Finite Group Databasebeta

← #10840271

Subgroups — #10840271

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3067C211yesmaximal, characteristic
2Dic3×#135C211yesmaximal, characteristic, complemented
3#72115C212yesmaximal, complemented
4#72114C212yesmaximal, complemented
5#72086C211yesmaximal, characteristic, complemented
6#72012C211yesmaximal, characteristic, complemented
7#72008C211yesmaximal, characteristic, complemented
8#70704C211yesmaximal, characteristic, complemented
9#70698C211yesmaximal, characteristic, complemented
10#70132C212yesmaximal, complemented
11#70131C212yesmaximal, complemented
12#24985—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—
28Dic3×Q16—21no—
29Dic3×SD16—22no—
30Dic3×D8—21no—
31#5119C2212yes—
32#5106C2211yescharacteristic
33#5106C2211yescharacteristic
34#5101C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4559C2212yes—
37#4558C2211yescharacteristic
38#4557C2212yes—
39#4557C2212yes—
40#4555C2212yes—
41#4555C2211yescharacteristic
42#3067D311yescharacteristic
43C4×#135—31no—
44#2080—32no—
45#2053—32no—
46#1976—31no—
47#1945—31no—
48#1906—32no—
49#1833—31no—
50#1827—32no—
51#1812—31no—
52#1764—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—21no—
92#837—21no—
93#837—42no—
94#835—22no—
95#835—21no—
96#834—42no—
97#834—42no—
98#832—22no—
99#832—21no—
100#818—21no—
101#818—21no—
102#817—21no—
103#817—21no—
104#815—21no—
105Dic3⋊C8—42no—
106C8×Dic3—21no—
107C8×Dic3—21no—
108C8×Dic3—21no—
109#575C2×D312yes—
110C2×#135C2×D311yescharacteristic
111C2×#131C2×D311yescharacteristic
112C4×#41—32no—
113#473—21no—
114#472—21no—
115#471—22no—
116C4×Q16—61no—
117C4×SD16—62no—
118C4×D8—61no—
119C2×#108C2×D311yescharacteristic
120#427—31no—
121#427—31no—
122#424—32no—
123#421—32no—
124C2×#104—31no—
125#416—31no—
126#416—32no—
127#413C2×D312yes—
128#410—32no—
129#410—32no—
130#405—31no—
131C2×C4×C8—31no—
132#344—32no—
133#341—31no—
134#339—31no—
135#339—32no—
136H(3,ℤ/4)—32no—
137H(3,ℤ/4)—32no—
138C3×#41—21no—
139C3×#41—21no—
140C3×#41—21no—
141C3×#41—22no—
142C3×#41—22no—
143C3×#41—22no—
144C3×#41—22no—
145C3×#41—21no—
146C2×C3×Q8C2×D412yes—
147C2×C3×Q8#4111yescharacteristic
148C2×C3×D4—21no—
149C2×C3×D4—22no—
150C2×C3×D4C2×D412yes—
151C2×C3×D4#4111yescharacteristic
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4C2411yescharacteristic
155C22×Dic3—21no—
156C22×Dic3—42no—
157C3×Q16—21no—
158C3×Q16—21no—
159C3×Q16—42no—
160C3×SD16—22no—
161C3×SD16—22no—
162C3×SD16—42no—
163C3×SD16—42no—
164C3×D8—21no—
165C3×D8—21no—
166C3×D8—42no—
167C3×M16—21no—
168C3×M16—22no—
169C3×M16—21no—
170C3×M16—22no—
171C3×M16—41no—
172C3×M16—41no—
173C2×C3×C8—21no—
174C2×C3×C8—22no—
175C2×C3×C8—41no—
176C2×C3×C8—41no—
177C2×C3×C8C2×D411yescharacteristic
178C2×C3×C8C2×D411yescharacteristic
179C2×C3×C8#4111yescharacteristic
180C2×C3×C8#4111yescharacteristic
181C2×C3×C8#4111yescharacteristic
182C2×C3×C8#4111yescharacteristic
183#217—22no—
184#217—21no—
185#217—22no—
186#217—42no—
187C4⋊Dic3—22no—
188C4⋊Dic3—22no—
189C4⋊Dic3—21no—
190C4⋊Dic3—42no—
191C4×Dic3—21no—
192C4×Dic3—21no—
193C4×Dic3—22no—
194C4×Dic3—22no—
195C4×Dic3—42no—
196C4×Dic3#4111yescharacteristic
197C4×Dic3#4111yescharacteristic
198#208—21no—
199#208—21no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—21no—
204C2×#41C22×D311yescharacteristic
205C2×#41C22×D312yes—
206#137—22no—
207#137—22no—
208AGL(1,ℤ/8)—22no—
209AGL(1,ℤ/8)—22no—
210#135—21no—
211#135—21no—
212#135—21no—
213#135—21no—
214#135—42no—
215C2×Q16#6711yescharacteristic
216C2×SD16#6712yes—
217C2×D8#6711yescharacteristic
218#131—21no—
219#131—21no—
220#131—21no—
221#131—21no—
222C2×M16—21no—
223C2×M16—31no—
224C2×M16C22×D311yescharacteristic
225C2×M16C22×D312yes—
226C22×C8—21no—
227C22×C8—31no—
228C22×C8C22×D311yescharacteristic
229C4×Q8—62no—
230C4×D4—62no—
231C4×D4—122no—
232#117—32no—
233#117—31no—
234#117—32no—
235#117—62no—
236C2×C42—31no—
237C2×C42—62no—
238#108—21no—
239#108—21no—
240#107—61no—
241#107—61no—
242#106—61no—
243#106—61no—
244C4⋊C8—122no—
245#104—61no—
246#104—61no—
247#104—122no—
248#103—61no—
249#103—62no—
250#102—61no—
251#102—62no—
252#101—22no—
253#100—22no—
254#99—122no—
255#99—122no—
256#97—61no—
257C4×C8—61no—
258C4×C8—61no—
259C4×C8—61no—
260C23×C3—22no—
261C23×C3—21no—
262C23×C3H(5,𝔽2)11yescharacteristic
263C3×Q8—22no—
264C3×Q8—21no—
265C3×Q8—21no—
266C3×Q8—22no—
267C3×D4—22no—
268C3×D4—21no—
269C3×D4—21no—
270C3×D4—22no—
271C3×D4—41no—
272C3×D4—42no—
273C3×D4—41no—
274C3×D4—42no—
275C2×C3×C4—22no—
276C2×C3×C4—21no—
277C2×C3×C4—41no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—41no—
281C2×C3×C4C22×D411yescharacteristic
282C2×C3×C4C2×#4111yescharacteristic
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4H(5,𝔽2)11yescharacteristic
287C2×Dic3—21no—
288C2×Dic3—21no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—41no—
293C2×Dic3—84no—
294C3×C8—41no—
295C3×C8—42no—
296C3×C8#12311yescharacteristic
297C3×C8#12311yescharacteristic
298C3×C8#12311yescharacteristic
299C3×C8#12311yescharacteristic
300C3⋊C8—41no—
301C3⋊C8—41no—
302#41—21no—
303#41—22no—
304#41—22no—
305#41—22no—
306#41—22no—
307#41—21no—
308#41—21no—
309#41—21no—
310C2×Q8#23711yescharacteristic
311C2×Q8C2×#6712yes—
312C2×D4—21no—
313C2×D4—22no—
314C2×D4#23711yescharacteristic
315C2×D4C2×#6712yes—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—61no—
319C22×C4—122no—
320C22×C4C23×D311yescharacteristic
321Q16—21no—
322Q16—21no—
323Q16—42no—
324SD16—22no—
325SD16—22no—
326SD16—42no—
327SD16—42no—
328D8—21no—
329D8—21no—
330D8—42no—
331M16—21no—
332M16—22no—
333M16—21no—
334M16—22no—
335M16—41no—
336M16—41no—
337M16—61no—
338M16—61no—
339C2×C8—21no—
340C2×C8—22no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—61no—
344C2×C8—61no—
345C2×C8—61no—
346C2×C8—61no—
347C2×C8#23711yescharacteristic
348C2×C8#23711yescharacteristic
349C2×C8#23711yescharacteristic
350C2×C8#23711yescharacteristic
351C2×C8C2×#6711yescharacteristic
352C2×C8C2×#6711yescharacteristic
353C4⋊C4—61no—
354C4⋊C4—62no—
355C4⋊C4—62no—
356C4⋊C4—122no—
357C22⋊C4—61no—
358C22⋊C4—62no—
359C22⋊C4—62no—
360C22⋊C4—122no—
361C42—31no—
362C42—31no—
363C42—61no—
364C42—61no—
365C42—62no—
366C42—62no—
367C42—122no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—42no—
371C22×C3—82no—
372C22×C3—81no—
373C22×C3#54011yescharacteristic
374C22×C3#54611yescharacteristic
375C22×C3#55311yescharacteristic
376C3×C4—42no—
377C3×C4—41no—
378C3×C4C2×#12311yescharacteristic
379C3×C4#52911yescharacteristic
380C3×C4#54611yescharacteristic
381C3×C4#55511yescharacteristic
382Dic3—22no—
383Dic3—41no—
384Dic3—82no—
385Dic3—81no—
386Dic3—82no—
387C23—22no—
388C23—21no—
389C23#100911yescharacteristic
390Q8—22no—
391Q8—21no—
392Q8—22no—
393Q8—21no—
394D4—22no—
395D4—21no—
396D4—22no—
397D4—21no—
398D4—41no—
399D4—42no—
400D4—41no—
401D4—42no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—122no—
411C2×C4—122no—
412C2×C4—121no—
413C2×C4—122no—
414C2×C4—244no—
415C2×C4C2×#23711yescharacteristic
416C2×C4C2×#23711yescharacteristic
417C2×C4C2×#23711yescharacteristic
418C2×C4#100411yescharacteristic
419C2×C4D3×#4111yescharacteristic
420C2×C4C22×#6711yescharacteristic
421C8—41no—
422C8—42no—
423C8—121no—
424C8—121no—
425C8#93511yescharacteristic
426C8#93511yescharacteristic
427C8#93511yescharacteristic
428C8#93511yescharacteristic
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—82no—
433C2×C3—81no—
434C2×C3#242011yescharacteristic
435C22—41no—
436C22—41no—
437C22—42no—
438C22—82no—
439C22—81no—
440C22#561811yescharacteristic
441C22#561911yescharacteristic
442C22#582911yescharacteristic
443C4—42no—
444C4—41no—
445C4—62no—
446C4—121no—
447C4—242no—
448C4—242no—
449C4—241no—
450C4#560011yescharacteristic
451C4#561611yescharacteristic
452C4C2×#93511yescharacteristic
453C4#580611yescharacteristic
454C3#2498511yescharacteristic, complemented, Sylow
455C2—21no—
456C2—21no—
457C2—21no—
458C2—82no—
459C2—81no—
460C2#8264311yescharacteristic
subgroupG/H
centerC4#5616
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3067C2
socleC2×C3#2420
hypercenter#3067D3
hypocenterC3#24985
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. Dic3×#135C2semidirect
3. #72115C2semidirect
4. #72114C2semidirect
5. #72086C2semidirect
6. #72012C2semidirect
7. #72008C2semidirect
8. #70704C2semidirect
9. #70698C2semidirect
10. #70132C2semidirect
11. #70131C2semidirect
454. C3#24985semidirect

Maximal subgroups

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

subgroupindex
C3×#30672
Dic3×#1352
#721152
#721142
#720862
#720122
#720082
#707042
#706982
#701322
#701312
#249853