Finite Group Databasebeta

← #10696022

Subgroups — #10696022

Order 768.

496 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 85 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×#2584C211yesmaximal, characteristic
2#83021C211yesmaximal, characteristic
3#82448C211yesmaximal, characteristic
4#71735C211yesmaximal, characteristic
5#71623C211yesmaximal, characteristic
6#69493C211yesmaximal, characteristic
7#67621C211yesmaximal, characteristic
8#10011—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#398—21no—
13C3×#395—21no—
14C3×#389—21no—
15C3×#389—21no—
16C3×#389—22no—
17C3×C4×C4⋊C4C2211yescharacteristic
18#5228—21no—
19#5069—21no—
20#5064—21no—
21#5063—22no—
22#5031—21no—
23#5030—22no—
24#5020—21no—
25#4971—21no—
26#4971—21no—
27#4970—21no—
28#4926C2211yescharacteristic
29#4926C2211yescharacteristic
30C4×#208C2211yescharacteristic
31#4543—22no—
32#4535—22no—
33#4523—22no—
34#3236—31no—
35#3113—31no—
36#2584D311yescharacteristic
37#1986—31no—
38#1528—31no—
39#1491—31no—
40#1285—31no—
41C22×C3×Q8D411yescharacteristic
42C3×#125—21no—
43C3×#125—21no—
44C3×#125—21no—
45C3×#125—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—22no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C42—21no—
60C2×C3×C42D412yes—
61C2×C3×C42C2311yescharacteristic
62C2×#216—22no—
63#926—21no—
64#924—21no—
65#924—21no—
66C2×#208—21no—
67#870—21no—
68#869D411yescharacteristic
69#869D411yescharacteristic
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×#95—22no—
73C3×#95—22no—
74C3×#95—21no—
75#835—22no—
76#830—21no—
77#830—21no—
78#810—22no—
79#810—22no—
80#810—22no—
81#808—22no—
82#808—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)D412yes—
86#803—21no—
87#803—21no—
88#803D412yes—
89C4×C3⋊C8—21no—
90#533—31no—
91C2×#125C2×D311yescharacteristic
92C2×C4×Q8C2×D311yescharacteristic
93#501—61no—
94#491—61no—
95#484—61no—
96#483—61no—
97#483—61no—
98#479—62no—
99#475—61no—
100#462—61no—
101#451—61no—
102#441—62no—
103#432—31no—
104#432—31no—
105C4×M16—31no—
106#398—21no—
107#395—21no—
108#389—21no—
109#389—21no—
110#389—22no—
111C4×C4⋊C4C2×D311yescharacteristic
112#339—62no—
113#330—62no—
114Q8⋊C8—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—42no—
118C22×C3×C4—22no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C2×Dic6—21no—
127C2×Dic6—21no—
128C3×C4⋊C4—24no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—44no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—82no—
143C3×C42—42no—
144C3×C42—41no—
145C3×C42—41no—
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42#4111yescharacteristic
150#217—41no—
151#216—44no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158C4×Dic3—41no—
159#208—22no—
160#208—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—41no—
164C22×Q8#6711yescharacteristic
165C2×Q16—62no—
166C2×M16—61no—
167C4⋊Q8—31no—
168C4⋊Q8—31no—
169#125—21no—
170#125—21no—
171#125—21no—
172#125—21no—
173#125—61no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—61no—
181C4×Q8—61no—
182#117—61no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4#6711yescharacteristic
190C2×C4⋊C4#6711yescharacteristic
191C2×C4⋊C4#6711yescharacteristic
192C2×C42—21no—
193C2×C42#6712yes—
194C2×C42C22×D311yescharacteristic
195#107—62no—
196#106—62no—
197#106—62no—
198C4⋊C8—32no—
199C4⋊C8—61no—
200#103—62no—
201#103—62no—
202#103—62no—
203#103—62no—
204C22⋊C8—61no—
205C22⋊C8—61no—
206#97—32no—
207#97—61no—
208#97—61no—
209C4×C8—61no—
210#95—22no—
211#95—22no—
212#95—21no—
213C23×C3#12011yescharacteristic
214C3×Q8—22no—
215C3×Q8—22no—
216C3×Q8—41no—
217C3×Q8—41no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—22no—
221C2×C3×C4—24no—
222C2×C3×C4—22no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12711yescharacteristic
242C2×Dic3—41no—
243C2×Dic3—41no—
244Dic6—42no—
245Dic6—42no—
246C3⋊C8—42no—
247C3⋊C8—42no—
248C3⋊C8—42no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—42no—
252C2×Q8—61no—
253C2×Q8—61no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4C2×#6711yescharacteristic
260C22×C4C2×#6711yescharacteristic
261C22×C4C2×#6711yescharacteristic
262Q16—124no—
263M16—62no—
264M16—121no—
265C2×C8—62no—
266C2×C8—62no—
267C2×C8—121no—
268C4⋊C4—24no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—44no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—82no—
287C4⋊C4—121no—
288C4⋊C4—121no—
289C22⋊C4—121no—
290C42—42no—
291C42—41no—
292C42—41no—
293C42—121no—
294C42D3×D411yescharacteristic
295C42D3×D411yescharacteristic
296C42D3×D411yescharacteristic
297C42#23711yescharacteristic
298C22×C3—21no—
299C22×C3—21no—
300C22×C3#39211yescharacteristic
301C22×C3#45511yescharacteristic
302C22×C3#45611yescharacteristic
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—44no—
309C3×C4—42no—
310C3×C4—81no—
311C3×C4—81no—
312C3×C4—81no—
313C3×C4#45612yes—
314Dic3—81no—
315Dic3—81no—
316C23#95311yescharacteristic
317Q8—22no—
318Q8—22no—
319Q8—41no—
320Q8—41no—
321Q8—122no—
322Q8—122no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4#93711yescharacteristic
344C2×C4#93711yescharacteristic
345C2×C4#93811yescharacteristic
346C2×C4#93911yescharacteristic
347C2×C4#93911yescharacteristic
348C2×C4#94011yescharacteristic
349C8—122no—
350C8—122no—
351C8—122no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3#200511yescharacteristic
355C2×C3#200711yescharacteristic
356C2×C3#201011yescharacteristic
357C22—21no—
358C22—21no—
359C22#522311yescharacteristic
360C22#524211yescharacteristic
361C22#524611yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—44no—
368C4—42no—
369C4—81no—
370C4—81no—
371C4—81no—
372C4—241no—
373C4—241no—
374C4#509512yes—
375C3#1001111yescharacteristic, complemented, Sylow
376C2—21no—
377C2—21no—
378C2#7171511yescharacteristic
379C2#7179511yescharacteristic
380C2#7188111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2584C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#2584D3
hypocenterC3#10011
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2584

Complements

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

NH ≅ G/Ntype
375. C3#10011semidirect

Maximal subgroups

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

subgroupindex
C3×#25842
#830212
#824482
#717352
#716232
#694932
#676212
#100113