Finite Group Databasebeta

← #10691044

Subgroups — #10691044

Order 768.

432 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79060C211yesmaximal, characteristic
2#79000C211yesmaximal, characteristic
3#70779C211yesmaximal, characteristic, complemented
4#70755C211yesmaximal, characteristic
5C2×#4486C211yesmaximal, characteristic, complemented
6C3×#1301C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#9168—31nomaximal, Sylow
9C4×#235C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14#5133—21no—
15#5121—21no—
16#5110—21no—
17#5101—21no—
18#4924C2211yescharacteristic
19C2×C4⋊(C3⋊C8)C2211yescharacteristic
20#4851—21no—
21#4850C412yes—
22#4846—21no—
23#4842C412yes—
24#4840—22no—
25#4835—21no—
26#4833—21no—
27#4486—21no—
28#4486—21no—
29#4481—22no—
30#4459—21no—
31#4456—21no—
32#3079—31no—
33#2925—31no—
34#1548—31no—
35C2×#332—31no—
36#1477—31no—
37#1301D311yescharacteristic, complemented
38#1292—31no—
39C2×#235D411yescharacteristic
40C22×Dic6—21no—
41C2×C3×M16—21no—
42C22×C3×C8—21no—
43C2×C3×C42C2311yescharacteristic
44C4×#67—21no—
45C4×#67—21no—
46#926D411yescharacteristic
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×#208—21no—
50C22×C3⋊C8—21no—
51C4×D12C2×C411yescharacteristic
52C4×D12D411yescharacteristic
53#872—21no—
54C42×D3—21no—
55#869—21no—
56#869—21no—
57#869—21no—
58#869C2×C412yes—
59C4×Dic6C2×C411yescharacteristic
60C4×Dic6D411yescharacteristic
61C3×C4⋊C8—21no—
62C3×C4⋊C8—21no—
63C3×C4⋊C8—21no—
64C3×C4⋊C8C2×C412yes—
65C3×C4⋊C8D411yescharacteristic
66C3×C4⋊C8D411yescharacteristic
67C3×C22⋊C8—21no—
68C3×C4×C8—21no—
69#821—22no—
70(C2×D3)⋊C8—22no—
71(C2×D3)⋊C8—22no—
72#818—22no—
73#817—22no—
74#816—22no—
75#815—22no—
76C8×Dic3—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81C2×C4⋊Q8—31no—
82C4×#41—31no—
83#477—32no—
84#476—32no—
85#475—61no—
86#474—61no—
87#433—62no—
88#432—61no—
89#431—61no—
90#431C2×D311yescharacteristic
91#428—61no—
92#423—31no—
93#423C2×D311yescharacteristic
94C2×C4⋊C8—31no—
95C2×C4⋊C8C2×D311yescharacteristic
96#417—61no—
97#408—61no—
98#405—61no—
99#333—62no—
100#332—61no—
101#332—61no—
102Q8⋊C8—61no—
103D4⋊C8—61no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C2×#67—21no—
108C22×Dic3—41no—
109#235—21no—
110#235—21no—
111#235—21no—
112#235—21no—
113C2×D12C22⋊C411yescharacteristic
114C2×C4×D3—21no—
115C2×C4×D3—21no—
116C2×C4×D3—21no—
117C2×Dic6—22no—
118C2×Dic6—41no—
119C2×Dic6—81no—
120C2×Dic6—81no—
121C2×Dic6C22⋊C411yescharacteristic
122C3×M16—41no—
123C3×M16—41no—
124C2×C3×C8—22no—
125C2×C3×C8—21no—
126C2×C3×C8—21no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C3×C42C22×C411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134#217—21no—
135(C2×D3)⋊C4—21no—
136(C2×D3)⋊C4—21no—
137C4⋊Dic3—22no—
138C4⋊Dic3—22no—
139C4⋊Dic3—41no—
140C4⋊Dic3—41no—
141C4⋊Dic3C22⋊C411yescharacteristic
142C4⋊Dic3C22⋊C411yescharacteristic
143Dic3⋊C4—21no—
144Dic3⋊C4—21no—
145C4×Dic3—21no—
146C4×Dic3—21no—
147C4×Dic3—21no—
148#208—41no—
149#208—41no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—81no—
153C2×#41—31no—
154C22×Q8—61no—
155C2×M16—21no—
156C2×M16—61no—
157C22×C8—21no—
158C22×C8—61no—
159C4⋊Q8—32no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4×Q8—31no—
164C4×Q8—31no—
165C4×D4—31no—
166C4×D4—31no—
167C4×D4—61no—
168C4×D4—61no—
169#117—31no—
170#117—61no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C42—61no—
174C2×C42C22×D311yescharacteristic
175#107—62no—
176#106—62no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—32no—
181C4⋊C8—61no—
182C4⋊C8—61no—
183C4⋊C8—61no—
184C4⋊C8C4×D312yes—
185C4⋊C8D1211yescharacteristic
186C4⋊C8D1211yescharacteristic
187#103—62no—
188#102—62no—
189C22⋊C8—21no—
190C22⋊C8—62no—
191C22⋊C8—62no—
192#97—62no—
193C4×C8—21no—
194C4×C8—62no—
195C23×C3#12311yescharacteristic
196C22×D3—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—41no—
200C2×C3×C4C2×C22⋊C411yescharacteristic
201C2×C3×C4C4×D411yescharacteristic
202C2×C3×C4C4×D411yescharacteristic
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#13511yescharacteristic
207C2×C3×C4#13511yescharacteristic
208C2×C3×C4#13711yescharacteristic
209C2×C3×C4#13711yescharacteristic
210#67—41no—
211#67—41no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—42no—
216C2×Dic3—82no—
217D12—21no—
218D12—21no—
219C4×D3—41no—
220C4×D3—41no—
221C4×D3—41no—
222C4×D3—41no—
223C4×D3—81no—
224Dic6—21no—
225Dic6—21no—
226Dic6—81no—
227Dic6—81no—
228Dic6—82no—
229C3×C8—42no—
230C3×C8—41no—
231C3×C8—41no—
232C3×C8—81no—
233C3⋊C8—42no—
234C3⋊C8—81no—
235#41—61no—
236#41—61no—
237#41—61no—
238#41—61no—
239C2×Q8—31no—
240C2×Q8—62no—
241C2×Q8—121no—
242C2×Q8—241no—
243C2×Q8—241no—
244C2×D4—31no—
245C2×D4—61no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—121no—
250C22×C4#23511yescharacteristic
251C22×C4D3×D411yescharacteristic
252C22×C4#23911yescharacteristic
253M16—41no—
254M16—41no—
255M16—121no—
256M16—121no—
257C2×C8—22no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—62no—
264C2×C8—121no—
265C2×C8—241no—
266C4⋊C4—31no—
267C4⋊C4—31no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—62no—
271C4⋊C4—62no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C22⋊C4—61no—
275C22⋊C4—61no—
276C22⋊C4—61no—
277C42—61no—
278C42—61no—
279C42—61no—
280C42C2×C4×D311yescharacteristic
281C42C2×D1211yescharacteristic
282C42D3×D411yescharacteristic
283C42D3×D411yescharacteristic
284C22×C3—21no—
285C22×C3—21no—
286C22×C3#38611yescharacteristic
287C22×C3#48511yescharacteristic
288C22×C3#48511yescharacteristic
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—41no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4#41611yescharacteristic
296C3×C4#41911yescharacteristic
297C3×C4#45211yescharacteristic
298C3×C4#45211yescharacteristic
299C3×C4#47111yescharacteristic
300C3×C4#47211yescharacteristic
301Dic3—41no—
302Dic3—41no—
303Dic3—41no—
304Dic3—82no—
305C23—61no—
306C23#89411yescharacteristic
307Q8—61no—
308Q8—61no—
309Q8—242no—
310Q8—241no—
311Q8—241no—
312D4—61no—
313D4—61no—
314D4—121no—
315D4—121no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—41no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—122no—
327C2×C4—241no—
328C2×C4—242no—
329C2×C4C4×D1211yescharacteristic
330C2×C4D3×C22⋊C411yescharacteristic
331C2×C4#88211yescharacteristic
332C2×C4#88411yescharacteristic
333C2×C4#89311yescharacteristic
334C2×C4#89511yescharacteristic
335C2×C4#91211yescharacteristic
336C2×C4#91511yescharacteristic
337C2×C4#91611yescharacteristic
338C2×C4#91811yescharacteristic
339C8—42no—
340C8—41no—
341C8—41no—
342C8—81no—
343C8—122no—
344C8—241no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3#187311yescharacteristic
348C2×C3#190211yescharacteristic
349C2×C3#190711yescharacteristic
350D3—81no—
351C22—21no—
352C22—21no—
353C22—121no—
354C22—121no—
355C22—121no—
356C22#466511yescharacteristic
357C22#486811yescharacteristic
358C22#488811yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—121no—
363C4—121no—
364C4—121no—
365C4—242no—
366C4#477211yescharacteristic
367C4#478311yescharacteristic
368C4#480211yescharacteristic
369C4#480811yescharacteristic
370C4#489711yescharacteristic
371C4#489811yescharacteristic
372C3#916811yescharacteristic, complemented, Sylow
373C2—21no—
374C2—21no—
375C2—241no—
376C2#6887911yescharacteristic
377C2#6894811yescharacteristic
378C2#6903711yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1301C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1301D3
hypocenterC3#9168
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
3. #70779C2semidirect
5. C2×#4486C2semidirect
6. C3×#1301C2semidirect
7. #67711C2semidirect
37. #1301D3semidirect
372. C3#9168semidirect

Maximal subgroups

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

subgroupindex
#790602
#790002
#707792
#707552
C2×#44862
C3×#13012
#677112
#91683