Finite Group Databasebeta

← #10712067

Subgroups — #10712067

Order 768.

470 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5144C211yesmaximal, characteristic
2C2×#4901C211yesmaximal, characteristic, complemented
3C3×#2113C211yesmaximal, characteristic, complemented
4C2×#4561C211yesmaximal, characteristic, complemented
5C2×#4515C211yesmaximal, characteristic, complemented
6#67810C211yesmaximal, characteristic, complemented
7#67804C211yesmaximal, characteristic, complemented
8#12980—31nomaximal, Sylow
9C2×C3×#131C2211yescharacteristic
10C2×#950C411yescharacteristic
11C2×#901C2211yescharacteristic
12C2×C3×M32C2211yescharacteristic
13C22×C3×C16C2211yescharacteristic
14C2×#837C411yescharacteristic
15#5144—22no—
16#5144—22no—
17C2×#825C411yescharacteristic
18C2×#822C411yescharacteristic
19C2×C8×Dic3C2211yescharacteristic
20C2×#812C2211yescharacteristic
21C22×C3⋊C16C2211yescharacteristic
22#4901—22no—
23#4901—22no—
24C2×#798—21no—
25C2×D3×C16—21no—
26C3×C22⋊C16—21no—
27C3×C22⋊C16—21no—
28#4561—22no—
29#4515—22no—
30C2×#504—31no—
31C2×#443—31no—
32C2×#350—31no—
33C2×#350—31no—
34#2113D311yescharacteristic, complemented
35#1376—31no—
36#1374—31no—
37C2×C3×#41C2×C411yescharacteristic
38C2×#235C2×C411yescharacteristic
39C3×#131—21no—
40C3×#131—21no—
41C3×#131—21no—
42C3×#131—21no—
43C2×C3×M16—21no—
44C2×C3×M16C2×C411yescharacteristic
45C22×C3×C8—21no—
46C22×C3×C8C2311yescharacteristic
47#950—22no—
48#950—22no—
49C2×#216C811yescharacteristic
50C2×#215C811yescharacteristic
51C2×#214C811yescharacteristic
52C2×#213C811yescharacteristic
53C2×C4×Dic3C2×C411yescharacteristic
54C2×#208C2×C411yescharacteristic
55C22×C3⋊C8C2×C411yescharacteristic
56#901—21no—
57#901—21no—
58#901D412yes—
59#901D412yes—
60C2×ΓL(1,𝔽25)—21no—
61C2×D3×C8—21no—
62C3×M32—21no—
63C3×M32—21no—
64C2×C3×C16—21no—
65C2×C3×C16—21no—
66C2×C3×C16—21no—
67C2×C3×C16D411yescharacteristic
68C2×C3×C16D411yescharacteristic
69#837—22no—
70#825—22no—
71#822—22no—
72C8×Dic3—21no—
73C8×Dic3—21no—
74#812—21no—
75#812D412yes—
76C2×C3⋊C16—21no—
77C2×C3⋊C16—21no—
78C2×C3⋊C16—21no—
79C2×C3⋊C16—21no—
80#798—24no—
81D3×C16—24no—
82C2×#135—31no—
83C2×#131—31no—
84C2×#131C2×D311yescharacteristic
85#504—62no—
86#504—62no—
87C2×M32—31no—
88C2×M32—61no—
89C2×M32C2×D311yescharacteristic
90C22×C16—31no—
91C22×C16—61no—
92C22×C16C2×D311yescharacteristic
93#443—62no—
94#443—62no—
95C2×#108—31no—
96C2×#104—31no—
97C2×#104—31no—
98C2×C4×C8—31no—
99#350—62no—
100#350—62no—
101C22⋊C16—21no—
102C22⋊C16—21no—
103C3×#41—21no—
104C3×#41—21no—
105C3×#41—21no—
106C3×#41—21no—
107C2×C3×Q8C2×C811yescharacteristic
108C2×C3×D4—21no—
109C2×C3×D4C2×C811yescharacteristic
110C22×C3×C4—21no—
111C22×C3×C4C22×C411yescharacteristic
112C2×#67—21no—
113C22×Dic3—21no—
114#235—21no—
115#235—21no—
116#235C22⋊C412yes—
117#235M1612yes—
118C2×D12C2×C811yescharacteristic
119C2×C4×D3—21no—
120C2×Dic6C2×C811yescharacteristic
121C3×M16—21no—
122C3×M16—21no—
123C3×M16—41no—
124C3×M16—41no—
125C2×C3×C8—21no—
126C2×C3×C8—41no—
127C2×C3×C8—41no—
128C2×C3×C8C2×D411yescharacteristic
129C2×C3×C8C2×D411yescharacteristic
130C2×C3×C8C2×D411yescharacteristic
131C2×C3×C8C2×D411yescharacteristic
132C2×C3×C8#4111yescharacteristic
133C2×C3×C8#4111yescharacteristic
134#216—22no—
135#215—22no—
136#214—22no—
137#213—22no—
138C4×Dic3—21no—
139C4×Dic3—21no—
140#208—21no—
141#208C22⋊C412yes—
142C2×C3⋊C8—21no—
143C2×C3⋊C8—21no—
144C2×C3⋊C8—21no—
145C2×C3⋊C8C2×C811yescharacteristic
146C2×C3⋊C8C2×C811yescharacteristic
147ΓL(1,𝔽25)—22no—
148ΓL(1,𝔽25)—22no—
149D3×C8—22no—
150D3×C8—22no—
151C3×C16—21no—
152C3×C16—21no—
153C3×C16—41no—
154C3⋊C16—22no—
155C3⋊C16—41no—
156C2×#41—31no—
157C2×#41C4×D311yescharacteristic
158#135—62no—
159#135—62no—
160C2×Q16—31no—
161C2×SD16—31no—
162C2×SD16—31no—
163C2×D8—31no—
164#131—21no—
165#131—21no—
166#131—21no—
167#131—21no—
168#131—32no—
169#131—32no—
170#131—61no—
171#131—61no—
172C2×M16—21no—
173C2×M16—31no—
174C2×M16—61no—
175C2×M16C4×D311yescharacteristic
176C22×C8—21no—
177C22×C8—31no—
178C22×C8—61no—
179C22×C8C22×D311yescharacteristic
180C2×C42—31no—
181M32—21no—
182M32—21no—
183M32—32no—
184M32—64no—
185M32—61no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—64no—
190C2×C16—61no—
191C2×C16—61no—
192C2×C16—61no—
193C2×C16—61no—
194C2×C16#6711yescharacteristic
195C2×C16#6711yescharacteristic
196#108—62no—
197#104—62no—
198#104—62no—
199C4×C8—61no—
200C4×C8—61no—
201C23×C3—21no—
202C23×C3#13111yescharacteristic
203C22×D3—21no—
204C3×Q8—21no—
205C3×Q8—21no—
206C3×D4—21no—
207C3×D4—21no—
208C3×D4—41no—
209C3×D4—41no—
210C2×C3×C4—21no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4C2×C22⋊C411yescharacteristic
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4C22×C811yescharacteristic
217C2×C3×C4C2×M1611yescharacteristic
218C2×C3×C4#13111yescharacteristic
219#67—22no—
220#67—22no—
221C2×Dic3—21no—
222C2×Dic3—22no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225D12—21no—
226D12C22⋊C812yes—
227C4×D3—22no—
228C4×D3—22no—
229Dic6—21no—
230Dic6C22⋊C812yes—
231C3×C8—41no—
232C3×C8#12011yescharacteristic
233C3×C8#12111yescharacteristic
234C3×C8#12111yescharacteristic
235C3×C8#12311yescharacteristic
236C3⋊C8—22no—
237C3⋊C8—21no—
238C3⋊C8—21no—
239#41—21no—
240#41—21no—
241#41—21no—
242#41—21no—
243#41—32no—
244#41—32no—
245#41—61no—
246#41—61no—
247C2×Q8—31no—
248C2×Q8D3×C811yescharacteristic
249C2×D4—21no—
250C2×D4—31no—
251C2×D4—61no—
252C2×D4D3×C811yescharacteristic
253C22×C4—21no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4C2×C4×D311yescharacteristic
257Q16—62no—
258SD16—62no—
259SD16—62no—
260D8—62no—
261M16—21no—
262M16—21no—
263M16—32no—
264M16—41no—
265M16—41no—
266M16—61no—
267M16—62no—
268M16—62no—
269C2×C8—21no—
270C2×C8—31no—
271C2×C8—31no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—61no—
275C2×C8—61no—
276C2×C8—61no—
277C2×C8—62no—
278C2×C8—62no—
279C2×C8#23511yescharacteristic
280C2×C8D3×D411yescharacteristic
281C2×C8D3×D411yescharacteristic
282C2×C8D3×D411yescharacteristic
283C2×C8#23711yescharacteristic
284C2×C8C2×#6711yescharacteristic
285C42—61no—
286C42—61no—
287C16—21no—
288C16—21no—
289C16—41no—
290C16—62no—
291C16—121no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3#40811yescharacteristic
298C22×C3C8×D411yescharacteristic
299C22×C3#43511yescharacteristic
300C2×D3—22no—
301C2×D3—41no—
302C2×D3—41no—
303C3×C4—41no—
304C3×C4#38611yescharacteristic
305C3×C4C2×C22⋊C811yescharacteristic
306C3×C4C8×D411yescharacteristic
307C3×C4#43511yescharacteristic
308Dic3—22no—
309Dic3—42no—
310C23—21no—
311C23—61no—
312C23#90111yescharacteristic
313Q8—21no—
314Q8—21no—
315Q8—32no—
316Q8—61no—
317D4—21no—
318D4—21no—
319D4—32no—
320D4—41no—
321D4—41no—
322D4—62no—
323D4—62no—
324D4—61no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—61no—
329C2×C4—62no—
330C2×C4—62no—
331C2×C4—62no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4D3×C22⋊C411yescharacteristic
335C2×C4#67⋊C411yescharacteristic
336C2×C4C2×D3×C811yescharacteristic
337C2×C4D3×M1611yescharacteristic
338C2×C4#90711yescharacteristic
339C2×C4C4×#6711yescharacteristic
340C8—41no—
341C8—62no—
342C8—61no—
343C8—61no—
344C8#88311yescharacteristic
345C8#88411yescharacteristic
346C8#93711yescharacteristic
347C8#93811yescharacteristic
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—81no—
351C2×C3#186911yescharacteristic
352C2×C3#217011yescharacteristic
353C2×C3#217111yescharacteristic
354D3—42no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—62no—
361C22—121no—
362C22—121no—
363C22#472711yescharacteristic
364C22#472811yescharacteristic
365C22C8×#6711yescharacteristic
366C4—41no—
367C4—62no—
368C4—122no—
369C4D3×C22⋊C811yescharacteristic
370C4#67⋊C811yescharacteristic
371C4#495611yescharacteristic
372C4#512711yescharacteristic
373C3#1298011yescharacteristic, complemented, Sylow
374C2—21no—
375C2—21no—
376C2—81no—
377C2—122no—
378C2#6995711yescharacteristic
379C2#7145211yescharacteristic
380C2#7145311yescharacteristic
subgroupG/H
centerC2×C8C2×#67
commutatorC2×C3×C4C22×C8
FrattiniC22×C8C22×D3
FittingC3×#2113C2
socleC22×C3C8×D4
prime radicalC2×#950C4
cosocleC22×C3×C8C23
hypercenter#2113D3
hypocenterC3#12980
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C8 ◁ C22×C16 ◁ #2113

Complements

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

NH ≅ G/Ntype
2. C2×#4901C2semidirect
3. C3×#2113C2semidirect
4. C2×#4561C2semidirect
5. C2×#4515C2semidirect
6. #67810C2semidirect
7. #67804C2semidirect
34. #2113D3semidirect
373. C3#12980semidirect

Maximal subgroups

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

subgroupindex
C2×#51442
C2×#49012
C3×#21132
C2×#45612
C2×#45152
#678102
#678042
#129803