Finite Group Databasebeta

← #10696014

Subgroups — #10696014

Order 768.

534 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 83 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#83369C211yesmaximal, characteristic, complemented
2#83357C211yesmaximal, characteristic
3#73735C211yesmaximal, characteristic
4C3×#1986C211yesmaximal, characteristic, complemented
5#69860C211yesmaximal, characteristic
6#69857C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#10010—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#875C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5094—21no—
15#5090—21no—
16#5089—21no—
17#5086—21no—
18#5050—21no—
19#5050—21no—
20#5046—21no—
21#5043—21no—
22#5030—22no—
23#5028—22no—
24#4988—21no—
25#4973—21no—
26#4940—22no—
27#4926C2211yescharacteristic
28#4926C2211yescharacteristic
29#4674—21no—
30#4673—21no—
31C3×#339—22no—
32#4464—22no—
33#4456—22no—
34#3215—31no—
35#3113—31no—
36#2592—31no—
37#1986D311yescharacteristic, complemented
38#1527—31no—
39#1491—31no—
40#1285—31no—
41C22×D12D411yescharacteristic
42C22×Dic6D411yescharacteristic
43C22×Dic6D411yescharacteristic
44C2×C3×M16—21no—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×#117—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C42D412yes—
55C2×C3×C42C2311yescharacteristic
56C2×#216—22no—
57C2×#216—22no—
58C2×#215—22no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—22no—
63C2×C4⋊Dic3D411yescharacteristic
64C2×Dic3⋊C4—21no—
65C2×C4×Dic3—21no—
66#875—21no—
67#875—21no—
68#875—21no—
69#875—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C4×Dic6—21no—
73C4×Dic6—21no—
74C3×#97—21no—
75C3×C4×C8—21no—
76C3×#95—21no—
77#835—22no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80#830—21no—
81#830—21no—
82#810—22no—
83#810—22no—
84#809—22no—
85#808—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)D412yes—
88#803—21no—
89#803D412yes—
90#533C2×D311yescharacteristic
91C2×#124—31no—
92C2×C4×Q8—31no—
93#497—61no—
94#489—61no—
95#483—61no—
96#475—61no—
97#464—62no—
98#462—61no—
99#451—61no—
100#451—61no—
101#450—61no—
102#441—62no—
103#432—31no—
104#432—31no—
105C4×M16C2×D311yescharacteristic
106#397—61no—
107#393—61no—
108#391—61no—
109#390—61no—
110#388—62no—
111C4×C4⋊C4C2×D311yescharacteristic
112#339—22no—
113#329—62no—
114Q8⋊C8—62no—
115C23×D3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C22×C3×C4—22no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—42no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—42no—
133C2×Dic6—42no—
134C3×M16—22no—
135C3×M16—41no—
136C2×C3×C8—41no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—42no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42—42no—
146C3×C42—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151#216—44no—
152#216—82no—
153#215—82no—
154(C2×D3)⋊C4—44no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—41no—
157(C2×D3)⋊C4—41no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—82no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162Dic3⋊C4—41no—
163Dic3⋊C4—41no—
164C4×Dic3—41no—
165C4×Dic3—41no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C22×Q8—31no—
169C22×Q8—31no—
170C22×D4—31no—
171C2×Q16—62no—
172C2×Q16—62no—
173C2×SD16—62no—
174C2×M16—21no—
175C4⋊Q8D1211yescharacteristic
176C4⋊Q8D1211yescharacteristic
177#125—21no—
178#124—61no—
179#124—61no—
180#124—61no—
181#124—61no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—61no—
187C4×Q8—61no—
188C4×Q8—61no—
189C4×Q8—61no—
190#117—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—61no—
194C2×C22⋊C4—62no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—62no—
198C2×C42—61no—
199C2×C42#6712yes—
200C2×C42C22×D311yescharacteristic
201#106—62no—
202C4⋊C8—32no—
203C4⋊C8—61no—
204#103—62no—
205#103—62no—
206#103—62no—
207#102—62no—
208C22⋊C8—61no—
209C22⋊C8—61no—
210#97—21no—
211#97—32no—
212#97—61no—
213C4×C8—21no—
214#95—21no—
215#95—62no—
216#95—62no—
217C23×C3#12011yescharacteristic
218C22×D3—42no—
219C22×D3—42no—
220C22×D3—41no—
221C22×D3—41no—
222C22×D3—41no—
223C3×Q8—42no—
224C3×Q8—42no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—22no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12711yescharacteristic
240C2×Dic3—22no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—42no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248D12—42no—
249D12—81no—
250Dic6—22no—
251Dic6—22no—
252Dic6—41no—
253Dic6—41no—
254Dic6—42no—
255Dic6—81no—
256C3×C8—42no—
257C3⋊C8—42no—
258C3⋊C8—42no—
259C24—61no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—61no—
263C2×Q8—61no—
264C2×Q8—61no—
265C2×Q8—61no—
266C2×Q8—122no—
267C2×Q8—122no—
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—122no—
271C22×C4—22no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4D3×D411yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4C2×#6711yescharacteristic
278Q16—124no—
279Q16—242no—
280SD16—242no—
281M16—22no—
282M16—41no—
283C2×C8—41no—
284C2×C8—62no—
285C2×C8—62no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C22⋊C4—41no—
298C22⋊C4—124no—
299C22⋊C4—121no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C22⋊C4—121no—
303C22⋊C4—242no—
304C42—42no—
305C42—41no—
306C42—121no—
307C42—121no—
308C42C2×D1211yescharacteristic
309C42#23511yescharacteristic
310C42D3×D411yescharacteristic
311C42D3×D411yescharacteristic
312C22×C3—21no—
313C22×C3—21no—
314C22×C3#39211yescharacteristic
315C22×C3#45511yescharacteristic
316C22×C3#45611yescharacteristic
317C2×D3—82no—
318C2×D3—82no—
319C2×D3—81no—
320C2×D3—81no—
321C2×D3—81no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—82no—
328C3×C4—81no—
329C3×C4—81no—
330C3×C4#45512yes—
331Dic3—42no—
332Dic3—81no—
333Dic3—81no—
334C23—122no—
335C23—122no—
336C23—121no—
337C23—121no—
338C23—121no—
339C23#93711yescharacteristic
340Q8—42no—
341Q8—42no—
342Q8—62no—
343Q8—62no—
344Q8—121no—
345Q8—121no—
346Q8—122no—
347Q8—241no—
348D4—122no—
349D4—241no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—62no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—122no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4#88211yescharacteristic
368C2×C4#88211yescharacteristic
369C2×C4#88411yescharacteristic
370C2×C4#88411yescharacteristic
371C2×C4#93011yescharacteristic
372C2×C4#94011yescharacteristic
373C8—42no—
374C8—122no—
375C8—122no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3#200411yescharacteristic
379C2×C3#200711yescharacteristic
380C2×C3#201011yescharacteristic
381D3—161no—
382C22—21no—
383C22—21no—
384C22—242no—
385C22—242no—
386C22—241no—
387C22—241no—
388C22—241no—
389C22#496011yescharacteristic
390C22#506111yescharacteristic
391C22#509511yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—82no—
398C4—81no—
399C4—81no—
400C4—122no—
401C4—241no—
402C4—241no—
403C4#482512yes—
404C3#1001011yescharacteristic, complemented, Sylow
405C2—21no—
406C2—21no—
407C2—481no—
408C2#6969411yescharacteristic
409C2#7038611yescharacteristic
410C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1986C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1986D3
hypocenterC3#10010
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
1. #83369C2semidirect
4. C3×#1986C2semidirect
6. #69857C2semidirect
7. #67621C2semidirect
37. #1986D3semidirect
404. C3#10010semidirect

Maximal subgroups

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

subgroupindex
#833692
#833572
#737352
C3×#19862
#698602
#698572
#676212
#100103