Finite Group Databasebeta

← #10714600

Subgroups — #10714600

Order 768.

776 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2679C211yesmaximal, characteristic
2#71669C212yesmaximal, complemented
3C2×#4536C212yesmaximal, complemented
4#69642C211yesmaximal, characteristic, complemented
5#67641C211yesmaximal, characteristic, complemented
6#13419—31nomaximal, Sylow
7C2×C3×#120—22no—
8C2×C3×#120C2212yes—
9#5798C2211yescharacteristic
10C3×#394—21no—
11C3×#394—22no—
12C3×#390—21no—
13C3×#390C2211yescharacteristic
14#5247—22no—
15C2×#834C2212yes—
16C2×#833C2211yescharacteristic
17#4964—21no—
18#4536—24no—
19#4527—22no—
20#4523—21no—
21#2679D311yescharacteristic
22C2×AΣL(1,𝔽16)—32no—
23#1889—32no—
24#1856—31no—
25#1393—31no—
26C25×C3D412yes—
27C22×C3×D4—21no—
28C22×C3×D4—21no—
29C22×C3×D4—22no—
30C22×C3×D4C2311yescharacteristic
31C3×#120—22no—
32C3×#120—22no—
33C3×#120—22no—
34C3×#120—22no—
35C3×#120—48no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4D412yes—
43C2×C3×C42D411yescharacteristic
44C2×#217—22no—
45C2×#217—22no—
46C2×#217—42no—
47C2×#217—42no—
48D4×Dic3—41no—
49D4×Dic3—41no—
50#926—21no—
51C2×#208—21no—
52C3×#95—21no—
53#834—22no—
54#834—22no—
55#833—21no—
56#833D412yescomplemented
57#808—41no—
58#807—41no—
59C2×#120—22no—
60C2×#120C2×D312yes—
61#518—31no—
62#428—61no—
63C2×#100—31no—
64C2×#99—32no—
65#394—21no—
66#394—22no—
67#390—21no—
68#390C2×D311yescharacteristic
69#379—62no—
70AΣL(1,𝔽16)—64no—
71#342—62no—
72#339—61no—
73C24×C3—22no—
74C24×C3—22no—
75C24×C3—22no—
76C24×C3—42no—
77C24×C3—42no—
78C24×C3—44no—
79C24×C3—82no—
80C24×C3C2×D412yes—
81C2×C3×D4—24no—
82C2×C3×D4—41no—
83C2×C3×D4—41no—
84C2×C3×D4—44no—
85C2×C3×D4—44no—
86C2×C3×D4—42no—
87C2×C3×D4—42no—
88C2×C3×D4—81no—
89C2×C3×D4—81no—
90C2×C3×D4—82no—
91C2×C3×D4—81no—
92C2×C3×D4C2×D411yescharacteristic
93C2×C3×D4#4111yescharacteristic
94C2×C3×D4#4112yes—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—22no—
98C22×C3×C4C2×D411yescharacteristic
99C22×Dic3—22no—
100C22×Dic3—22no—
101C22×Dic3—42no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—81no—
104C3×C22⋊C4—22no—
105C3×C22⋊C4—22no—
106C3×C22⋊C4—44no—
107C3×C22⋊C4—44no—
108C3×C22⋊C4—44no—
109C3×C22⋊C4—42no—
110C3×C22⋊C4—42no—
111C3×C42—41no—
112#217—42no—
113#217—41no—
114#217—42no—
115#217—42no—
116#217—84no—
117#217—84no—
118C4⋊Dic3—21no—
119C4⋊Dic3—21no—
120C4×Dic3—41no—
121#208—22no—
122#208—41no—
123C2×C3⋊C8—41no—
124C25#6712yes—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4C22×D311yescharacteristic
129C2×M16—61no—
130#120—22no—
131#120—22no—
132#120—22no—
133#120—22no—
134#120—48no—
135C4×D4—121no—
136C4×D4—121no—
137#117—61no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4#6711yescharacteristic
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—62no—
145C2×C22⋊C4—62no—
146C2×C22⋊C4—122no—
147C2×C22⋊C4—122no—
148C2×C22⋊C4#6712yes—
149C2×C42#6711yescharacteristic
150#107—121no—
151#106—121no—
152#100—32no—
153#100—61no—
154#99—62no—
155#99—62no—
156#95—21no—
157C23×C3—22no—
158C23×C3—22no—
159C23×C3—24no—
160C23×C3—42no—
161C23×C3—42no—
162C23×C3—42no—
163C23×C3—42no—
164C23×C3—42no—
165C23×C3—44no—
166C23×C3—44no—
167C23×C3—44no—
168C23×C3—42no—
169C23×C3—42no—
170C23×C3—42no—
171C23×C3—82no—
172C23×C3—84no—
173C23×C3—82no—
174C23×C3—84no—
175C23×C3—88no—
176C23×C3—82no—
177C23×C3#12011yescharacteristic
178C23×C3#12112yes—
179C23×C3#12312yes—
180C3×D4—41no—
181C3×D4—41no—
182C3×D4—42no—
183C3×D4—84no—
184C3×D4—84no—
185C3×D4—84no—
186C3×D4—84no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—22no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—41no—
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12411yescharacteristic
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—82no—
202C2×Dic3—82no—
203C2×Dic3—82no—
204C2×Dic3—82no—
205C2×Dic3—82no—
206C3⋊C8—42no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24—42no—
211C24—44no—
212C24—42no—
213C24—82no—
214C24C2×#6712yes—
215C2×D4—24no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—81no—
223C2×D4—81no—
224C2×D4—82no—
225C2×D4—81no—
226C2×D4D3×D411yescharacteristic
227C2×D4#23712yes—
228C2×D4#23711yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—62no—
233C22×C4—62no—
234C22×C4—122no—
235C22×C4C2×#6711yescharacteristic
236M16—62no—
237M16—121no—
238C2×C8—121no—
239C4⋊C4—41no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—81no—
243C22⋊C4—22no—
244C22⋊C4—22no—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—122no—
251C22⋊C4—121no—
252C22⋊C4—122no—
253C22⋊C4—122no—
254C22⋊C4—244no—
255C22⋊C4—244no—
256C42—41no—
257C42—121no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—42no—
263C22×C3—42no—
264C22×C3—44no—
265C22×C3—44no—
266C22×C3—42no—
267C22×C3—42no—
268C22×C3—48no—
269C22×C3—44no—
270C22×C3—82no—
271C22×C3—82no—
272C22×C3—82no—
273C22×C3—84no—
274C22×C3—84no—
275C22×C3—84no—
276C22×C3—82no—
277C22×C3—82no—
278C22×C3#39411yescharacteristic
279C22×C3B(4,𝔽2)11yescharacteristic
280C22×C3#45811yescharacteristic
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—42no—
284C3×C4—82no—
285C3×C4—81no—
286Dic3—81no—
287Dic3—81no—
288Dic3—162no—
289C23—22no—
290C23—22no—
291C23—24no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—44no—
298C23—44no—
299C23—44no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—82no—
304C23—84no—
305C23—82no—
306C23—84no—
307C23—88no—
308C23—82no—
309C23#93512yes—
310C23#93912yes—
311C23#95311yescharacteristic
312D4—41no—
313D4—41no—
314D4—42no—
315D4—84no—
316D4—84no—
317D4—84no—
318D4—84no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—242no—
332C2×C4—242no—
333C2×C4—242no—
334C2×C4—242no—
335C2×C4—242no—
336C2×C4#93611yescharacteristic
337C2×C4#93811yescharacteristic
338C8—122no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—42no—
342C2×C3—42no—
343C2×C3—84no—
344C2×C3#204411yescharacteristic
345C2×C3#219611yescharacteristic
346C2×C3#219911yescharacteristic
347C22—21no—
348C22—21no—
349C22—22no—
350C22—22no—
351C22—42no—
352C22—42no—
353C22—44no—
354C22—44no—
355C22—42no—
356C22—42no—
357C22—48no—
358C22—44no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22—84no—
363C22—84no—
364C22—84no—
365C22—82no—
366C22—82no—
367C22#522211yescharacteristic
368C22#524311yescharacteristic
369C22#524411yescharacteristic
370C4—21no—
371C4—21no—
372C4—42no—
373C4—82no—
374C4—81no—
375C4—241no—
376C4—241no—
377C4—482no—
378C3#1341911yescharacteristic, complemented, Sylow
379C2—21no—
380C2—21no—
381C2—42no—
382C2—42no—
383C2—84no—
384C2#7185011yescharacteristic
385C2#7190911yescharacteristic
386C2#7195011yescharacteristic
subgroupG/H
centerC22#5244
commutator, cosocleC22×C3×D4C23
FrattiniC22×D4C22×D3
Fitting, prime radicalC3×#2679C2
socleC22×C3B(4,𝔽2)
hypercenter#2679D3
hypocenterC3#13419
LCSG ▷ C22×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2679

Complements

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

NH ≅ G/Ntype
2. #71669C2semidirect
3. C2×#4536C2semidirect
4. #69642C2semidirect
5. #67641C2semidirect
56. #833D4semidirect
378. C3#13419semidirect

Maximal subgroups

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

subgroupindex
C3×#26792
#716692
C2×#45362
#696422
#676412
#134193