Finite Group Databasebeta

← #10692846

Subgroups — #10692846

Order 768.

568 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 103 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×#2973C211yesmaximal, characteristic, complemented
2#82381C211yesmaximal, characteristic
3#71603C211yesmaximal, characteristic, complemented
4#70808C212yesmaximal, complemented
5#67749C212yesmaximal, complemented
6#9447—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#940C411yescharacteristic
9C2×#936C411yescharacteristic
10C2×#875C2211yescharacteristic
11C3×#435—24no—
12C3×#432C2212yes—
13C3×#407—21no—
14C2×C3×C22⋊C8—21no—
15#5213—21no—
16#5213—21no—
17#4988C411yescharacteristic
18#4973C411yescharacteristic
19#4961—21no—
20#4961—21no—
21C4×C4⋊Dic3C2211yescharacteristic
22#4926C2212yes—
23#4525—21no—
24#4525—21no—
25#4464—24no—
26#2973D311yescharacteristic, complemented
27#2544—31no—
28#1533—32no—
29#1466—31no—
30#1285—32no—
31C22×C3×D4C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C22×#67—21no—
35C22×#67—21no—
36C22×D12C2×C411yescharacteristic
37C22×Dic6C2×C411yescharacteristic
38C2×C3×M16—22no—
39C22×C3×C8—22no—
40C3×C4×D4—22no—
41C3×C4×D4—22no—
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C22⋊C4D411yescharacteristic
44C2×C3×C22⋊C4D411yescharacteristic
45C2×C3×C42C2311yescharacteristic
46C2×#217—21no—
47C2×#217—21no—
48#940—42no—
49#936—42no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×C4⋊Dic3—21no—
55C2×C4×Dic3C2×C411yescharacteristic
56C2×C4×Dic3C2×C411yescharacteristic
57#875—22no—
58#875—22no—
59C3×C4⋊C8—22no—
60C3×C22⋊C8—22no—
61C3×C22⋊C8—22no—
62C3×C22⋊C8—22no—
63C3×#97D414yescomplemented
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67#830—22no—
68C4⋊(C3⋊C8)—22no—
69#803—22no—
70C2×#127—31no—
71C2×#124—31no—
72C2×#124—31no—
73C2×C4×D4C2×D311yescharacteristic
74#435—24no—
75#432—32no—
76#432C2×D312yes—
77#407—21no—
78C2×C22⋊C8—21no—
79#394—61no—
80#394—61no—
81#391—31no—
82#390—31no—
83#386—61no—
84#386—61no—
85C4×C4⋊C4—31no—
86#329—64no—
87#323—61no—
88#323—61no—
89C24×C3C22⋊C411yescharacteristic
90C24×C3C22⋊C411yescharacteristic
91C23×D3—21no—
92C2×C3×D4—22no—
93C2×C3×D4—22no—
94C2×C3×D4—41no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—81no—
100C22×C3×C4—81no—
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C2×#67—42no—
107C2×#67—42no—
108C2×#67—42no—
109C2×#67—42no—
110C2×#67—81no—
111C2×#67—81no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—21no—
115C2×D12—22no—
116C2×D12—41no—
117C2×D12—41no—
118C2×Dic6—22no—
119C2×Dic6—41no—
120C2×Dic6—41no—
121C3×M16—82no—
122C2×C3×C8—24no—
123C2×C3×C8—82no—
124C3×C4⋊C4—22no—
125C3×C22⋊C4—22no—
126C3×C22⋊C4—22no—
127C3×C42C2×D412yes—
128C3×C42#4112yes—
129#217—42no—
130#217—42no—
131(C2×D3)⋊C4—41no—
132(C2×D3)⋊C4—42no—
133(C2×D3)⋊C4—41no—
134(C2×D3)⋊C4—81no—
135(C2×D3)⋊C4—81no—
136C4⋊Dic3—42no—
137C4×Dic3—41no—
138C4×Dic3—41no—
139C2×C3⋊C8—42no—
140C22×Q8—31no—
141C22×D4—31no—
142C22×D4—61no—
143C22×D4—61no—
144C22×D4C4×D311yescharacteristic
145C23×C4#6711yescharacteristic
146C23×C4#6711yescharacteristic
147C2×M16—22no—
148C22×C8—22no—
149#127—122no—
150#124—62no—
151#124—62no—
152#124—122no—
153C4×D4—22no—
154C4×D4—22no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4C4×D311yescharacteristic
157C2×C22⋊C4—61no—
158C2×C22⋊C4—61no—
159C2×C22⋊C4—61no—
160C2×C22⋊C4—61no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4D1211yescharacteristic
164C2×C22⋊C4D1211yescharacteristic
165C2×C42—31no—
166C2×C42—31no—
167C2×C42C22×D311yescharacteristic
168C4⋊C8—22no—
169C4⋊C8—62no—
170C22⋊C8—22no—
171C22⋊C8—22no—
172C22⋊C8—22no—
173C22⋊C8—62no—
174#97—62no—
175#97#6714yescomplemented
176#95—61no—
177#95—61no—
178#95—61no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—41no—
185C23×C3—41no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3#10012yes—
189C23×C3C2×C22⋊C411yescharacteristic
190C22×D3—41no—
191C22×D3—41no—
192C22×D3—41no—
193C22×D3—41no—
194C22×D3—41no—
195C22×D3—41no—
196C22×D3—41no—
197C3×D4—44no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—81no—
209C2×C3×C4—81no—
210C2×C3×C4C4×D412yes—
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12112yes—
213C2×C3×C4#12311yescharacteristic
214#67—84no—
215#67—84no—
216C2×Dic3—22no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225D12—82no—
226Dic6—82no—
227C3×C8—44no—
228C3⋊C8—82no—
229C24—61no—
230C24(C2×D3)⋊C411yescharacteristic
231C24(C2×D3)⋊C411yescharacteristic
232C2×Q8—62no—
233C2×Q8—121no—
234C2×Q8—121no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—41no—
238C2×D4—62no—
239C2×D4—121no—
240C2×D4—121no—
241C2×D4—122no—
242C2×D4—122no—
243C2×D4—122no—
244C2×D4—122no—
245C2×D4—241no—
246C2×D4—241no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4(C2×D3)⋊C411yescharacteristic
257C22×C4(C2×D3)⋊C411yescharacteristic
258C22×C4C2×C4×D311yescharacteristic
259C22×C4C2×D1211yescharacteristic
260C22×C4C2×#6711yescharacteristic
261M16—82no—
262C2×C8—24no—
263C2×C8—82no—
264C2×C8—122no—
265C4⋊C4—22no—
266C4⋊C4—122no—
267C22⋊C4—22no—
268C22⋊C4—22no—
269C22⋊C4—122no—
270C22⋊C4—122no—
271C22⋊C4—121no—
272C22⋊C4—122no—
273C22⋊C4—121no—
274C22⋊C4—241no—
275C22⋊C4—241no—
276C42—121no—
277C42—121no—
278C42#23512yes—
279C42C2×#6712yes—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—42no—
284C22×C3—42no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—81no—
291C22×C3—81no—
292C22×C3#38611yescharacteristic
293C22×C3C2×#10011yescharacteristic
294C22×C3#41311yescharacteristic
295C2×D3—81no—
296C2×D3—81no—
297C2×D3—81no—
298C2×D3—81no—
299C2×D3—81no—
300C2×D3—81no—
301C2×D3—81no—
302C3×C4—22no—
303C3×C4—22no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—41no—
307C3×C4—41no—
308Dic3—42no—
309Dic3—81no—
310Dic3—81no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—121no—
321C23—121no—
322C23—121no—
323C23—121no—
324C23—121no—
325C23—121no—
326C23—121no—
327C23#82312yes—
328C23C2×(C2×D3)⋊C411yescharacteristic
329Q8—242no—
330D4—44no—
331D4—242no—
332D4—244no—
333D4—244no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—62no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4C4×#6712yes—
356C2×C4#92911yescharacteristic
357C2×C4#93012yes—
358C2×C4#95311yescharacteristic
359C8—44no—
360C8—242no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—42no—
364C2×C3—81no—
365C2×C3#188411yescharacteristic
366C2×C3#190011yescharacteristic
367C2×C3#191111yescharacteristic
368D3—161no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—42no—
373C22—42no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—81no—
380C22—81no—
381C22—241no—
382C22—241no—
383C22—241no—
384C22—241no—
385C22—241no—
386C22—241no—
387C22—241no—
388C22C2×#82311yescharacteristic
389C22#513211yescharacteristic
390C22#521311yescharacteristic
391C4—22no—
392C4—22no—
393C4—21no—
394C4—21no—
395C4—41no—
396C4—41no—
397C4—122no—
398C4—241no—
399C4—241no—
400C3#944711yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—42no—
404C2—81no—
405C2—481no—
406C2#7165211yescharacteristic
407C2#7200211yescharacteristic
408C2#7202111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2973C2
socleC22×C3#386
prime radicalC2×#940C4
cosocleC2×C3×C42C23
hypercenter#2973D3
hypocenterC3#9447
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
1. C3×#2973C2semidirect
3. #71603C2semidirect
4. #70808C2semidirect
5. #67749C2semidirect
26. #2973D3semidirect
63. C3×#97D4semidirect
175. #97#67semidirect
400. C3#9447semidirect

Maximal subgroups

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

subgroupindex
C3×#29732
#823812
#716032
#708082
#677492
#94473