Finite Group Databasebeta

← #10691740

Subgroups — #10691740

Order 768.

492 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79305C211yesmaximal, characteristic
2#79008C211yesmaximal, characteristic, complemented
3#70521C211yesmaximal, characteristic
4#70509C211yesmaximal, characteristic, complemented
5C2×#4485C211yesmaximal, characteristic
6C3×#1301C211yesmaximal, characteristic, complemented
7#67744C211yesmaximal, characteristic, complemented
8#9274—31nomaximal, Sylow
9C2×#874C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×#823—21no—
15C2×#823—21no—
16#5102—21no—
17#5101—21no—
18C2×C4⋊(C3⋊C8)C2211yescharacteristic
19C4×#208C2211yescharacteristic
20#4848—22no—
21#4831—21no—
22#4830—21no—
23#4828—22no—
24#4731C414yes—
25#4694—21no—
26#4693—21no—
27#4486—21no—
28#4485—21no—
29#4485—21no—
30#4481—21no—
31#3208—31no—
32#2962—31no—
33#1965—31no—
34#1550—31no—
35C2×#331—31no—
36#1301D311yescharacteristic, complemented
37#1283—31no—
38C22×D12—21no—
39C22×D12—21no—
40C22×D12C2×C412yes—
41C2×C3×M16—21no—
42C22×C3×C8—21no—
43C2×C3×C42C2311yescharacteristic
44#926—21no—
45#924—21no—
46#924C2×C412yes—
47C2×#208D411yescharacteristic
48C2×#208D411yescharacteristic
49C22×C3⋊C8—21no—
50C2×#204—24no—
51#874—21no—
52#874—21no—
53#874—21no—
54#874—21no—
55#870—21no—
56#869D411yescharacteristic
57#869D411yescharacteristic
58C4×Dic6—21no—
59C4×Dic6D412yes—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×C4⋊C8—21no—
63C3×C4⋊C8—21no—
64C3×C4⋊C8D411yescharacteristic
65C3×C4⋊C8D411yescharacteristic
66C3×C22⋊C8C2×C412yes—
67C3×C4×C8—21no—
68#823—42no—
69#823—42no—
70#821—24no—
71#815—22no—
72Dic3⋊C8—22no—
73Dic3⋊C8—22no—
74C8×Dic3—22no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77#803—21no—
78C4×C3⋊C8—21no—
79#533—31no—
80C2×#127—31no—
81#492—61no—
82#486—61no—
83#460—62no—
84#450—34no—
85#446—62no—
86#446—61no—
87C8×Q8—61no—
88#431C2×D311yescharacteristic
89#424—61no—
90#423C2×D311yescharacteristic
91C2×C4⋊C8—31no—
92C2×C4⋊C8C2×D311yescharacteristic
93C2×#100—61no—
94C2×#100—61no—
95#405—61no—
96C4×M16—31no—
97#333—61no—
98#332—61no—
99#331—61no—
100#331—61no—
101C23×D3—22no—
102C22×C3×C4C22×C411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C2×D12—22no—
106C2×D12—22no—
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—82no—
116C2×D12—81no—
117C2×D12—81no—
118C2×Dic6—21no—
119C2×Dic6C22⋊C412yes—
120C3×M16—41no—
121C3×M16—41no—
122C2×C3×C8—22no—
123C2×C3×C8—21no—
124C2×C3×C8—21no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C2×C3×C8—41no—
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132#217—22no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3C22⋊C412yes—
137Dic3⋊C4—22no—
138Dic3⋊C4—22no—
139C4×Dic3—22no—
140#208—21no—
141#208—21no—
142#208—21no—
143#208SD1612yescomplemented
144C2×C3⋊C8—21no—
145C2×C3⋊C8—21no—
146C2×C3⋊C8—41no—
147C2×C3⋊C8—81no—
148#204—84no—
149C22×D4—32no—
150C22×D4—61no—
151C22×D4—61no—
152C2×SD16—64no—
153C2×M16—21no—
154C2×M16—31no—
155C2×M16—31no—
156C22×C8—21no—
157C22×C8—61no—
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160#127—61no—
161#127—61no—
162#127—61no—
163#127—61no—
164#125—61no—
165C22⋊Q8—32no—
166C22⋊Q8—61no—
167C4×Q8—32no—
168C4×Q8—61no—
169#117—61no—
170C2×C42C22×D311yescharacteristic
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8—62no—
176C4⋊C8—62no—
177C4⋊C8—61no—
178C4⋊C8—61no—
179C4⋊C8D1211yescharacteristic
180C4⋊C8D1211yescharacteristic
181#102—64no—
182#100—122no—
183#100—122no—
184C22⋊C8C4×D312yes—
185#97—62no—
186#97—61no—
187C4×C8—21no—
188C4×C8—62no—
189C4×C8—61no—
190C23×C3C4×D411yescharacteristic
191C22×D3—42no—
192C22×D3—42no—
193C22×D3—42no—
194C22×D3—82no—
195C22×D3—82no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—41no—
199C2×C3×C4C2×C22⋊C411yescharacteristic
200C2×C3×C4C4×D411yescharacteristic
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12311yescharacteristic
205C2×C3×C4C2×SD1611yescharacteristic
206C2×C3×C4#13511yescharacteristic
207C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
208C2×C3×C4#13711yescharacteristic
209C2×Dic3—22no—
210C2×Dic3—22no—
211D12—44no—
212D12—82no—
213D12—82no—
214D12—81no—
215D12—81no—
216D12—81no—
217D12—81no—
218D12—81no—
219D12—81no—
220Dic6—41no—
221Dic6—41no—
222Dic6—42no—
223C3×C8—42no—
224C3×C8—41no—
225C3×C8—41no—
226C3×C8—81no—
227C3⋊C8—22no—
228C3⋊C8—41no—
229C3⋊C8—81no—
230C24—62no—
231C2×Q8—32no—
232C2×Q8—61no—
233C2×D4—62no—
234C2×D4—62no—
235C2×D4—121no—
236C2×D4—121no—
237C2×D4—121no—
238C2×D4—121no—
239C2×D4—121no—
240C2×D4—121no—
241C2×D4—121no—
242C2×D4—121no—
243C2×D4—241no—
244C2×D4—241no—
245C2×D4—242no—
246C22×C4C2×C4×D311yescharacteristic
247C22×C4D3×D411yescharacteristic
248C22×C4#23911yescharacteristic
249SD16—244no—
250M16—32no—
251M16—41no—
252M16—41no—
253M16—61no—
254M16—61no—
255M16—61no—
256C2×C8—22no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—61no—
263C2×C8—61no—
264C2×C8—121no—
265C2×C8—241no—
266C4⋊C4—32no—
267C4⋊C4—62no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—62no—
271C4⋊C4—61no—
272C22⋊C4—62no—
273C42—62no—
274C42C2×D1211yescharacteristic
275C42#23511yescharacteristic
276C42D3×D411yescharacteristic
277C42D3×D411yescharacteristic
278C22×C3—21no—
279C22×C3—21no—
280C22×C3#38611yescharacteristic
281C22×C3C4×SD1611yescharacteristic
282C22×C3#44011yescharacteristic
283C2×D3—82no—
284C2×D3—82no—
285C2×D3—82no—
286C2×D3—162no—
287C2×D3—162no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4#44811yescharacteristic
292C3×C4#44911yescharacteristic
293C3×C4#46011yescharacteristic
294C3×C4#46411yescharacteristic
295C3×C4#47111yescharacteristic
296C3×C4#47211yescharacteristic
297Dic3—42no—
298Dic3—42no—
299C23—122no—
300C23—122no—
301C23—122no—
302C23—242no—
303C23—242no—
304C23#89311yescharacteristic
305Q8—122no—
306Q8—121no—
307Q8—121no—
308D4—124no—
309D4—242no—
310D4—242no—
311D4—241no—
312D4—241no—
313D4—241no—
314D4—241no—
315D4—241no—
316D4—241no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—62no—
321C2×C4—62no—
322C2×C4C4×D1211yescharacteristic
323C2×C4D3×C22⋊C411yescharacteristic
324C2×C4#88211yescharacteristic
325C2×C4#88411yescharacteristic
326C2×C4#89411yescharacteristic
327C2×C4#89511yescharacteristic
328C2×C4#91111yescharacteristic
329C2×C4D3×SD1611yescharacteristic
330C2×C4#91611yescharacteristic
331C2×C4#91811yescharacteristic
332C8—42no—
333C8—41no—
334C8—41no—
335C8—62no—
336C8—81no—
337C8—121no—
338C8—241no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3#188011yescharacteristic
342C2×C3#190711yescharacteristic
343C2×C3#191411yescharacteristic
344D3—162no—
345C22—21no—
346C22—21no—
347C22—242no—
348C22—242no—
349C22—242no—
350C22—482no—
351C22—482no—
352C22#466511yescharacteristic
353C22#485211yescharacteristic
354C22#488911yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—122no—
359C4—122no—
360C4#476211yescharacteristic
361C4#478111yescharacteristic
362C4#479811yescharacteristic
363C4#480611yescharacteristic
364C4#489711yescharacteristic
365C4#489811yescharacteristic
366C3#927411yescharacteristic, complemented, Sylow
367C2—21no—
368C2—21no—
369C2—482no—
370C2#6890911yescharacteristic
371C2#6894811yescharacteristic
372C2#6904011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1301C2
socleC22×C3#386
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#1301D3
hypocenterC3#9274
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
2. #79008C2semidirect
4. #70509C2semidirect
6. C3×#1301C2semidirect
7. #67744C2semidirect
36. #1301D3semidirect
143. #208SD16semidirect
366. C3#9274semidirect

Maximal subgroups

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

subgroupindex
#793052
#790082
#705212
#705092
C2×#44852
C3×#13012
#677442
#92743