Finite Group Databasebeta

← #13058577

Subgroups — #13058577

Order 1344.

540 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66677C211yesmaximal, characteristic
2C2×D21⋊Q8C211yesmaximal, characteristic, complemented
3#10621377C211yesmaximal, characteristic, complemented
4C7×#888C211yesmaximal, characteristic, complemented
5C3×#6459C211yesmaximal, characteristic, complemented
6#10621017C211yesmaximal, characteristic, complemented
7#10620995C211yesmaximal, characteristic, complemented
8#10620981C211yesmaximal, characteristic, complemented
9#10620973C211yesmaximal, characteristic, complemented
10#10620965C211yesmaximal, characteristic, complemented
11#10620918C211yesmaximal, characteristic, complemented
12#10620916C211yesmaximal, characteristic, complemented
13#10620915C211yesmaximal, characteristic, complemented
14#10620911C211yesmaximal, characteristic, complemented
15#10620891C211yesmaximal, characteristic, complemented
16#90415—31nomaximal, Hall
17C2×D84C2211yescharacteristic
18C2×C4×D21C2211yescharacteristic
19C2×C7×Dic6C2211yescharacteristic
20C2×C7×Dic6C2211yescharacteristic
21C2×C3×C4×D7C2211yescharacteristic
22C2×C3×Dic14C2211yescharacteristic
23C2×#4248C2211yescharacteristic
24C2×#4248C2211yescharacteristic
25C2×#4246C2211yescharacteristic
26C2×#4246C2211yescharacteristic
27C2×D21⋊C4C2211yescharacteristic
28C2×D21⋊C4C2211yescharacteristic
29C2×D21⋊C4C2211yescharacteristic
30#66677—24no—
31D21⋊Q8—24no—
32C3×C7×C4⋊C4C2211yescharacteristic
33#66631C2211yescharacteristic
34#66631C2211yescharacteristic
35#66631C2211yescharacteristic
36#66629C2211yescharacteristic
37C7×C4⋊Dic3C2211yescharacteristic
38C7×Dic3⋊C4C2211yescharacteristic
39C7×Dic3⋊C4C2211yescharacteristic
40C4×C7×Dic3C2211yescharacteristic
41C3×(C2×D7)⋊C4C2211yescharacteristic
42C3×(C2×D7)⋊C4C2211yescharacteristic
43C3×Dic7⋊C4C2211yescharacteristic
44C3×Dic7⋊C4C2211yescharacteristic
45#66577C2211yescharacteristic
46Dic7⋊Dic3C2211yescharacteristic
47Dic7⋊Dic3C2211yescharacteristic
48#66575C2211yescharacteristic
49#66575C2211yescharacteristic
50#66575C2211yescharacteristic
51(C2×D7)⋊Dic3C2211yescharacteristic
52(C2×D7)⋊Dic3C2211yescharacteristic
53Dic3×Dic7C2211yescharacteristic
54C2×#1161—31no—
55C2×Q8×D7—31no—
56C7×C4⋊Q8—31no—
57#6510—31no—
58#6507—31no—
59#6459—31no—
60#6459—31no—
61#6459—31no—
62#6459D311yescharacteristic, complemented
63#6457—31no—
64#6457—31no—
65#6457—31no—
66#6457—31no—
67#6438—31no—
68#6435—31no—
69#5628—71nomaximal, Hall
70C22×D21C2311yescharacteristic
71C22×D21C2311yescharacteristic
72C22×C3×D7C2311yescharacteristic
73C2×C3×C4×C7C2311yescharacteristic
74C2×C3×C4×C7C2311yescharacteristic
75C2×C3×C4×C7C2311yescharacteristic
76C2×Dic21C2311yescharacteristic
77C2×Dic21C2311yescharacteristic
78D84—22no—
79C4×D21—22no—
80C2×C7×Dic3C2311yescharacteristic
81C2×C7×Dic3C2311yescharacteristic
82C2×C7×Dic3C2311yescharacteristic
83C2×C7×Dic3C2311yescharacteristic
84C7×Dic6—22no—
85C7×Dic6—22no—
86C2×C3×Dic7C2311yescharacteristic
87C2×C3×Dic7C2311yescharacteristic
88C2×C3×Dic7C2311yescharacteristic
89C3×C4×D7—22no—
90C3×Dic14—22no—
91#4248—22no—
92#4248—22no—
93#4246—22no—
94#4246—22no—
95D21⋊C4—22no—
96D21⋊C4—22no—
97D21⋊C4D414yes—
98C2×C7×Q8—31no—
99C2×C7×Q8—31no—
100#1161—64no—
101Q8×D7—64no—
102C2×D28—31no—
103C2×D28—31no—
104C2×D28—31no—
105C2×C4×D7—31no—
106C2×C4×D7—31no—
107C2×C4×D7—31no—
108C2×C4×D7—31no—
109C2×C4×D7C2×D311yescharacteristic
110C2×Dic14—31no—
111C2×Dic14—31no—
112C2×Dic14C2×D311yescharacteristic
113C7×C4⋊C4—31no—
114C7×C4⋊C4—31no—
115C7×C4⋊C4—31no—
116C7×C4⋊C4C2×D311yescharacteristic
117C42×C7—31no—
118(C2×D7)⋊C4—31no—
119(C2×D7)⋊C4—31no—
120(C2×D7)⋊C4—31no—
121(C2×D7)⋊C4—31no—
122(C2×D7)⋊C4—31no—
123(C2×D7)⋊C4—31no—
124(C2×D7)⋊C4—31no—
125(C2×D7)⋊C4—31no—
126(C2×D7)⋊C4C2×D311yescharacteristic
127(C2×D7)⋊C4C2×D311yescharacteristic
128Dic7⋊C4—31no—
129Dic7⋊C4—31no—
130Dic7⋊C4—31no—
131Dic7⋊C4—31no—
132Dic7⋊C4C2×D311yescharacteristic
133Dic7⋊C4C2×D311yescharacteristic
134C4×Dic7—31no—
135C2×D3×Q8—71no—
136C2×#235—71no—
137C3×C22⋊Q8—71no—
138#946—71no—
139#924—71no—
140#897—71no—
141#894—71no—
142#894—71no—
143#892—71no—
144#888—71no—
145#888D711yescharacteristic, complemented
146#885—71no—
147#885—71no—
148#883—71no—
149#883—71no—
150C22×C3×C7C2411yescharacteristic
151C2×D21—21no—
152C2×D21—21no—
153C2×D21—21no—
154C2×D21—21no—
155C2×D21—21no—
156C2×D21C2×D412yes—
157C2×C3×D7—21no—
158C2×C3×D7—21no—
159C2×C3×D7—21no—
160C3×C4×C7—21no—
161C3×C4×C7—21no—
162C3×C4×C7—21no—
163Dic21—21no—
164Dic21—21no—
165C3×Dic7—21no—
166C3×Dic7—21no—
167C3×Dic7C2×D412yes—
168C7×Dic3—21no—
169C7×Dic3—21no—
170C7×Dic3—21no—
171C7×Dic3C2×D412yes—
172#536—211noSylow
173C22×D7—31no—
174C22×D7—31no—
175C22×D7C22×D311yescharacteristic
176C7×Q8—62no—
177C7×Q8—62no—
178C2×C4×C7—31no—
179C2×C4×C7—31no—
180C2×C4×C7—31no—
181C2×C4×C7—31no—
182C2×C4×C7C22×D311yescharacteristic
183C2×C4×C7C22×D311yescharacteristic
184C2×C4×C7C22×D311yescharacteristic
185C2×Dic7—31no—
186C2×Dic7—31no—
187C2×Dic7C22×D311yescharacteristic
188C2×Dic7C22×D311yescharacteristic
189C2×Dic7C22×D311yescharacteristic
190D28—62no—
191D28—62no—
192D28—62no—
193C4×D7—22no—
194C4×D7—34no—
195C4×D7—62no—
196C4×D7—62no—
197C4×D7—62no—
198Dic14—22no—
199Dic14—62no—
200Dic14—62no—
201C2×C3×Q8—71no—
202C22×C3×C4—71no—
203C2×#67—71no—
204C2×#67—71no—
205D3×Q8—144no—
206#235—144no—
207C2×D12—71no—
208C2×C4×D3—71no—
209C2×C4×D3—71no—
210C2×C4×D3—71no—
211C2×C4×D3—71no—
212C2×Dic6—71no—
213C2×Dic6—71no—
214C2×Dic6C2×D711yescharacteristic
215C2×Dic6C2×D711yescharacteristic
216C3×C4⋊C4—71no—
217C3×C4⋊C4—71no—
218C3×C4⋊C4C2×D711yescharacteristic
219C3×C22⋊C4—71no—
220C3×C22⋊C4—71no—
221#217—71no—
222#217—71no—
223(C2×D3)⋊C4—71no—
224(C2×D3)⋊C4—71no—
225(C2×D3)⋊C4—71no—
226(C2×D3)⋊C4—71no—
227(C2×D3)⋊C4—71no—
228(C2×D3)⋊C4—71no—
229C4⋊Dic3—71no—
230C4⋊Dic3—71no—
231C4⋊Dic3C2×D711yescharacteristic
232Dic3⋊C4—71no—
233Dic3⋊C4—71no—
234Dic3⋊C4C2×D711yescharacteristic
235Dic3⋊C4C2×D711yescharacteristic
236C4×Dic3—71no—
237C4×Dic3C2×D711yescharacteristic
238C2×C3×C7C22×D411yescharacteristic
239C2×C3×C72−1+411yescharacteristic
240C2×C3×C72−1+411yescharacteristic
241D21—22no—
242D21—41no—
243C3×D7—41no—
244C2×#41—211no—
245C22×Q8—211no—
246C4⋊Q8—31no—
247C4⋊Q8—211no—
248#124—211no—
249#124—211no—
250#123—211no—
251#123—211no—
252#123—211no—
253#123—211no—
254C22⋊Q8—71no—
255C22⋊Q8—211no—
256C22⋊Q8—211no—
257C22⋊Q8—211no—
258#117—211no—
259C22×C7C23×D311yescharacteristic
260C2×D7—21no—
261C2×D7—21no—
262C2×D7—21no—
263C2×D7—32no—
264C2×D7—61no—
265C2×D7—61no—
266C2×D7—61no—
267C2×D7—61no—
268C2×D7—61no—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—21no—
272C4×C7—32no—
273C4×C7—61no—
274C4×C7—61no—
275C4×C7—61no—
276Dic7—21no—
277Dic7—21no—
278Dic7—61no—
279Dic7—61no—
280Dic7D3×D412yes—
281C23×C3—71no—
282C22×D3—71no—
283C22×D3—71no—
284C3×Q8—142no—
285C2×C3×C4—71no—
286C2×C3×C4—71no—
287C2×C3×C4—71no—
288C2×C3×C4—142no—
289C2×C3×C4C22×D711yescharacteristic
290C2×C3×C4C22×D711yescharacteristic
291C2×C3×C4C22×D711yescharacteristic
292#67—142no—
293#67—142no—
294C2×Dic3—71no—
295C2×Dic3—71no—
296C2×Dic3C22×D711yescharacteristic
297C2×Dic3C22×D711yescharacteristic
298C2×Dic3C22×D711yescharacteristic
299C2×Dic3C22×D711yescharacteristic
300D12—142no—
301C4×D3—74no—
302C4×D3—142no—
303C4×D3—142no—
304C4×D3—142no—
305Dic6—22no—
306Dic6—22no—
307Dic6—142no—
308Dic6—142no—
309C3×C7#53611yescharacteristic, complemented, Hall
310#41—424no—
311C2×Q8—31no—
312C2×Q8—31no—
313C2×Q8—71no—
314C2×Q8—211no—
315C2×Q8—211no—
316C2×Q8—424no—
317C2×D4—211no—
318C2×D4—211no—
319C2×D4—211no—
320C22×C4—71no—
321C22×C4—211no—
322C22×C4—211no—
323C22×C4—211no—
324C22×C4—211no—
325C4⋊C4—31no—
326C4⋊C4—31no—
327C4⋊C4—31no—
328C4⋊C4—71no—
329C4⋊C4—71no—
330C4⋊C4—211no—
331C4⋊C4—211no—
332C4⋊C4—211no—
333C4⋊C4—211no—
334C4⋊C4D3×D711yescharacteristic
335C22⋊C4—71no—
336C22⋊C4—71no—
337C22⋊C4—211no—
338C22⋊C4—211no—
339C22⋊C4—211no—
340C22⋊C4—211no—
341C22⋊C4—211no—
342C22⋊C4—211no—
343C22⋊C4—211no—
344C22⋊C4—211no—
345C42—31no—
346C42—211no—
347C2×C7C2×D3×D411yescharacteristic
348C2×C7#100711yescharacteristic
349C2×C7#101011yescharacteristic
350D7—41no—
351D7—62no—
352D7—121no—
353C22×C3—141no—
354C22×C3—141no—
355C22×C3—141no—
356C22×C3C23×D711yescharacteristic
357C2×D3—72no—
358C2×D3—141no—
359C2×D3—141no—
360C2×D3—141no—
361C2×D3—141no—
362C2×D3—141no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—72no—
367C3×C4—141no—
368C3×C4—141no—
369Dic3—21no—
370Dic3—21no—
371Dic3—21no—
372Dic3—141no—
373Dic3—141no—
374Dic3D4×D712yes—
375C23—71no—
376C23—211no—
377C23—211no—
378Q8—62no—
379Q8—62no—
380Q8—142no—
381Q8—422no—
382Q8—422no—
383D4—422no—
384D4—422no—
385D4—422no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—31no—
389C2×C4—31no—
390C2×C4—71no—
391C2×C4—71no—
392C2×C4—71no—
393C2×C4—142no—
394C2×C4—214no—
395C2×C4—211no—
396C2×C4—211no—
397C2×C4—422no—
398C2×C4—422no—
399C2×C4—422no—
400C2×C4C2×D3×D711yescharacteristic
401C2×C4C2×D3×D711yescharacteristic
402C2×C4C2×D3×D711yescharacteristic
403C7#562811yescharacteristic, complemented, Sylow
404C2×C3—281no—
405C2×C3C2×D4×D711yescharacteristic
406C2×C3#655111yescharacteristic
407C2×C3#655111yescharacteristic
408D3—142no—
409D3—281no—
410C22—141no—
411C22—141no—
412C22—141no—
413C22—212no—
414C22—421no—
415C22—421no—
416C22—421no—
417C22—421no—
418C22—421no—
419C22C22×D3×D711yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—32no—
424C4—61no—
425C4—61no—
426C4—61no—
427C4—72no—
428C4—141no—
429C4—141no—
430C4—421no—
431C4—421no—
432C3#9041511yescharacteristic, complemented, Sylow
433C2—281no—
434C2—422no—
435C2—841no—
436C2#1062154711yescharacteristic
437C2D3×D4×D711yescharacteristic
438C2#1062157611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
prime radicalC2×#66677C2
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#536
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×D21⋊Q8C2semidirect
3. #10621377C2semidirect
4. C7×#888C2semidirect
5. C3×#6459C2semidirect
6. #10621017C2semidirect
7. #10620995C2semidirect
8. #10620981C2semidirect
9. #10620973C2semidirect
10. #10620965C2semidirect
11. #10620918C2semidirect
12. #10620916C2semidirect
13. #10620915C2semidirect
14. #10620911C2semidirect
15. #10620891C2semidirect
62. #6459D3semidirect
145. #888D7semidirect
309. C3×C7#536semidirect
403. C7#5628semidirect
432. C3#90415semidirect

Maximal subgroups

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

subgroupindex
C2×#666772
C2×D21⋊Q82
#106213772
C7×#8882
C3×#64592
#106210172
#106209952
#106209812
#106209732
#106209652
#106209182
#106209162
#106209152
#106209112
#106208912
#904153
#56287