Finite Group Databasebeta

← #11784138

Subgroups — #11784138

Order 1152.

717 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64110C211yesmaximal, characteristic
2C3×#5101C211yesmaximal, characteristic, complemented
3C2×C3×#815C211yesmaximal, characteristic, complemented
4C2×#63570C211yesmaximal, characteristic
5#10591105C211yesmaximal, characteristic
6#10589166C211yesmaximal, characteristic, complemented
7#10589161C211yesmaximal, characteristic, complemented
8#69888—31nomaximal
9#69839—31nomaximal
10C2×#3828C2211yescharacteristic
11C22×C32×C8C2211yescharacteristic
12C2×#3748C2211yescharacteristic
13#64110—22no—
14#64110C414yescomplemented
15C2×#3743—21no—
16C2×C8×C3⋊D3—21no—
17C3×#926C2211yescharacteristic
18C2×C3×C4×Dic3C2211yescharacteristic
19C2×C3×#208C2211yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21C3×#815—22no—
22C3×#815—21no—
23C3×C8×Dic3—21no—
24#63570C414yescomplemented
25#63567C412yescomplemented
26#63564C412yescomplemented
27C2×#901—31no—
28C2×#901—31no—
29C2×#901—61no—
30C3×#405—31no—
31C2×C3×#97—31no—
32C2×#825—31no—
33#5113—31no—
34#5101D311yescharacteristic, complemented
35C2×#815D311yescharacteristic, complemented
36#5002—31no—
37C2×#805—31no—
38#4558—31no—
39#4552—31no—
40#4532—31no—
41#4523—31no—
42C22×C32×C4C2311yescharacteristic
43C2×#638—21no—
44#3828—21no—
45#3828—21no—
46#3828C2×C412yes—
47#3828C2×C412yes—
48C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
49C2×C4×C3⋊D3—21no—
50C2×#634C2×C411yescharacteristic
51C22×C3×Dic3—21no—
52C2×C32×C8C2×C412yes—
53C2×C32×C8D411yescharacteristic
54C2×C32×C8D411yescharacteristic
55C2×C32×C8D412yes—
56#3748—21no—
57#3748C2×C412yes—
58C2×C3⋊(C3⋊C8)—21no—
59#3743—24no—
60C8×C3⋊D3—24no—
61C3×#217—21no—
62C3×C4⋊Dic3C2×C412yes—
63C3×C4×Dic3—21no—
64C3×C4×Dic3—21no—
65C3×C4×Dic3C2×C412yes—
66C3×#208—21no—
67C3×#208C2×C412yes—
68C2×C3×C3⋊C8—21no—
69C2×C3×C3⋊C8—21no—
70C2×C3×C3⋊C8—21no—
71C2×C3×C3⋊C8C2×C412yes—
72#1765—91noSylow
73C2×#235—31no—
74C2×#235—31no—
75C2×#235—61no—
76C2×C3×M16—31no—
77C22×C3×C8—21no—
78C22×C3×C8—31no—
79C22×C3×C8C2×D311yescharacteristic
80C22×C3×C8C2×D311yescharacteristic
81C3×#117—31no—
82C2×C3×C42—31no—
83#926C2×D311yescharacteristic
84C2×C4×Dic3C2×D311yescharacteristic
85C2×#208—31no—
86C2×#208—31no—
87C2×#208—61no—
88C2×#208C2×D311yescharacteristic
89C22×C3⋊C8C2×D311yescharacteristic
90#901—34no—
91#901—34no—
92#901—64no—
93#901—64no—
94#901—62no—
95#901—62no—
96C2×ΓL(1,𝔽25)—62no—
97C2×ΓL(1,𝔽25)—61no—
98C2×ΓL(1,𝔽25)—61no—
99C2×D3×C8—62no—
100C2×D3×C8—61no—
101C2×D3×C8—61no—
102C3×#97—62no—
103C3×#97—61no—
104C3×C4×C8—61no—
105#825—34no—
106#821—32no—
107#816—32no—
108#815—22no—
109#815—21no—
110C8×Dic3—21no—
111#810—32no—
112#809—32no—
113#805—34no—
114C23×C32#4111yescharacteristic
115C22×C3⋊D3—21no—
116C2×C32×C4C22×C411yescharacteristic
117C2×C32×C4C22×C411yescharacteristic
118C2×C32×C4C22×C411yescharacteristic
119C2×C32×C4C2×D411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4#4111yescharacteristic
122#638—22no—
123#638—22no—
124C2×C3⋊Dic3—21no—
125C4⋊(C3⋊D3)—21no—
126C4⋊(C3⋊D3)C4212yes—
127C4×C3⋊D3—22no—
128C4×C3⋊D3—22no—
129#634—21no—
130#634C4212yes—
131C2×C3×Dic3—22no—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—42no—
134C32×C8C22⋊C412yes—
135C32×C8C22⋊C412yes—
136C3⋊(C3⋊C8)—22no—
137C3×C3⋊C8—22no—
138C3×C3⋊C8—22no—
139C2×#131—91no—
140C2×#104—91no—
141#416—91no—
142#405—31no—
143C2×#97—31no—
144#341—91no—
145#339—91no—
146C22×C3×C4—21no—
147C22×C3×C4—61no—
148C22×C3×C4C22×D311yescharacteristic
149C22×C3×C4C22×D311yescharacteristic
150C2×#67—62no—
151C2×#67—61no—
152C2×#67—61no—
153C22×Dic3—21no—
154#235—32no—
155#235—32no—
156#235—32no—
157#235—32no—
158#235—62no—
159#235—62no—
160#235—62no—
161#235—62no—
162#235—61no—
163#235—61no—
164#235—61no—
165#235—61no—
166C2×D12—31no—
167C2×D12—31no—
168C2×D12—61no—
169C2×C4×D3—62no—
170C2×C4×D3—61no—
171C2×C4×D3—61no—
172C2×Dic6—31no—
173C2×Dic6—31no—
174C2×Dic6—61no—
175C3×M16—32no—
176C3×M16—61no—
177C2×C3×C8—22no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—22no—
181C2×C3×C8—32no—
182C2×C3×C8—61no—
183C2×C3×C8—61no—
184C2×C3×C8—61no—
185C2×C3×C8C4×D312yes—
186C2×C3×C8C4×D312yes—
187C2×C3×C8D1211yescharacteristic
188C2×C3×C8D1211yescharacteristic
189C2×C3×C8D1212yes—
190C2×C3×C8#6711yescharacteristic
191C2×C3×C8#6711yescharacteristic
192C2×C3×C8#6712yes—
193C3×C4⋊C4—32no—
194C3×C22⋊C4—61no—
195C3×C42—32no—
196C3×C42—61no—
197C3×C42—61no—
198#217—21no—
199C4⋊Dic3C4×D312yes—
200C4×Dic3—21no—
201C4×Dic3—21no—
202C4×Dic3C4×D312yes—
203#208—21no—
204#208—32no—
205#208—32no—
206#208—62no—
207#208—62no—
208#208—61no—
209#208—61no—
210#208C4×D312yes—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—62no—
215C2×C3⋊C8—61no—
216C2×C3⋊C8—61no—
217C2×C3⋊C8C4×D312yes—
218ΓL(1,𝔽25)—68no—
219ΓL(1,𝔽25)—64no—
220ΓL(1,𝔽25)—64no—
221D3×C8—68no—
222D3×C8—64no—
223D3×C8—64no—
224C22×C32—22no—
225C22×C32#11711yescharacteristic
226C22×C32C4×D411yescharacteristic
227C22×C32C4×D411yescharacteristic
228C2×C3⋊D3—22no—
229C2×C3⋊D3—42no—
230C32×C4C2×C4211yescharacteristic
231C32×C4C2×C22⋊C411yescharacteristic
232C32×C4C4×D411yescharacteristic
233C32×C4C4×D411yescharacteristic
234C3⋊Dic3—22no—
235C3×Dic3—42no—
236C3×Dic3—42no—
237C2×#41—91no—
238#131—94no—
239#131—182no—
240C2×M16—31no—
241C2×M16—91no—
242C2×M16—181no—
243C22×C8—31no—
244C22×C8—181no—
245C22×C8D3211yescharacteristic
246#117—31no—
247C2×C42—31no—
248#104—94no—
249#103—92no—
250#102—92no—
251#97—62no—
252#97—61no—
253C4×C8—61no—
254C23×C3—21no—
255C23×C3#23511yescharacteristic
256C23×C3#23511yescharacteristic
257C22×D3—62no—
258C22×D3—61no—
259C22×D3—61no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—62no—
267C2×C3×C4—62no—
268C2×C3×C4—122no—
269C2×C3×C4C2×C4×D311yescharacteristic
270C2×C3×C4C2×C4×D311yescharacteristic
271C2×C3×C4C2×C4×D311yescharacteristic
272C2×C3×C4C2×C4×D311yescharacteristic
273C2×C3×C4C2×C4×D311yescharacteristic
274C2×C3×C4C2×C4×D311yescharacteristic
275C2×C3×C4C2×D1211yescharacteristic
276C2×C3×C4C2×D1211yescharacteristic
277C2×C3×C4#23511yescharacteristic
278C2×C3×C4#23511yescharacteristic
279C2×C3×C4C2×#6711yescharacteristic
280C2×C3×C4C2×#6711yescharacteristic
281#67—64no—
282#67—64no—
283#67—62no—
284#67—62no—
285#67—62no—
286#67—62no—
287C2×Dic3—22no—
288C2×Dic3—22no—
289C2×Dic3—42no—
290C2×Dic3—62no—
291C2×Dic3—61no—
292C2×Dic3—61no—
293D12—32no—
294D12—32no—
295D12—62no—
296D12—62no—
297D12—61no—
298D12—61no—
299C4×D3—64no—
300C4×D3—64no—
301C4×D3—62no—
302C4×D3—62no—
303C4×D3—62no—
304C4×D3—62no—
305Dic6—32no—
306Dic6—32no—
307Dic6—62no—
308Dic6—62no—
309Dic6—61no—
310Dic6—61no—
311C3×C8—22no—
312C3×C8—22no—
313C3×C8—62no—
314C3×C8—62no—
315C3×C8(C2×D3)⋊C412yes—
316C3×C8(C2×D3)⋊C412yes—
317C3×C8(C2×D3)⋊C412yes—
318C3×C8(C2×D3)⋊C412yes—
319C3⋊C8—22no—
320C3⋊C8—22no—
321C3⋊C8—64no—
322C3⋊C8—62no—
323C3⋊C8—62no—
324C2×C32—21no—
325C2×C32—21no—
326C2×C32C4×C22⋊C411yescharacteristic
327C2×C32#44412yes—
328C3⋊D3—42no—
329#41—92no—
330#41—92no—
331#41—181no—
332#41—181no—
333C2×Q8—91no—
334C2×D4—91no—
335C2×D4—181no—
336C22×C4—61no—
337C22×C4—181no—
338C22×C4C2×D3211yescharacteristic
339M16—32no—
340M16—61no—
341M16—92no—
342M16—181no—
343M16—184no—
344C2×C8—32no—
345C2×C8—61no—
346C2×C8—61no—
347C2×C8—61no—
348C2×C8—181no—
349C2×C8—184no—
350C2×C8#62412yes—
351C2×C8#62612yes—
352C2×C8#62611yescharacteristic
353C2×C8#62611yescharacteristic
354C4⋊C4—32no—
355C22⋊C4—61no—
356C42—32no—
357C42—61no—
358C42—61no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—42no—
365C22×C3#87211yescharacteristic
366C22×C3#87211yescharacteristic
367C22×C3C4×D1211yescharacteristic
368C22×C3C4×D1211yescharacteristic
369C22×C3C4×#6711yescharacteristic
370C22×C3C4×#6711yescharacteristic
371C2×D3—64no—
372C2×D3—62no—
373C2×D3—62no—
374C2×D3—124no—
375C2×D3—122no—
376C2×D3—122no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—122no—
382C3×C4—122no—
383C3×C4C42×D311yescharacteristic
384C3×C4C42×D311yescharacteristic
385C3×C4C4×D1211yescharacteristic
386C3×C4C4×D1211yescharacteristic
387C3×C4C2×(C2×D3)⋊C411yescharacteristic
388C3×C4C2×(C2×D3)⋊C411yescharacteristic
389C3×C4C4×#6711yescharacteristic
390C3×C4C4×#6711yescharacteristic
391Dic3—42no—
392Dic3—42no—
393Dic3—64no—
394Dic3—62no—
395Dic3—62no—
396C32#176511yescharacteristic, complemented, Sylow
397C23—181no—
398C23#379811yescharacteristic
399Q8—92no—
400Q8—181no—
401D4—92no—
402D4—182no—
403D4—182no—
404D4—181no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—122no—
408C2×C4—181no—
409C2×C4—182no—
410C2×C4—182no—
411C2×C4#379811yescharacteristic
412C2×C4C4×D3211yescharacteristic
413C2×C4C4×D3211yescharacteristic
414C2×C4C2×#62411yescharacteristic
415C2×C4C2×#62611yescharacteristic
416C2×C4C2×#62611yescharacteristic
417C8—62no—
418C8—62no—
419C8—182no—
420C8#372212yes—
421C8#372212yes—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—22no—
428C2×C3—41no—
429C2×C3—41no—
430C2×C3#471712yes—
431C2×C3C4×(C2×D3)⋊C411yescharacteristic
432C2×C3C4×(C2×D3)⋊C411yescharacteristic
433C2×C3#514512yes—
434D3—124no—
435D3—122no—
436D3—122no—
437C22—22no—
438C22—182no—
439C22—362no—
440C22#6387411yescharacteristic
441C22C4×#62611yescharacteristic
442C22C4×#62611yescharacteristic
443C4—122no—
444C4—122no—
445C4—182no—
446C4C4×#62411yescharacteristic
447C4C4×#62611yescharacteristic
448C4C4×#62611yescharacteristic
449C4C2×#372211yescharacteristic
450C3—21no—
451C3#6983911yescharacteristic, complemented
452C3#6988811yescharacteristic, complemented
453C2—21no—
454C2—21no—
455C2—362no—
456C2#1059092412yes—
457C2C4×#372211yescharacteristic
subgroupG/H
centerC2×C4C2×#626
commutatorC32×C4C2×C42
FrattiniC22×C4C2×D32
FittingC22×C32×C8C22
socleC22×C32C4×D4
prime radicalC2×#3828C22
cosocleC22×C32×C4C23
hypercenterC22×C8D32
hypocenterC32#1765
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
2. C3×#5101C2semidirect
3. C2×C3×#815C2semidirect
6. #10589166C2semidirect
7. #10589161C2semidirect
14. #64110C4semidirect
24. #63570C4semidirect
25. #63567C4semidirect
26. #63564C4semidirect
34. #5101D3semidirect
35. C2×#815D3semidirect
396. C32#1765semidirect
451. C3#69839semidirect
452. C3#69888semidirect

Maximal subgroups

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

subgroupindex
C2×#641102
C3×#51012
C2×C3×#8152
C2×#635702
#105911052
#105891662
#105891612
#698883
#698393