Finite Group Databasebeta

← #10694274

Subgroups — #10694274

Order 768.

610 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83251C211yesmaximal, characteristic
2C2×C3×#332C211yesmaximal, characteristic, complemented
3C2×#4547C211yesmaximal, characteristic, complemented
4#70445C211yesmaximal, characteristic, complemented
5#70441C211yesmaximal, characteristic, complemented
6C2×#4485C211yesmaximal, characteristic
7C2×#4485C211yesmaximal, characteristic
8#9704—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#947—21no—
11C2×#947—21no—
12C2×#874C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×#815—21no—
16C2×Dic3⋊C8—22no—
17#4988—22no—
18#4988C414yes—
19C42×Dic3C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×#332—22no—
23#4550—21no—
24#4547—22no—
25#4485—22no—
26#4485—22no—
27#2858—31no—
28#1842—31no—
29#1767—31no—
30C2×#332—31no—
31C2×#332D311yescharacteristic, complemented
32C2×#331—31no—
33C2×#331—31no—
34C22×C3×Q8—21no—
35C22×D12—21no—
36C22×D12—21no—
37C22×D12C2×C412yes—
38C22×C3×C8—21no—
39C22×C3×C8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4C2×C412yes—
46C2×C3×C42C2311yescharacteristic
47#947—42no—
48#947—42no—
49#947—42no—
50#947—42no—
51C2×(C2×D3)⋊C4—24no—
52C2×(C2×D3)⋊C4—24no—
53C2×C4×Dic3C2×C412yes—
54C2×C4×Dic3D412yes—
55C2×C4×Dic3D411yescharacteristic
56C2×C4×Dic3D411yescharacteristic
57C22×C3⋊C8—21no—
58C22×C3⋊C8—21no—
59#874—21no—
60#874—21no—
61#874—21no—
62#874—21no—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8D412yes—
67#815—42no—
68Dic3⋊C8—44no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)D412yes—
73C2×C4⋊Q8C2×D311yescharacteristic
74C2×#127—31no—
75C2×#124—61no—
76C2×#124—61no—
77C2×C4⋊C8—31no—
78C2×C4⋊C8—31no—
79C2×C4⋊C8—62no—
80C2×C4⋊C8C2×D311yescharacteristic
81C2×C4⋊C8C2×D311yescharacteristic
82C2×#97—61no—
83#390—34no—
84#390—62no—
85C43—31no—
86#336—61no—
87#332—22no—
88#332—62no—
89#331—62no—
90#331—62no—
91C23×D3—22no—
92C2×C3×Q8—41no—
93C2×C3×Q8—41no—
94C2×C3×Q8—41no—
95C2×C3×Q8—41no—
96C2×C3×Q8—41no—
97C2×C3×Q8—41no—
98C22×C3×C4—22no—
99C22×C3×C4C22×C411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×Dic3C22⋊C412yes—
103C22×Dic3C22⋊C412yes—
104C2×D12—22no—
105C2×D12—22no—
106C2×D12—44no—
107C2×D12—41no—
108C2×D12—41no—
109C2×D12—41no—
110C2×D12—41no—
111C2×D12—41no—
112C2×D12—41no—
113C2×D12—41no—
114C2×D12—41no—
115C2×D12—42no—
116C2×D12—42no—
117C2×C3×C8—22no—
118C2×C3×C8—41no—
119C2×C3×C8—41no—
120C2×C3×C8—41no—
121C2×C3×C8—42no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129C3×C42#4111yescharacteristic
130(C2×D3)⋊C4—48no—
131(C2×D3)⋊C4—48no—
132C4×Dic3—22no—
133C4×Dic3—21no—
134C4×Dic3—21no—
135C4×Dic3—22no—
136C4×Dic3—42no—
137C4×Dic3D812yes—
138C4×Dic3SD1612yes—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—41no—
141C2×C3⋊C8—41no—
142C2×C3⋊C8—41no—
143C2×C3⋊C8—42no—
144C22×Q8—21no—
145C22×D4—32no—
146C22×D4—61no—
147C22×D4—61no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—61no—
151C22×C8—61no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156#127—61no—
157#127—61no—
158#127—61no—
159#127—61no—
160#124—122no—
161#124—122no—
162#124—122no—
163#124—122no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4C4×D312yes—
166C2×C22⋊C4—64no—
167C2×C22⋊C4—64no—
168C2×C42—31no—
169C2×C42—31no—
170C2×C42—32no—
171C2×C42—32no—
172C2×C42C22×D311yescharacteristic
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—32no—
177C4⋊C8—61no—
178C4⋊C8—61no—
179C4⋊C8—61no—
180C4⋊C8—124no—
181C4⋊C8#6712yes—
182#97—122no—
183C23×C3C2×C22⋊C411yescharacteristic
184C22×D3—42no—
185C22×D3—44no—
186C22×D3—42no—
187C22×D3—44no—
188C22×D3—42no—
189C3×Q8—81no—
190C3×Q8—81no—
191C3×Q8—81no—
192C3×Q8—81no—
193C2×C3×C4—22no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4C4×D411yescharacteristic
198C2×C3×C4C4×D411yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12411yescharacteristic
202C2×C3×C4#12711yescharacteristic
203C2×C3×C4C2×D811yescharacteristic
204C2×C3×C4C2×SD1611yescharacteristic
205C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
206C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
207C2×C3×C4AGL(1,ℤ/8)12yes—
208C2×C3×C4#13712yes—
209C2×Dic3—22no—
210C2×Dic3—22no—
211C2×Dic3—22no—
212C2×Dic3—22no—
213C2×Dic3—22no—
214C2×Dic3#10214yes—
215D12—82no—
216D12—82no—
217D12—81no—
218D12—81no—
219D12—81no—
220D12—81no—
221D12—82no—
222D12—81no—
223D12—81no—
224C3×C8—42no—
225C3×C8—81no—
226C3⋊C8—42no—
227C3⋊C8—81no—
228C24—62no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×D4—62no—
236C2×D4—62no—
237C2×D4—124no—
238C2×D4—121no—
239C2×D4—121no—
240C2×D4—121no—
241C2×D4—121no—
242C2×D4—121no—
243C2×D4—121no—
244C2×D4—122no—
245C2×D4—122no—
246C2×D4—121no—
247C2×D4—121no—
248C22×C4—22no—
249C22×C4—32no—
250C22×C4—32no—
251C22×C4C2×C4×D311yescharacteristic
252C22×C4D3×D411yescharacteristic
253C22×C4D3×D411yescharacteristic
254C2×C8—22no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—42no—
258C2×C8—41no—
259C2×C8—62no—
260C2×C8—121no—
261C2×C8—121no—
262C2×C8—122no—
263C2×C8—121no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C22⋊C4—128no—
269C22⋊C4—128no—
270C42—32no—
271C42—32no—
272C42—61no—
273C42—62no—
274C42—61no—
275C42—62no—
276C42—122no—
277C42#23511yescharacteristic
278C42D3×D411yescharacteristic
279C42#23711yescharacteristic
280C42C2×#6711yescharacteristic
281C22×C3#39011yescharacteristic
282C22×C3#41312yes—
283C22×C3C2×#10211yescharacteristic
284C22×C3#41712yes—
285C22×C3#41811yescharacteristic
286C2×D3—84no—
287C2×D3—84no—
288C2×D3—82no—
289C2×D3—82no—
290C2×D3—82no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—82no—
294C3×C4#45911yescharacteristic
295C3×C4#45911yescharacteristic
296C3×C4#46011yescharacteristic
297C3×C4#46011yescharacteristic
298C3×C4#48912yes—
299C3×C4#49611yescharacteristic
300C3×C4#49711yescharacteristic
301Dic3—24no—
302Dic3—42no—
303C23—122no—
304C23—124no—
305C23—122no—
306C23—122no—
307C23—124no—
308C23D3×C22⋊C411yescharacteristic
309Q8—81no—
310Q8—81no—
311Q8—81no—
312Q8—81no—
313D4—242no—
314D4—242no—
315D4—241no—
316D4—241no—
317D4—241no—
318D4—241no—
319D4—241no—
320D4—241no—
321D4—242no—
322C2×C4—22no—
323C2×C4—34no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—62no—
328C2×C4—62no—
329C2×C4—62no—
330C2×C4—62no—
331C2×C4—62no—
332C2×C4#67⋊C411yescharacteristic
333C2×C4#88411yescharacteristic
334C2×C4#88511yescharacteristic
335C2×C4D3×D811yescharacteristic
336C2×C4#91111yescharacteristic
337C2×C4D3×SD1611yescharacteristic
338C2×C4#91411yescharacteristic
339C2×C4#91412yes—
340C2×C4#91812yes—
341C2×C4C4×#6711yescharacteristic
342C2×C4#93911yescharacteristic
343C2×C4#94011yescharacteristic
344C8—42no—
345C8—81no—
346C8—122no—
347C8—241no—
348C2×C3#154712yes—
349C2×C3#155712yes—
350C2×C3#195911yescharacteristic
351C2×C3#196711yescharacteristic
352C2×C3#198211yescharacteristic
353D3—162no—
354C22—242no—
355C22—244no—
356C22—244no—
357C22—242no—
358C22—242no—
359C22#475112yes—
360C22D3×#10211yescharacteristic
361C22#477011yescharacteristic
362C22#480312yes—
363C22#495711yescharacteristic
364C4—21no—
365C4—21no—
366C4—64no—
367C4—82no—
368C4—122no—
369C4#476211yescharacteristic
370C4#478611yescharacteristic
371C4#479412yes—
372C4#515011yescharacteristic
373C4#516311yescharacteristic
374C4#517611yescharacteristic
375C4#519011yescharacteristic
376C3#970411yescharacteristic, complemented, Sylow
377C2—482no—
378C2#6842112yes—
379C2#6842912yes—
380C2#7009911yescharacteristic
381C2#7016111yescharacteristic
382C2#7022711yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#332C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenterC2×#332D3
hypocenterC3#9704
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. C2×C3×#332C2semidirect
3. C2×#4547C2semidirect
4. #70445C2semidirect
5. #70441C2semidirect
31. C2×#332D3semidirect
376. C3#9704semidirect

Maximal subgroups

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

subgroupindex
#832512
C2×C3×#3322
C2×#45472
#704452
#704412
C2×#44852
#97043