Finite Group Databasebeta

← #11784151

Subgroups — #11784151

Order 1152.

649 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
1C2×#63815C211yesmaximal, characteristic
2#10592821C211yesmaximal, characteristic, complemented
3C2×C3×#825C211yesmaximal, characteristic
4#10591084C211yesmaximal, characteristic
5C2×#63554C211yesmaximal, characteristic, complemented
6C3×#4558C211yesmaximal, characteristic, complemented
7#10589162C211yesmaximal, characteristic, complemented
8#70370—31nomaximal
9#68927—31nomaximal
10C2×C3×#235C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12#64135C2211yescharacteristic
13C22×C3⋊(C3⋊C8)C2211yescharacteristic
14C2×C3×C4×Dic3C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C2×#3711—21no—
18#63815—22no—
19#63815C414yescomplemented
20C2×D3×C3⋊C8—21no—
21#63642—21no—
22C8×C3⋊Dic3—21no—
23C3×#825C414yescomplemented
24#63566C412yescomplemented
25#63563C412yescomplemented
26#63554—22no—
27C2×#948—31no—
28C2×#907—31no—
29C2×C3×#104—31no—
30#5233—31no—
31C2×#825D311yescharacteristic
32#5122—31no—
33#5122—31no—
34#5122—61no—
35#5113—31no—
36C2×#815—31no—
37C2×#803—31no—
38C3×#341—31no—
39#4558D311yescharacteristic, complemented
40#4555—31no—
41#4523—31no—
42C22×C32×C4C2311yescharacteristic
43C2×C3×#67—21no—
44C22×C3×Dic3—21no—
45C3×#235—21no—
46C3×#235—21no—
47C3×#235C2×C412yes—
48C3×#235C2×C412yes—
49C2×C3×D12C2×C411yescharacteristic
50C2×C3×C4×D3—21no—
51C2×C3×Dic6C2×C411yescharacteristic
52C32×M16—21no—
53C32×M16C2×C412yes—
54C2×C32×C8—21no—
55#3757—21no—
56C4⋊(C3⋊Dic3)C2×C412yes—
57C4×C3⋊Dic3—21no—
58C2×C3⋊(C3⋊C8)C2×C412yes—
59C2×C3⋊(C3⋊C8)D412yes—
60C2×C3⋊(C3⋊C8)D411yescharacteristic
61C2×C3⋊(C3⋊C8)D411yescharacteristic
62C3×C4×Dic3—21no—
63C3×C4×Dic3C2×C412yes—
64C3×#208—21no—
65C3×#208C2×C412yes—
66C2×C3×C3⋊C8—21no—
67C2×C3×C3⋊C8—21no—
68C2×C3×C3⋊C8—21no—
69C2×C3×C3⋊C8C2×C412yes—
70#3711—24no—
71D3×C3⋊C8—24no—
72#1765—91noSylow
73C2×C3×#41—31no—
74C2×#235C2×D311yescharacteristic
75C2×C3×M16—21no—
76C2×C3×M16—31no—
77C2×C3×M16C2×D311yescharacteristic
78C2×C3×M16C2×D311yescharacteristic
79C22×C3×C8—31no—
80C2×C3×C42—31no—
81#948—34no—
82#948—62no—
83#926—31no—
84#926—31no—
85#926—61no—
86C2×C4×Dic3C2×D311yescharacteristic
87C2×#208—61no—
88C2×#208C2×D311yescharacteristic
89C22×C3⋊C8—31no—
90C22×C3⋊C8—31no—
91C22×C3⋊C8—61no—
92C22×C3⋊C8—61no—
93C22×C3⋊C8C2×D311yescharacteristic
94#907—34no—
95#907—62no—
96C2×ΓL(1,𝔽25)—61no—
97C2×D3×C8—61no—
98C3×#104—34no—
99#835—32no—
100#832—32no—
101#825Dic314yescomplemented
102#821—32no—
103#816—32no—
104#815—62no—
105#815—61no—
106#815—61no—
107#815—62no—
108C8×Dic3—61no—
109C8×Dic3—61no—
110C8×Dic3—62no—
111#803—62no—
112C23×C32#4111yescharacteristic
113C22×C3×D3—21no—
114C2×C32×C4C22×C411yescharacteristic
115C2×C32×C4C22×C411yescharacteristic
116C2×C32×C4C22×C411yescharacteristic
117C2×C32×C4C2×D411yescharacteristic
118C2×C32×C4C2×D411yescharacteristic
119C2×C32×C4#4111yescharacteristic
120C2×C3⋊Dic3—22no—
121C3×#67—22no—
122C3×#67—22no—
123C2×C3×Dic3—21no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—42no—
126C3×D12—21no—
127C3×D12C4212yes—
128C3×C4×D3—22no—
129C3×C4×D3—22no—
130C3×Dic6—21no—
131C3×Dic6C4212yes—
132C32×C8—22no—
133C3⋊(C3⋊C8)C22⋊C412yes—
134C3⋊(C3⋊C8)C22⋊C412yes—
135C3×C3⋊C8—22no—
136C3×C3⋊C8—22no—
137C2×#131—91no—
138C2×#104—31no—
139#416—91no—
140#405—91no—
141C2×#97—91no—
142#341—31no—
143#339—91no—
144C3×#41—32no—
145C3×#41—32no—
146C3×#41—61no—
147C3×#41—61no—
148C2×C3×Q8—31no—
149C2×C3×D4—31no—
150C2×C3×D4—61no—
151C22×C3×C4—21no—
152C22×C3×C4—61no—
153C22×C3×C4—61no—
154C22×C3×C4C22×D311yescharacteristic
155C22×C3×C4C22×D311yescharacteristic
156C2×#67—21no—
157C22×Dic3—21no—
158#235—21no—
159#235—21no—
160#235C4×D312yes—
161#235C2×Dic312yes—
162C2×D12C4×D311yescharacteristic
163C2×C4×D3—21no—
164C2×Dic6C4×D311yescharacteristic
165C3×M16—22no—
166C3×M16—21no—
167C3×M16—21no—
168C3×M16—22no—
169C3×M16—32no—
170C3×M16—61no—
171C3×M16C4×D312yes—
172C3×M16C2×Dic312yes—
173C2×C3×C8—22no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8—32no—
177C2×C3×C8—61no—
178C2×C3×C8—61no—
179C2×C3×C8—61no—
180C3×C42—32no—
181C3×C42—61no—
182#217—62no—
183#217—61no—
184#217—61no—
185C4⋊Dic3—32no—
186C4⋊Dic3—32no—
187C4⋊Dic3—62no—
188C4×Dic3—21no—
189C4×Dic3—62no—
190C4×Dic3—61no—
191C4×Dic3—61no—
192C4×Dic3C2×Dic312yes—
193#208—21no—
194#208—64no—
195#208C4×D312yes—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—21no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—32no—
200C2×C3⋊C8—32no—
201C2×C3⋊C8—32no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—32no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—61no—
208C2×C3⋊C8—61no—
209C2×C3⋊C8—62no—
210C2×C3⋊C8—62no—
211C2×C3⋊C8—64no—
212C2×C3⋊C8C4×D312yes—
213ΓL(1,𝔽25)—64no—
214D3×C8—64no—
215C22×C32—22no—
216C22×C32#11711yescharacteristic
217C22×C32C4×D411yescharacteristic
218C22×C32C4×D411yescharacteristic
219C2×C3×D3—22no—
220C2×C3×D3—42no—
221C32×C4C2×C4211yescharacteristic
222C32×C4C2×C22⋊C411yescharacteristic
223C32×C4C4×D411yescharacteristic
224C32×C4C4×D411yescharacteristic
225C3⋊Dic3—42no—
226C3×Dic3—22no—
227C3×Dic3—42no—
228C2×#41—31no—
229#131—94no—
230#131—182no—
231C2×M16—31no—
232C2×M16—181no—
233C2×M16D3211yescharacteristic
234C22×C8—31no—
235C22×C8—91no—
236C22×C8—181no—
237#117—91no—
238C2×C42—31no—
239#104—34no—
240#103—92no—
241#102—92no—
242#97—182no—
243#97—181no—
244C4×C8—181no—
245C23×C3—21no—
246C23×C3—61no—
247C23×C3#23711yescharacteristic
248C23×C3#23911yescharacteristic
249C22×D3—21no—
250C3×Q8—32no—
251C3×Q8—61no—
252C3×D4—32no—
253C3×D4—62no—
254C3×D4—62no—
255C3×D4—61no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—61no—
263C2×C3×C4—62no—
264C2×C3×C4—62no—
265C2×C3×C4—62no—
266C2×C3×C4—122no—
267C2×C3×C4C2×C4×D311yescharacteristic
268C2×C3×C4C2×C4×D311yescharacteristic
269C2×C3×C4C2×C4×D311yescharacteristic
270C2×C3×C4C2×C4×D311yescharacteristic
271C2×C3×C4C2×C4×D311yescharacteristic
272C2×C3×C4D3×D411yescharacteristic
273C2×C3×C4D3×D411yescharacteristic
274C2×C3×C4D3×D411yescharacteristic
275C2×C3×C4D3×D411yescharacteristic
276C2×C3×C4#23711yescharacteristic
277C2×C3×C4#23711yescharacteristic
278C2×C3×C4C22×Dic311yescharacteristic
279#67—22no—
280#67—22no—
281C2×Dic3—21no—
282C2×Dic3—22no—
283C2×Dic3—42no—
284C2×Dic3—64no—
285C2×Dic3—62no—
286C2×Dic3—62no—
287D12—21no—
288D12C4×Dic312yes—
289C4×D3—22no—
290C4×D3—22no—
291Dic6—21no—
292Dic6C4×Dic312yes—
293C3×C8—24no—
294C3×C8—22no—
295C3×C8—22no—
296C3×C8—62no—
297C3×C8—62no—
298C3⋊C8—22no—
299C3⋊C8—22no—
300C3⋊C8—32no—
301C3⋊C8—32no—
302C3⋊C8—32no—
303C3⋊C8—32no—
304C3⋊C8—62no—
305C3⋊C8—62no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32C4×C22⋊C411yescharacteristic
309C2×C32#44412yes—
310C3×D3—42no—
311#41—32no—
312#41—32no—
313#41—61no—
314#41—61no—
315C2×Q8—31no—
316C2×D4—31no—
317C2×D4—61no—
318C22×C4—61no—
319C22×C4—61no—
320C22×C4C2×D3211yescharacteristic
321M16—21no—
322M16—32no—
323M16—61no—
324M16—184no—
325M16D3×Dic312yes—
326C2×C8—21no—
327C2×C8—32no—
328C2×C8—61no—
329C2×C8—61no—
330C2×C8—61no—
331C2×C8—92no—
332C2×C8—92no—
333C2×C8—91no—
334C2×C8—91no—
335C2×C8—184no—
336C4⋊C4—92no—
337C22⋊C4—181no—
338C42—32no—
339C42—61no—
340C42—181no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—42no—
347C22×C3—62no—
348C22×C3—122no—
349C22×C3#87711yescharacteristic
350C22×C3#67⋊C411yescharacteristic
351C22×C3#89211yescharacteristic
352C22×C3#89311yescharacteristic
353C22×C3#89311yescharacteristic
354C22×C3D4×Dic311yescharacteristic
355C2×D3—22no—
356C2×D3—42no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—62no—
362C3×C4—122no—
363C3×C4C42×D311yescharacteristic
364C3×C4D3×C22⋊C411yescharacteristic
365C3×C4D3×C22⋊C411yescharacteristic
366C3×C4#67⋊C411yescharacteristic
367C3×C4#67⋊C411yescharacteristic
368C3×C4#67⋊C411yescharacteristic
369C3×C4C2×C4×Dic311yescharacteristic
370C3×C4D4×Dic311yescharacteristic
371Dic3—22no—
372Dic3—42no—
373Dic3—124no—
374Dic3—122no—
375Dic3—122no—
376C32#176511yescharacteristic, complemented, Sylow
377C23—61no—
378C23#379611yescharacteristic
379Q8—32no—
380Q8—61no—
381D4—32no—
382D4—62no—
383D4—62no—
384D4—61no—
385C2×C4—61no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—122no—
390C2×C4—182no—
391C2×C4C4×D3211yescharacteristic
392C2×C4C4×D3211yescharacteristic
393C2×C4#380211yescharacteristic
394C2×C4C2×D3×Dic311yescharacteristic
395C2×C4#380511yescharacteristic
396C2×C4#381111yescharacteristic
397C8—22no—
398C8—62no—
399C8—62no—
400C8—92no—
401C8—92no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—22no—
407C2×C3—21no—
408C2×C3—41no—
409C2×C3—41no—
410C2×C3—122no—
411C2×C3#466611yescharacteristic
412C2×C3#482312yes—
413C2×C3#489412yes—
414C2×C3Dic3×C22⋊C411yescharacteristic
415D3—42no—
416C22—22no—
417C22—62no—
418C22—122no—
419C22#6387811yescharacteristic
420C22#6388111yescharacteristic
421C22#6389811yescharacteristic
422C4—62no—
423C4—122no—
424C4—362no—
425C4C4×D3×Dic311yescharacteristic
426C4#6390211yescharacteristic
427C4#6392111yescharacteristic
428C4#67⋊Dic311yescharacteristic
429C3—21no—
430C3#6892711yescharacteristic, complemented
431C3#7037011yescharacteristic, complemented
432C2—21no—
433C2—21no—
434C2—122no—
435C2#1059097812yes—
436C2#1059136811yescharacteristic
subgroupG/H
centerC2×C4#3811
commutatorC32×C4C2×C42
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32C4×D4
prime radicalC2×C3×#235C22
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1765
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. #10592821C2semidirect
5. C2×#63554C2semidirect
6. C3×#4558C2semidirect
7. #10589162C2semidirect
19. #63815C4semidirect
23. C3×#825C4semidirect
24. #63566C4semidirect
25. #63563C4semidirect
39. #4558D3semidirect
101. #825Dic3semidirect
376. C32#1765semidirect
430. C3#68927semidirect
431. C3#70370semidirect

Maximal subgroups

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

subgroupindex
C2×#638152
#105928212
C2×C3×#8252
#105910842
C2×#635542
C3×#45582
#105891622
#703703
#689273