Finite Group Databasebeta

← #10712065

Subgroups — #10712065

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×#4907C211yesmaximal, characteristic, complemented
2C2×#4700C211yesmaximal, characteristic
3C2×#4515C211yesmaximal, characteristic, complemented
4C2×#4515C211yesmaximal, characteristic, complemented
5#71432C211yesmaximal, characteristic, complemented
6C3×#1376C211yesmaximal, characteristic, complemented
7#67657C211yesmaximal, characteristic, complemented
8#12980—31nomaximal, Sylow
9C2×#904C411yescharacteristic
10C2×#901C2211yescharacteristic
11C2×#901C2211yescharacteristic
12C2×C3×M32C2211yescharacteristic
13C2×C3×M32C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15C2×#819C411yescharacteristic
16C22×C3⋊C16C2211yescharacteristic
17C22×C3⋊C16C2211yescharacteristic
18C2×#805C411yescharacteristic
19C2×#805C411yescharacteristic
20#4907—22no—
21#4907—22no—
22C2×#798—21no—
23C2×D3×C16—21no—
24#4700—22no—
25#4700—22no—
26#4515—22no—
27#4515—22no—
28(C2×D3)⋊C16—21no—
29(C2×D3)⋊C16—21no—
30C2×#504—31no—
31C2×#443—31no—
32C2×#350—31no—
33C2×#350—31no—
34#2113—31no—
35#1376D311yescharacteristic, complemented
36#1374—31no—
37C2×#235C2×C411yescharacteristic
38C2×#235C2×C411yescharacteristic
39C22×C3×C8C2×C411yescharacteristic
40C22×C3×C8C2311yescharacteristic
41C2×C3×C42C2×C411yescharacteristic
42C2×#208C2×C411yescharacteristic
43C2×#208C2×C411yescharacteristic
44C2×Dic12C811yescharacteristic
45#904—22no—
46#904—22no—
47C2×D24C811yescharacteristic
48C2×#204C811yescharacteristic
49C2×#204C811yescharacteristic
50#901—21no—
51#901—21no—
52#901—21no—
53#901—21no—
54#901—21no—
55#901—21no—
56#901D412yes—
57#901D412yes—
58C2×ΓL(1,𝔽25)—21no—
59C2×ΓL(1,𝔽25)—21no—
60C2×D3×C8—21no—
61C2×D3×C8—21no—
62C3×M32—21no—
63C3×M32—21no—
64C3×M32—21no—
65C3×M32D412yes—
66C2×C3×C16—21no—
67C2×C3×C16—21no—
68C3×C4×C8—21no—
69C3×C4×C8—21no—
70#819—22no—
71C2×C3⋊C16—21no—
72C2×C3⋊C16—21no—
73C2×C3⋊C16—21no—
74C2×C3⋊C16—21no—
75C2×C3⋊C16—21no—
76C2×C3⋊C16D411yescharacteristic
77C2×C3⋊C16D411yescharacteristic
78#805—22no—
79#805—22no—
80#798—24no—
81D3×C16—24no—
82C2×#135—31no—
83C2×#131—31no—
84C2×#131—31no—
85#504—62no—
86#504—62no—
87C2×M32—61no—
88C2×M32C2×D311yescharacteristic
89C2×M32C2×D311yescharacteristic
90C22×C16—31no—
91C22×C16—31no—
92C22×C16—61no—
93#443—62no—
94#443—62no—
95C2×#108—31no—
96C2×#104—31no—
97C2×#104—31no—
98C2×C4×C8C2×D311yescharacteristic
99#350—62no—
100#350—62no—
101C22⋊C16—61no—
102C22⋊C16—61no—
103C22×C3×C4—21no—
104C22×C3×C4C22×C411yescharacteristic
105C2×#67—21no—
106C2×#67—21no—
107#235—21no—
108#235—21no—
109#235—21no—
110#235—21no—
111#235—21no—
112#235—21no—
113#235C22⋊C412yes—
114#235M1612yes—
115C2×D12C2×C811yescharacteristic
116C2×D12C2×C811yescharacteristic
117C2×C4×D3—21no—
118C2×C4×D3—21no—
119C2×Dic6C2×C811yescharacteristic
120C2×Dic6C2×C811yescharacteristic
121C2×C3×C8—21no—
122C2×C3×C8—21no—
123C2×C3×C8C2×C811yescharacteristic
124C2×C3×C8C2×C811yescharacteristic
125C2×C3×C8C2×D411yescharacteristic
126C2×C3×C8C2×D411yescharacteristic
127C2×C3×C8C2×D411yescharacteristic
128C2×C3×C8C2×D411yescharacteristic
129C2×C3×C8#4111yescharacteristic
130C2×C3×C8#4111yescharacteristic
131C3×C42—21no—
132C3×C42—21no—
133#208—21no—
134#208—21no—
135#208—21no—
136#208C22⋊C412yes—
137C2×C3⋊C8—21no—
138C2×C3⋊C8—21no—
139Dic12—22no—
140D24—22no—
141#204—22no—
142#204—22no—
143ΓL(1,𝔽25)—22no—
144ΓL(1,𝔽25)—22no—
145ΓL(1,𝔽25)—41no—
146ΓL(1,𝔽25)—41no—
147D3×C8—22no—
148D3×C8—22no—
149D3×C8—41no—
150D3×C8—41no—
151C3×C16—22no—
152C3×C16—41no—
153C3⋊C16—21no—
154C3⋊C16—21no—
155C3⋊C16—41no—
156C2×#41—31no—
157C2×#41—31no—
158#135—62no—
159#135—62no—
160C2×Q16—31no—
161C2×SD16—31no—
162C2×SD16—31no—
163C2×D8—31no—
164#131—32no—
165#131—32no—
166#131—61no—
167#131—61no—
168#131—61no—
169#131—61no—
170#131—61no—
171#131—61no—
172C2×M16—31no—
173C2×M16—31no—
174C2×M16—61no—
175C2×M16—61no—
176C22×C8—61no—
177C22×C8—61no—
178C22×C8C4×D311yescharacteristic
179C22×C8C22×D311yescharacteristic
180C2×C42C4×D311yescharacteristic
181M32—21no—
182M32—21no—
183M32—21no—
184M32—64no—
185M32D1212yes—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—31no—
189C2×C16—31no—
190C2×C16—64no—
191C2×C16—61no—
192C2×C16—61no—
193C2×C16—61no—
194C2×C16—61no—
195C2×C16—61no—
196#108—62no—
197#104—62no—
198#104—62no—
199C4×C8—21no—
200C4×C8—21no—
201C23×C3#13111yescharacteristic
202C22×D3—21no—
203C22×D3—21no—
204C2×C3×C4—22no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4C2×C22⋊C411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4C22×C811yescharacteristic
211C2×C3×C4C2×M1611yescharacteristic
212C2×C3×C4#13111yescharacteristic
213#67—22no—
214#67—22no—
215#67—41no—
216#67—41no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219D12—21no—
220D12—21no—
221D12—21no—
222D12C22⋊C812yes—
223C4×D3—22no—
224C4×D3—22no—
225C4×D3—41no—
226C4×D3—41no—
227Dic6—21no—
228Dic6—21no—
229Dic6—21no—
230Dic6C22⋊C812yes—
231C3×C8—21no—
232C3×C8—21no—
233C3×C8#12011yescharacteristic
234C3×C8#12111yescharacteristic
235C3×C8#12111yescharacteristic
236C3×C8#12311yescharacteristic
237C3⋊C8—22no—
238C3⋊C8—41no—
239#41—32no—
240#41—32no—
241#41—61no—
242#41—61no—
243#41—61no—
244#41—61no—
245#41—61no—
246#41—61no—
247C2×Q8—31no—
248C2×Q8—31no—
249C2×D4—31no—
250C2×D4—31no—
251C2×D4—61no—
252C2×D4—61no—
253C22×C4—21no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4C2×C4×D311yescharacteristic
257Q16—62no—
258SD16—62no—
259SD16—62no—
260D8—62no—
261M16—32no—
262M16—61no—
263M16—62no—
264M16—62no—
265M16—61no—
266M16—61no—
267M16—121no—
268M16—121no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—61no—
272C2×C8—62no—
273C2×C8—62no—
274C2×C8—61no—
275C2×C8—121no—
276C2×C8—121no—
277C2×C8D3×C811yescharacteristic
278C2×C8D3×C811yescharacteristic
279C2×C8C2×D1211yescharacteristic
280C2×C8#23511yescharacteristic
281C2×C8D3×D411yescharacteristic
282C2×C8D3×D411yescharacteristic
283C2×C8D3×D411yescharacteristic
284C2×C8#23911yescharacteristic
285C42—21no—
286C42—21no—
287C16—22no—
288C16—41no—
289C16—61no—
290C16—61no—
291C16—121no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3#40811yescharacteristic
295C22×C3C8×D411yescharacteristic
296C22×C3#43511yescharacteristic
297C2×D3—22no—
298C2×D3—41no—
299C2×D3—41no—
300C2×D3—41no—
301C2×D3—41no—
302C2×D3—41no—
303C3×C4—42no—
304C3×C4#38611yescharacteristic
305C3×C4C2×C22⋊C811yescharacteristic
306C3×C4C8×D411yescharacteristic
307C3×C4#43511yescharacteristic
308Dic3—22no—
309Dic3—41no—
310C23—61no—
311C23—61no—
312C23#90711yescharacteristic
313Q8—32no—
314Q8—61no—
315Q8—61no—
316Q8—61no—
317D4—32no—
318D4—62no—
319D4—62no—
320D4—61no—
321D4—61no—
322D4—61no—
323D4—121no—
324D4—121no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—61no—
329C2×C4—62no—
330C2×C4—62no—
331C2×C4—61no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4C4×D1211yescharacteristic
335C2×C4D3×C22⋊C411yescharacteristic
336C2×C4#89311yescharacteristic
337C2×C4C2×D3×C811yescharacteristic
338C2×C4#90111yescharacteristic
339C2×C4D3×M1611yescharacteristic
340C8—21no—
341C8—21no—
342C8—62no—
343C8—121no—
344C8#88211yescharacteristic
345C8#88411yescharacteristic
346C8#89411yescharacteristic
347C8#89511yescharacteristic
348C2×C3—21no—
349C2×C3—21no—
350C2×C3#186911yescharacteristic
351C2×C3#217011yescharacteristic
352C2×C3#217111yescharacteristic
353D3—42no—
354D3—81no—
355C22—21no—
356C22—21no—
357C22—62no—
358C22—121no—
359C22—121no—
360C22—121no—
361C22—121no—
362C22—121no—
363C22#470611yescharacteristic
364C22#472711yescharacteristic
365C22#483411yescharacteristic
366C4—42no—
367C4—62no—
368C4—121no—
369C4#466511yescharacteristic
370C4C8×D1211yescharacteristic
371C4D3×C22⋊C811yescharacteristic
372C4#483611yescharacteristic
373C3#1298011yescharacteristic, complemented, Sylow
374C2—21no—
375C2—21no—
376C2—122no—
377C2—241no—
378C2#6866811yescharacteristic
379C2#6930011yescharacteristic
380C2#6930111yescharacteristic
subgroupG/H
centerC2×C8D3×D4
commutatorC2×C3×C4C22×C8
FrattiniC22×C8C22×D3
FittingC3×#1376C2
socleC22×C3C8×D4
prime radicalC2×#904C4
cosocleC22×C3×C8C23
hypercenter#1376D3
hypocenterC3#12980
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ #1376

Complements

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

NH ≅ G/Ntype
1. C2×#4907C2semidirect
3. C2×#4515C2semidirect
4. C2×#4515C2semidirect
5. #71432C2semidirect
6. C3×#1376C2semidirect
7. #67657C2semidirect
35. #1376D3semidirect
373. C3#12980semidirect

Maximal subgroups

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

subgroupindex
C2×#49072
C2×#47002
C2×#45152
#714322
C3×#13762
#676572
#129803