C3×D4—21no—
148C2×C3×D4—22no—
149C2×C3×D4C2×D412yes—
150C2×C3×D4#4111yescharacteristic
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4C2411yescharacteristic
154C22×Dic3—21no—
155C22×Dic3—42no—
156C3×Q16—21no—
157C3×Q16—21no—
158C3×Q16—42no—
159C3×SD16—22no—
160C3×SD16—22no—
161C3×SD16—42no—
162C3×SD16—42no—
163C3×D8—21no—
164C3×D8—21no—
165C3×D8—42no—
166C3×M16—21no—
167C3×M16—22no—
168C3×M16—21no—
169C3×M16—22no—
170C3×M16—41no—
171C3×M16—41no—
172C2×C3×C8—21no—
173C2×C3×C8—22no—
174C2×C3×C8—41no—
175C2×C3×C8—41no—
176C2×C3×C8C2×D411yescharacteristic
177C2×C3×C8C2×D411yescharacteristic
178C2×C3×C8#4111yescharacteristic
179C2×C3×C8#4111yescharacteristic
180C2×C3×C8#4111yescharacteristic
181C2×C3×C8#4111yescharacteristic
182#217—22no—
183#217—21no—
184#217—22no—
185#217—42no—
186C4⋊Dic3—22no—
187C4⋊Dic3—22no—
188C4⋊Dic3—21no—
189C4⋊Dic3—42no—
190C4×Dic3—21no—
191C4×Dic3—21no—
192C4×Dic3—22no—
193C4×Dic3—22no—
194C4×Dic3—42no—
195C4×Dic3#4112yes—
196#208—21no—
197#208—21no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202C2×#41C22×D311yescharacteristic
203C2×#41C22×D312yes—
204#137—22no—
205#137—22no—
206AGL(1,ℤ/8)—22no—
207AGL(1,ℤ/8)—22no—
208#135—21no—
209#135—21no—
210#135—21no—
211#135—21no—
212#135—42no—
213C2×Q16#6711yescharacteristic
214C2×SD16#6712yes—
215C2×D8#6711yescharacteristic
216#131—21no—
217#131—21no—
218#131—21no—
219#131—21no—
220C2×M16—21no—
221C2×M16—31no—
222C2×M16C22×D311yescharacteristic
223C2×M16C22×D312yes—
224C22×C8—21no—
225C22×C8—31no—
226C22×C8C22×D311yescharacteristic
227C4×Q8—62no—
228C4×D4—62no—
229C4×D4—122no—
230#117—32no—
231#117—31no—
232#117—32no—
233#117—62no—
234C2×C42—31no—
235C2×C42—62no—
236#108—21no—
237#108—21no—
238#107—61no—
239#107—61no—
240#106—61no—
241#106—61no—
242C4⋊C8—122no—
243#104—62no—
244#104—122no—
245#103—61no—
246#103—62no—
247#102—61no—
248#102—62no—
249#101—22no—
250#100—22no—
251#99—122no—
252#99—122no—
253#97—61no—
254#97—61no—
255#97—61no—
256C4×C8—61no—
257C23×C3—22no—
258C23×C3—21no—
259C23×C3H(5,𝔽2)11yescharacteristic
260C3×Q8—22no—
261C3×Q8—21no—
262C3×Q8—21no—
263C3×Q8—22no—
264C3×D4—22no—
265C3×D4—21no—
266C3×D4—21no—
267C3×D4—22no—
268C3×D4—41no—
269C3×D4—42no—
270C3×D4—41no—
271C3×D4—42no—
272C2×C3×C4—22no—
273C2×C3×C4—21no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—41no—
278C2×C3×C4C22×D411yescharacteristic
279C2×C3×C4C2×#4111yescharacteristic
280C2×C3×C4C2×#4111yescharacteristic
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C4C2×#4111yescharacteristic
283C2×C3×C4H(5,𝔽2)11yescharacteristic
284C2×Dic3—22no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—41no—
288C2×Dic3—81no—
289C2×Dic3—84no—
290C3×C8—41no—
291C3×C8—42no—
292C3×C8#12311yescharacteristic
293C3×C8#12311yescharacteristic
294C3×C8#12311yescharacteristic
295C3×C8#12311yescharacteristic
296C3⋊C8—41no—
297C3⋊C8—41no—
298#41—21no—
299#41—22no—
300#41—22no—
301#41—22no—
302#41—22no—
303#41—21no—
304#41—21no—
305#41—21no—
306C2×Q8#23711yescharacteristic
307C2×Q8C2×#6712yes—
308C2×D4—21no—
309C2×D4—22no—
310C2×D4#23711yescharacteristic
311C2×D4C2×#6712yes—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—61no—
315C22×C4—122no—
316C22×C4C23×D311yescharacteristic
317Q16—21no—
318Q16—21no—
319Q16—42no—
320SD16—22no—
321SD16—22no—
322SD16—42no—
323SD16—42no—
324D8—21no—
325D8—21no—
326D8—42no—
327M16—21no—
328M16—22no—
329M16—21no—
330M16—22no—
331M16—41no—
332M16—41no—
333M16—61no—
334M16—61no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—61no—
340C2×C8—61no—
341C2×C8—61no—
342C2×C8—61no—
343C2×C8#23711yescharacteristic
344C2×C8#23711yescharacteristic
345C2×C8#23711yescharacteristic
346C2×C8#23711yescharacteristic
347C2×C8C2×#6711yescharacteristic
348C2×C8C2×#6711yescharacteristic
349C4⋊C4—61no—
350C4⋊C4—62no—
351C4⋊C4—62no—
352C4⋊C4—122no—
353C22⋊C4—61no—
354C22⋊C4—62no—
355C22⋊C4—62no—
356C22⋊C4—122no—
357C42—32no—
358C42—61no—
359C42—61no—
360C42—62no—
361C42—62no—
362C42—122no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—42no—
366C22×C3—82no—
367C22×C3—81no—
368C22×C3#54011yescharacteristic
369C22×C3#54611yescharacteristic
370C22×C3#55311yescharacteristic
371C3×C4—42no—
372C3×C4—41no—
373C3×C4C2×#12311yescharacteristic
374C3×C4#52911yescharacteristic
375C3×C4#54611yescharacteristic
376C3×C4#55511yescharacteristic
377Dic3—42no—
378Dic3—82no—
379Dic3—81no—
380Dic3—82no—
381C23—22no—
382C23—21no—
383C23#100911yescharacteristic
384Q8—22no—
385Q8—21no—
386Q8—22no—
387Q8—21no—
388D4—22no—
389D4—21no—
390D4—22no—
391D4—21no—
392D4—41no—
393D4—42no—
394D4—41no—
395D4—42no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—62no—
403C2×C4—122no—
404C2×C4—121no—
405C2×C4—122no—
406C2×C4—241no—
407C2×C4—244no—
408C2×C4C2×#23711yescharacteristic
409C2×C4C2×#23711yescharacteristic
410C2×C4C2×#23711yescharacteristic
411C2×C4#100411yescharacteristic
412C2×C4D3×#4111yescharacteristic
413C2×C4C22×#6711yescharacteristic
414C8—41no—
415C8—42no—
416C8—121no—
417C8—121no—
418C8#93511yescharacteristic
419C8#93511yescharacteristic
420C8#93511yescharacteristic
421C8#93511yescharacteristic
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—82no—
426C2×C3—81no—
427C2×C3#242011yescharacteristic
428C22—41no—
429C22—41no—
430C22—42no—
431C22—82no—
432C22—81no—
433C22#561811yescharacteristic
434C22#561911yescharacteristic
435C22#582911yescharacteristic
436C4—42no—
437C4—41no—
438C4—122no—
439C4—242no—
440C4—242no—
441C4—241no—
442C4#560011yescharacteristic
443C4#561611yescharacteristic
444C4C2×#93511yescharacteristic
445C4#580611yescharacteristic
446C3#2498611yescharacteristic, complemented, Sylow
447C2—21no—
448C2—21no—
449C2—21no—
450C2—82no—
451C2—81no—
452C2#8264311yescharacteristic
subgroupG/H
centerC4#5616
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3067C2
socleC2×C3#2420
hypercenter#3067D3
hypocenterC3#24986
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #3067

Complements

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

NH ≅ G/Ntype
2. #83897C2semidirect
3. #72115C2semidirect
4. #72114C2semidirect
5. #72086C2semidirect
6. #72012C2semidirect
7. #72009C2semidirect
8. #70705C2semidirect
9. #70698C2semidirect
10. #70132C2semidirect
11. #70131C2semidirect
446. C3#24986semidirect

Maximal subgroups

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

subgroupindex
C3×#30672
#838972
#721152
#721142
#720862
#720122
#720092
#707052
#706982
#701322
#701312
#249863