Finite Group Databasebeta

← #10697619

Subgroups — #10697619

Order 768.

604 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 101 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
1#83259C211yesmaximal, characteristic
2#83222C211yesmaximal, characteristic, complemented
3#79059C211yesmaximal, characteristic
4C3×#1548C211yesmaximal, characteristic, complemented
5#70792C211yesmaximal, characteristic, complemented
6#70757C211yesmaximal, characteristic
7#67715C211yesmaximal, characteristic, complemented
8#10248—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#946—22no—
11C2×C4×D12C2211yescharacteristic
12#5470C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15#5228—22no—
16#5131—21no—
17#5121—21no—
18#5092—21no—
19#5090—22no—
20#5079—21no—
21#4993—22no—
22#4988—21no—
23#4975—21no—
24C4×C4⋊Dic3C2211yescharacteristic
25#4925C2211yescharacteristic
26#4847—22no—
27#4846—21no—
28#4842—21no—
29C3×#333—22no—
30#4467—22no—
31#4459—22no—
32#3209—31no—
33#3077—31no—
34#2542—31no—
35#1548D311yescharacteristic, complemented
36#1503—31no—
37#1478—31no—
38#1303—31no—
39C22×C3×Q8D412yes—
40C22×D12D411yescharacteristic
41C22×C4×D3—21no—
42C2×C3×M16—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50#946—48no—
51C2×#216—24no—
52C2×(C2×D3)⋊C4—24no—
53C2×(C2×D3)⋊C4—21no—
54#926—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3D411yescharacteristic
57#924—21no—
58#924—21no—
59C2×Dic3⋊C4—24no—
60C2×C4×Dic3—21no—
61C4×D12—21no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12—21no—
65#870—21no—
66#869D411yescharacteristic
67#869D411yescharacteristic
68C4×Dic6—21no—
69C4×Dic6—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×C22⋊C8—21no—
74C3×C4×C8—21no—
75#835—24no—
76C3⋊#95D412yes—
77#830—21no—
78#821—22no—
79#821—22no—
80#818—22no—
81#817—22no—
82C4⋊(C3⋊C8)D412yes—
83C4×C3⋊C8—21no—
84#533—31no—
85C2×C4⋊Q8C2×D311yescharacteristic
86C2×C22⋊Q8—62no—
87C2×C4×D4—31no—
88#494—61no—
89#487—61no—
90#478—62no—
91#476—61no—
92#474—61no—
93#462—62no—
94#451—62no—
95#431—31no—
96#431C2×D311yescharacteristic
97#428—61no—
98#423C2×D311yescharacteristic
99#418—61no—
100#399—62no—
101#390—61no—
102#384—61no—
103C4×C4⋊C4—31no—
104#333—22no—
105#328—62no—
106D4⋊C8—62no—
107C23×D3—21no—
108C2×C3×Q8—22no—
109C2×C3×Q8—22no—
110C2×C3×Q8—44no—
111C22×C3×C4—22no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—22no—
116C22×Dic3—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12Q1612yescomplemented
123C2×C4×D3—44no—
124C2×C4×D3—41no—
125C2×C4×D3—41no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C3×M16—41no—
129C3×M16—41no—
130C2×C3×C8—22no—
131C2×C3×C8—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140#217—41no—
141#216—84no—
142(C2×D3)⋊C4—48no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145C4⋊Dic3—24no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152Dic3⋊C4—48no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155C4×Dic3—41no—
156C4×Dic3—41no—
157C4×Dic3—41no—
158C2×C3⋊C8—22no—
159C22×Q8D1212yes—
160C22×D4—31no—
161C23×C4—61no—
162C2×Q16—64no—
163C2×M16—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—31no—
169C4⋊Q8—31no—
170#125—61no—
171C22⋊Q8—61no—
172C22⋊Q8—61no—
173C22⋊Q8—128no—
174C4×Q8—61no—
175C4×Q8—61no—
176C4×D4—61no—
177C4×D4—61no—
178C4×D4—61no—
179C4×D4—61no—
180#117—61no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—64no—
186C2×C22⋊C4—64no—
187C2×C22⋊C4—61no—
188C2×C42—61no—
189C2×C42C22×D311yescharacteristic
190#107—62no—
191#106—62no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—32no—
195C4⋊C8#6712yes—
196#103—64no—
197#102—62no—
198#102—62no—
199C22⋊C8—21no—
200C22⋊C8—61no—
201C4×C8—21no—
202C4×C8—61no—
203#95—32no—
204C23×C3#12011yescharacteristic
205C22×D3—22no—
206C22×D3—42no—
207C22×D3—42no—
208C22×D3—41no—
209C22×D3—41no—
210C3×Q8—44no—
211C3×Q8—82no—
212C2×C3×C4—22no—
213C2×C3×C4—41no—
214C2×C3×C4—44no—
215C2×C3×C4—42no—
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12711yescharacteristic
222C2×C3×C4C2×Q1611yescharacteristic
223C2×C3×C4#13511yescharacteristic
224C2×C3×C4#13711yescharacteristic
225C2×C3×C4#13711yescharacteristic
226C2×Dic3—24no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233D12—22no—
234D12—22no—
235D12—41no—
236D12—41no—
237C4×D3—82no—
238C4×D3—82no—
239Dic6—42no—
240Dic6—81no—
241C3×C8—42no—
242C3×C8—81no—
243C3⋊C8—42no—
244C24—61no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—44no—
248C2×Q8—61no—
249C2×Q8—61no—
250C2×D4—32no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C22×C4—22no—
257C22×C4—62no—
258C22×C4—61no—
259C22×C4—124no—
260C22×C4—121no—
261C22×C4—121no—
262C22×C4C2×D1211yescharacteristic
263C22×C4D3×D411yescharacteristic
264C22×C4D3×D411yescharacteristic
265Q16—244no—
266M16—41no—
267M16—41no—
268C2×C8—22no—
269C2×C8—41no—
270C2×C8—62no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—64no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—128no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—128no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C42—121no—
290C42—121no—
291C42—121no—
292C42#23511yescharacteristic
293C42D3×D411yescharacteristic
294C42D3×D411yescharacteristic
295C42C2×#6711yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3#39211yescharacteristic
299C22×C3#45111yescharacteristic
300C22×C3#45211yescharacteristic
301C2×D3—44no—
302C2×D3—42no—
303C2×D3—82no—
304C2×D3—81no—
305C2×D3—81no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—82no—
310C3×C4#45111yescharacteristic
311C3×C4#45211yescharacteristic
312C3×C4#45312yes—
313C3×C4#46711yescharacteristic
314C3×C4#47011yescharacteristic
315Dic3—44no—
316Dic3—81no—
317Dic3—81no—
318Dic3—81no—
319C23—62no—
320C23—122no—
321C23—122no—
322C23—121no—
323C23—121no—
324C23#88211yescharacteristic
325Q8—44no—
326Q8—82no—
327Q8—122no—
328Q8—241no—
329D4—62no—
330D4—62no—
331D4—121no—
332D4—121no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—64no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—242no—
345C2×C4—242no—
346C2×C4#88211yescharacteristic
347C2×C4#88411yescharacteristic
348C2×C4#88411yescharacteristic
349C2×C4#91511yescharacteristic
350C2×C4#91611yescharacteristic
351C2×C4D3×Q1611yescharacteristic
352C2×C4#91811yescharacteristic
353C2×C4#93011yescharacteristic
354C2×C4#93711yescharacteristic
355C2×C4#94011yescharacteristic
356C8—42no—
357C8—81no—
358C8—122no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3#200111yescharacteristic
362C2×C3#200211yescharacteristic
363C2×C3#200911yescharacteristic
364D3—82no—
365C22—21no—
366C22—21no—
367C22—124no—
368C22—122no—
369C22—242no—
370C22—241no—
371C22—241no—
372C22#480811yescharacteristic
373C22#480911yescharacteristic
374C22#496011yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—82no—
379C4—124no—
380C4—241no—
381C4—241no—
382C4—241no—
383C4#474712yes—
384C4#481311yescharacteristic
385C4#481611yescharacteristic
386C4#517311yescharacteristic
387C4#518611yescharacteristic
388C3#1024811yescharacteristic, complemented, Sylow
389C2—21no—
390C2—21no—
391C2—242no—
392C2#7008211yescharacteristic
393C2#7027311yescharacteristic
394C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1548D3
hypocenterC3#10248
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
2. #83222C2semidirect
4. C3×#1548C2semidirect
5. #70792C2semidirect
7. #67715C2semidirect
35. #1548D3semidirect
122. C2×D12Q16semidirect
388. C3#10248semidirect

Maximal subgroups

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

subgroupindex
#832592
#832222
#790592
C3×#15482
#707922
#707572
#677152
#102483