Finite Group Databasebeta

← #10877380

Subgroups — #10877380

Order 768.

626 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83787C211yesmaximal, characteristic
2#79252C211yesmaximal, characteristic, complemented
3C3×#1814C211yesmaximal, characteristic, complemented
4#72013C211yesmaximal, characteristic, complemented
5#70705C211yesmaximal, characteristic, complemented
6#70664C211yesmaximal, characteristic, complemented
7#70663C211yesmaximal, characteristic, complemented
8#70417C212yesmaximal
9#70405C212yesmaximal, complemented
10#69034C212yesmaximal, complemented
11#69024C212yesmaximal, complemented
12#27694—31nomaximal, Sylow
13C2×C3×#131C2211yescharacteristic
14#5832C2211yescharacteristic
15#5745C2212yes—
16C2×#901C2211yescharacteristic
17C2×C3×#108C2211yescharacteristic
18C3×#427C2211yescharacteristic
19C3×#421C2212yes—
20#5236C2212yes—
21#5234C2211yescharacteristic
22#5129C2211yescharacteristic
23#5127—22no—
24#5124C2212yes—
25#5119C2212yes—
26#5118C2211yescharacteristic
27#5111—22no—
28C2×#819C2211yescharacteristic
29#5106C2211yescharacteristic
30C2×#815C2211yescharacteristic
31#5006C2212yes—
32#4999C2212yes—
33#4893—22no—
34D3×#108—22no—
35C3×#344C2212yes—
36#4559C2212yes—
37#4555C2212yes—
38#4532C2212yes—
39#4531C2212yes—
40#2932—31no—
41#2916—31no—
42#2102—32no—
43#2098—32no—
44#1946—31no—
45#1946—31no—
46#1884—32no—
47#1868—32no—
48#1814—31no—
49#1814D311yescharacteristic, complemented
50#1812—31no—
51C2×C3×#41C2311yescharacteristic
52C2×#235C2311yescharacteristic
53C3×#131—21no—
54C3×#131—21no—
55C3×#131—21no—
56C3×#131—21no—
57C2×C3×M16—21no—
58C2×C3×M16C2312yes—
59C2×C3×M16C2311yescharacteristic
60C22×C3×C8—21no—
61C22×C3×C8C2311yescharacteristic
62C3×#117C2312yes—
63#947Q811yescharacteristic
64#944Q811yescharacteristic
65#940Q811yescharacteristic
66#939—22no—
67#936Q811yescharacteristic
68C4×#67—22no—
69#926C2312yes—
70C2×C4×Dic3C2311yescharacteristic
71C2×#208C2312yes—
72C2×#208C2311yescharacteristic
73C22×C3⋊C8C2311yescharacteristic
74#907—42no—
75D3×M16—24no—
76#901—21no—
77#901—21no—
78#901—21no—
79#901—21no—
80#901—42no—
81C2×ΓL(1,𝔽25)D412yes—
82C2×D3×C8D412yes—
83C3×#108—21no—
84C3×#108—21no—
85C3×#107—21no—
86C3×#106—21no—
87C3×#104—42no—
88#837—42no—
89#830—22no—
90#822—24no—
91(C2×D3)⋊C8—22no—
92#819—21no—
93#819—21no—
94#818—21no—
95#817—21no—
96#815—21no—
97#815—21no—
98Dic3⋊C8—22no—
99Dic3⋊C8—42no—
100#806—24no—
101C4⋊(C3⋊C8)—42no—
102#568—32no—
103C2×#131—31no—
104C2×#131C2×D311yescharacteristic
105#532—31no—
106#435—62no—
107#435—62no—
108#430—32no—
109#430—62no—
110C2×#108—31no—
111C2×#108—62no—
112C2×#108C2×D311yescharacteristic
113#427—31no—
114#427C2×D311yescharacteristic
115#424—32no—
116#424—32no—
117#423—31no—
118#421—32no—
119#421C2×D312yes—
120#410—32no—
121#408—31no—
122#408—31no—
123C2×#97—31no—
124#344—32no—
125#344C2×D312yes—
126#339—32no—
127#339—32no—
128H(3,ℤ/4)—32no—
129C3×#41—21no—
130C3×#41—21no—
131C3×#41—21no—
132C3×#41—21no—
133C2×C3×Q8C2×Q811yescharacteristic
134C2×C3×D4—21no—
135C2×C3×D4C2×Q811yescharacteristic
136C22×C3×C4—21no—
137C22×C3×C4C2411yescharacteristic
138C2×#67—22no—
139C2×#67#4112yes—
140C22×Dic3—21no—
141#235—21no—
142#235—21no—
143#235—21no—
144#235—21no—
145C2×D12C2×Q811yescharacteristic
146C2×C4×D3C2×D412yes—
147C2×Dic6C2×Q811yescharacteristic
148C3×M16—22no—
149C3×M16—21no—
150C3×M16—21no—
151C3×M16—22no—
152C3×M16—41no—
153C3×M16—41no—
154C2×C3×C8—21no—
155C2×C3×C8—22no—
156C2×C3×C8—41no—
157C2×C3×C8—41no—
158C2×C3×C8C2×D411yescharacteristic
159C2×C3×C8C2×D411yescharacteristic
160C2×C3×C8#4111yescharacteristic
161C2×C3×C8#4111yescharacteristic
162C2×C3×C8#4111yescharacteristic
163C2×C3×C8#4111yescharacteristic
164C3×C4⋊C4—22no—
165C3×C22⋊C4—22no—
166C3×C42—22no—
167#217—22no—
168#217—22no—
169(C2×D3)⋊C4—22no—
170C4⋊Dic3—22no—
171Dic3⋊C4—22no—
172C4×Dic3—21no—
173C4×Dic3—22no—
174C4×Dic3C2×Q811yescharacteristic
175C4×Dic3C2×Q811yescharacteristic
176#208—21no—
177#208—21no—
178#208—22no—
179#208—22no—
180C2×C3⋊C8—21no—
181C2×C3⋊C8—21no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—21no—
184C2×C3⋊C8C2×D412yes—
185ΓL(1,𝔽25)—22no—
186ΓL(1,𝔽25)—22no—
187ΓL(1,𝔽25)—44no—
188D3×C8—22no—
189D3×C8—22no—
190D3×C8—44no—
191C2×#41—31no—
192C2×#41C22×D311yescharacteristic
193#131—21no—
194#131—21no—
195#131—21no—
196#131—21no—
197#131—61no—
198#131—61no—
199#131—61no—
200#131—61no—
201#131—122no—
202#131—122no—
203C2×M16—21no—
204C2×M16—31no—
205C2×M16—32no—
206C2×M16—32no—
207C2×M16—64no—
208C2×M16C22×D311yescharacteristic
209C2×M16C22×D312yes—
210C22×C8—21no—
211C22×C8—32no—
212C22×C8—31no—
213C22×C8C22×D311yescharacteristic
214C4⋊Q8—31no—
215#127—31no—
216#124—31no—
217#124—31no—
218#121—62no—
219C4×D4—62no—
220#117—32no—
221#117C22×D312yes—
222C2×C42—31no—
223#108—21no—
224#108—21no—
225#108—64no—
226#108—61no—
227#108—61no—
228#107—21no—
229#107—61no—
230#106—21no—
231#106—61no—
232C4⋊C8—62no—
233C4⋊C8—122no—
234C4⋊C8—122no—
235#104—42no—
236#104—122no—
237#99—64no—
238C22⋊C8—62no—
239C22⋊C8—62no—
240#97—61no—
241#97—61no—
242C23×C3—21no—
243C23×C3C2×#4111yescharacteristic
244C22×D3C22⋊Q812yes—
245C3×Q8—21no—
246C3×Q8—21no—
247C3×D4—21no—
248C3×D4—21no—
249C3×D4—41no—
250C3×D4—41no—
251C2×C3×C4—21no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4C22×D411yescharacteristic
256C2×C3×C4C22×Q811yescharacteristic
257C2×C3×C4C2×#4111yescharacteristic
258C2×C3×C4C2×#4111yescharacteristic
259C2×C3×C4H(5,𝔽2)11yescharacteristic
260C2×C3×C4H(5,𝔽2)11yescharacteristic
261#67—22no—
262#67—22no—
263#67—82no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—42no—
267C2×Dic3—81no—
268C2×Dic3C22⋊Q812yes—
269D12—21no—
270D12—21no—
271C4×D3—22no—
272C4×D3—22no—
273Dic6—21no—
274Dic6—21no—
275C3×C8—41no—
276C3×C8—42no—
277C3×C8#12311yescharacteristic
278C3×C8#12311yescharacteristic
279C3×C8#12311yescharacteristic
280C3×C8#12311yescharacteristic
281C3⋊C8—22no—
282C3⋊C8—42no—
283C3⋊C8—41no—
284#41—21no—
285#41—21no—
286#41—21no—
287#41—21no—
288#41—61no—
289#41—61no—
290#41—61no—
291#41—61no—
292C2×Q8—31no—
293C2×Q8D3×Q811yescharacteristic
294C2×D4—21no—
295C2×D4—32no—
296C2×D4—31no—
297C2×D4—62no—
298C2×D4D3×Q811yescharacteristic
299C22×C4—21no—
300C22×C4—32no—
301C22×C4—61no—
302C22×C4C23×D311yescharacteristic
303M16—21no—
304M16—22no—
305M16—21no—
306M16—22no—
307M16—41no—
308M16—41no—
309M16—61no—
310M16—61no—
311M16—62no—
312M16—62no—
313M16—62no—
314M16—62no—
315M16—124no—
316C2×C8—21no—
317C2×C8—22no—
318C2×C8—32no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—62no—
322C2×C8—62no—
323C2×C8—61no—
324C2×C8—61no—
325C2×C8—62no—
326C2×C8—61no—
327C2×C8—124no—
328C2×C8#23511yescharacteristic
329C2×C8#23511yescharacteristic
330C2×C8D3×D411yescharacteristic
331C2×C8D3×D411yescharacteristic
332C2×C8#23711yescharacteristic
333C2×C8#23711yescharacteristic
334C4⋊C4—22no—
335C4⋊C4—62no—
336C4⋊C4—62no—
337C22⋊C4—22no—
338C22⋊C4—62no—
339C22⋊C4—62no—
340C22⋊C4—62no—
341C42—22no—
342C42—31no—
343C42—31no—
344C42—61no—
345C42—62no—
346C22×C3—41no—
347C22×C3—42no—
348C22×C3—41no—
349C22×C3C2×C22⋊Q811yescharacteristic
350C22×C3#55311yescharacteristic
351C22×C3#55311yescharacteristic
352C2×D3—22no—
353C2×D3—42no—
354C3×C4—41no—
355C3×C4—82no—
356C3×C4C2×#12311yescharacteristic
357C3×C4#53511yescharacteristic
358C3×C4#55411yescharacteristic
359C3×C4#55411yescharacteristic
360Dic3—22no—
361Dic3—41no—
362Dic3—41no—
363Dic3—82no—
364C23—21no—
365C23—32no—
366C23D3×#4111yescharacteristic
367Q8—21no—
368Q8—21no—
369Q8—61no—
370Q8—61no—
371D4—21no—
372D4—21no—
373D4—41no—
374D4—41no—
375D4—62no—
376D4—62no—
377D4—61no—
378D4—61no—
379D4—242no—
380C2×C4—21no—
381C2×C4—32no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—61no—
388C2×C4—61no—
389C2×C4—122no—
390C2×C4—241no—
391C2×C4C2×#23511yescharacteristic
392C2×C4C2×D3×D411yescharacteristic
393C2×C4C2×#23711yescharacteristic
394C2×C4#100411yescharacteristic
395C2×C4C2×D3×Q811yescharacteristic
396C2×C4#100911yescharacteristic
397C8—41no—
398C8—42no—
399C8—62no—
400C8—121no—
401C8—122no—
402C8#88311yescharacteristic
403C8#88311yescharacteristic
404C8#88311yescharacteristic
405C8#88311yescharacteristic
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—81no—
410C2×C3#248711yescharacteristic
411D3—42no—
412C22—41no—
413C22—42no—
414C22—41no—
415C22—62no—
416C22—122no—
417C22#553011yescharacteristic
418C22#561411yescharacteristic
419C22D3×C22⋊Q811yescharacteristic
420C4—41no—
421C4—62no—
422C4—82no—
423C4—121no—
424C4—121no—
425C4—242no—
426C4C2×#88311yescharacteristic
427C4#551111yescharacteristic
428C4#560711yescharacteristic
429C4#563511yescharacteristic
430C3#2769411yescharacteristic, complemented, Sylow
431C2—21no—
432C2—21no—
433C2—21no—
434C2—81no—
435C2—122no—
436C2#7391211yescharacteristic
subgroupG/H
centerC4#5607
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1814C2
socleC2×C3#2487
prime radical#5832C22
hypercenter#1814D3
hypocenterC3#27694
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1814

Complements

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

NH ≅ G/Ntype
2. #79252C2semidirect
3. C3×#1814C2semidirect
4. #72013C2semidirect
5. #70705C2semidirect
6. #70664C2semidirect
7. #70663C2semidirect
9. #70405C2semidirect
10. #69034C2semidirect
11. #69024C2semidirect
49. #1814D3semidirect
430. C3#27694semidirect

Maximal subgroups

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

subgroupindex
#837872
#792522
C3×#18142
#720132
#707052
#706642
#706632
#704172
#704052
#690342
#690242
#276943